outer envelope has been stripped in a stable way, $\zeta^{\rm dyn}_{Rd}$
decreases enough to lead to instability. This phenomenon is referred to as
“delayed” dynamical instability. A delayed instability may also arise as a
result of a thermal-timescale mass transfer. The system in which the donor
star is gradually increasing in size with respect to its Roche lobe may lose a
non-negligible amount of mass through the outer Langrangian points (L2/L3).
The matter escaping through outer Lagrangian points carries a lot of angular
momentum away from the binary, driving an increased shrinkage of the orbit
[436]. Understanding the conditions leading to such a delayed instability is
an area of active research [425, 424, 437, 426, 438]. The relatively long
duration on which the instability arises (hundreds of years) calls for the use
of detailed stellar evolution codes. However, most of such codes are limited
to 1D and an interacting binary (with stars highly distorted from their
spherical shapes) is clearly not a 1D problem. While simplifications are
unavoidable to make the task computationally feasible, it is certainly worth
the effort, given that most cases of unstable mass transfer evolution in the
massive-star regime are likely the result of such an instability [425]. This
stems from the fact that most massive donors have outer radiative envelopes
[101], for which an immediate (adiabatic) instability would only occur in
cases of extreme mass ratios [421, 424]. One consequence of a delayed
dynamical instability is that at the onset of the CE phase the binary is
already surrounded by a torus of non-accreted matter, which may affect the
observational signatures of optical transient associated with CE evolution
known as luminous red novae [439, 440, 441, 442, 443].
The above considerations still neglect the potential response of the accreting
star to the added mass. If the companion is donated mass on a timescale that
is shorter than its thermal readjustment, it is driven out of thermal
equilibrium. That is the case, for instance, if the binary enters a thermal
timescale mass transfer from a more massive donor to a less massive stellar
companion. Stellar response to mass gain is the reverse to that expected due
to mass loss: accreting star with a convective envelope shrinks, while that
with a radiative envelope expands and may fill its own Roche lobe leading to a
contact binary and possibly affecting the stability of the interaction [444,
407, 445, 408, 446, 447]. We caution that this is often not accounted for in
the binary mass transfer models which treat the accretor as a point mass or
are limited to non-conservative mass transfer.

In summary, the mass transfer is likely to be stable if the donor is radiative
($\zeta^{\rm dyn}_{Rd}>0$) and the binary mass ratio $q$ is not too extreme.
The exact value of the critical mass ratio for massive radiative donors may be
a strong function of the orbital separation [424]. Conversely, if the donor
has a deep outer convective envelope (characterized by $\zeta^{\rm
dyn}_{Rd}<0$) then in most cases the mass transfer will become dynamically
unstable. Which stars, and when during their evolution, develop deep
convective envelopes is a strong function of both birth mass and metallicity
(see Sec. VI.1 and references therein). In general, the higher the mass of a
star and the lower the metallicity, the smaller the parameter space for it to
initiate mass transfer at a convective-envelope state. In the context of the
evolution of BH binary progenitors this means that: i) high mass, low
metallicity BH progenitors evolving in binaries are more likely to avoid the
CE evolution and only interact through stable mass transfer; ii) low mass,
high metallicity BH progenitors evolving in binaries are more likely to engage
in CE evolution [448, 437]. This dichotomy may lead to systematic differences
in BH merger times and properties of BH mergers that originate from low and
high metallicity binary progenitors [449, 450, 451].
The major sources of uncertainty entering the determination of the stability
of binary interaction are related to the amount of mass and angular momentum
escaping the system with the non-accreted matter, the stellar (envelope)
structure of the donor (setting its response to mass loss), and the response
of the accretor star to the rapid gain of mass and angular momentum.

####  VI.3 Mass transfer: effect on the evolution

#####  VI.3.1 Impact on the orbit: stable mass transfer case

The orbital evolution of a binary that undergoes stable mass transfer
evolution can be described by investigating the change in its orbital angular
momentum $J_{orb}$. Using equation VI.2 one can derive the general formula:

$\rm 2\frac{\dot{J}_{orb}}{J_{orb}}=2\frac{\dot{M}_{\rm donor}}{M_{\rm
donor}}+2\frac{\dot{M}_{\rm accretor}}{M_{\rm accretor}}-\frac{\dot{M}_{\rm
donor}+\dot{M}_{\rm accretor}}{M_{\rm donor}+M_{\rm
accretor}}+\frac{\dot{a}}{a}+\frac{(-2e\dot{e})}{(1-e^{2})}$ (VI.4)

It is often assumed that at the onset of the Roche Lobe Overflow (RLOF) the
orbit is fully circularised by tides, i.e. $e\approx 0$ therefore allowing to
neglect the last term (see e.g. [413, 452, 453, 454, 455, 456, 457, 458] for
the more general case with non-zero $e$). The specific angular momentum of the
ejected matter, $\rm j_{loss}$, is often expressed in the units of the
specific angular momentum of the binary by introducing a parameter $\gamma$:

$\begin{split}\rm j_{loss}=\gamma\frac{J_{orb}}{M_{\rm donor}+M_{\rm
accretor}}=\frac{\dot{J}_{orb}}{\dot{M}_{\rm donor}+\dot{M}_{\rm
accretor}},\\\
\rm\rightarrow\frac{\dot{J}_{orb}}{J_{orb}}=\frac{\gamma(1-\beta)\dot{M}_{\rm
donor}}{M_{\rm donor}+M_{\rm accretor}}.\end{split}$ (VI.5)

The effect on the orbit then depends on the conservativeness
$\beta=-\dot{M}_{\rm accretor}/\dot{M}_{\rm donor}$ and $\gamma$ parameters:

$\rm\frac{\dot{a}}{a}=-2\frac{\dot{M}_{\rm donor}}{M_{\rm
donor}}\left(1-\beta\frac{M_{\rm donor}}{M_{\rm
accretor}}-(\gamma+\frac{1}{2})(1-\beta)\frac{M_{\rm donor}}{M_{\rm
donor}+M_{\rm accretor}}\right)$ (VI.6)

Figure 9:  Visual representation of the expression 1 -
$f(q,\beta,\gamma)=1-q\beta-(\gamma+\frac{1}{2})(1-\beta)\frac{q}{1+q}$ from
equation VI.6 that, for a given set of parameters $\beta$ and $\gamma$
determines the evolution of the orbital separation of the binary with mass
ratio $q$. If this expression is negative, the orbital separation decreases as
the donor loses mass. Black line corresponds to the conservative mass transfer
case. Colors indicate different values of $\beta$, different line styles
indicate different modes of angular momentum loss. Thick solid lines
correspond to isotropic re-emission ($\gamma=q$), thin solid lines to fast
isotropic wind ($\gamma=1/q$), colored ranges span the range of
$\gamma=\gamma_{\rm ring}$ of the circumbinary ring with $a_{\rm ring}$
between 1.8$a$ (dashed) and 2.6$a$ (dotted).

Because $\rm\dot{M}_{\rm donor}<0$, the orbit shrinks ($\rm\dot{a}<0$) when
the expression in the parenthesis in equation VI.6 is negative (see Figure 9
for the illustration). This is, for instance, the case for conservative mass
transfer ($\beta$=1, $\gamma$=0) as long as $q>1$. As soon as the mass ratio
reverses (and $q$ drops below $1$), the orbit begins to expand. The parameters
$\beta$ and $\gamma$, and the way how they depend on the properties of
particular stars and binaries, are usually poorly known. The value of $\gamma$
strongly depends on where the non-accreted matter is lost from the system. The
most commonly considered, idealized scenarios include the “fast (Jeans)
isotropic emission” from the vicinity of the donor (typical assumption for
mass loss through stellar wind), in which case $\gamma=\frac{M_{\rm
accretor}}{M_{\rm donor}}$ or the “isotropic re-emission” from the proximity
of the accretor, leading to $\gamma=\gamma_{\rm iso}=\frac{M_{\rm
donor}}{M_{\rm accretor}}$ (typical assumption for mass loss during non-
conservative mass transfer). The latter case is also though to be a good
approximation for the mass ejected through jets or fast disk-winds launched in
the proximity of a compact accretor in X-ray binaries. However, if (part of)
the ejecta is slow-moving, the non-accreted mass may be escaping with notably
higher specific angular momentum than in both the above cases. Such ejecta
continues to exert torques on the binary orbit and gain angular momentum while
moving outwards. This may happen, for instance, when part of the non-accreted
mass forms a curcumbinary disk or torus, as seems to be the case in the SS433
system, an X-ray binary with a stellar BH accretor that currently evolves
through a face of stable and largely non-conservative mass transfer [459, 460,
461]. A formation of a circumbinary disk may also occur when the Roche lobe of
the accretor becomes overfilled with mass, such that it reaches the critical
potential surface corresponding to the outer Lagrangian point $L_{2}$. The
matter will then flow through $L_{2}$ and, not having enough energy to escape,
starts orbiting the system on a Keplerian orbit at a certain distance $a_{\rm
ring}$ from its center of mass. In ejection through a circumbinary disk
scenario, the specific angular momentum of the escaping mass
$\gamma$=$\gamma_{\rm ring}$ depends on the $a_{\rm ring}/a$ ratio [462]:

$\rm\gamma_{\rm ring}=\frac{(M_{\rm donor}+M_{\rm accretor})^{2}}{M_{\rm
donor}M_{\rm accretor}}\sqrt[\ ]{\frac{a_{\rm
ring}}{a}}=\frac{(q+1)^{2}}{q}\sqrt[\ ]{\frac{a_{\rm ring}}{a}}$ (VI.7)

The three modes of angular momentum loss introduced above are compared in
Figure 9. For the illustration of the circumbinary ring angular momentum loss
scheme, we follow [462] and consider $a_{\rm ring}/a$ between $1.8$ and $2.6$
as plausible values.
It is worth noticing that the wind-type of mass loss typically leads to the
widening of the orbit (especially in the lack of wind accretion $\beta$=0).
Wind mass loss rate of massive stars $\dot{M}_{\rm donor;\ wind}$ is expected
to heavily depend on the metallicity (the higher the metallicity, the higher
$\dot{M}_{\rm donor;\ wind}$ [202, 204, 205, 463, 464]). This means that when
similar types of systems are compared, one can expect higher metallicity
binaries to be wider. One can also notice that if the non-accreted mass
carries sufficiently high specific angular momentum (as in the example of the
circumbinary ring), the orbit can shrink for a wide range of mass ratios and
conservativeness parameters.
Understanding the extent to which the orbital separation can decrease as a
result of stable mass transfer is crucial in the context of the formation of
merging BBH systems. For the BBH system to merge within the time shorter than
the age of the Universe, its separation needs to be smaller than a few 10
$\,\rm{R}_{\odot}$ (depending on the masses, see Section VII.2). Typical BH
progenitors reach radii $\gtrsim 100\ \,\rm{R}_{\odot}$ during their lifetime
(see Figure 8), so the formation of a BBH merger through isolated evolution
typically requires a mass transfer interaction that significantly shrinks the
orbital separation (see also Section X). Whether this is feasible with stable
mass transfer evolution is a matter of an ongoing debate in the literature
[465, 448, 437, 450, 449, 466]. An alternative, that was long considered the
dominant mechanism responsible for the drastic orbital tightening of the BBH
merger progenitors formed in isolated binaries, is the CE evolution.

#####  VI.3.2 Impact on the orbit: unstable (CE) case

Once a binary is deemed to undergo an unstable mass transfer and CE evolution,
the most important question that one has to answer is whether the system is
going to survive this interaction. If the answer is negative, the binary
components merge and a single object is formed, possibly with some unique or
exotic properties. Stellar mergers are hypothesized to lead to the formation
of, for instance, massive stars with strong magnetic fields [467], blue
stragglers and blue MS in clusters [468, 469, 470], Thorne-Zytkov objects
[471, 472, 473], isolated binary-origin BH with mass in the pair-instability
gap [474, 102, 105]. The details of the merging process as well as the fate
and properties of the merger product are uncertain. If the binary can avoid
the merger during the CE evolution, one needs to determine the post-
interaction properties of the system - in particular the final separation. CE
is a complex problem that is inherently challenging to model, as it requires
accounting for physical effects happening on very different spatial and
temporal scales [475, 433, 476, 434]. In the big picture, the outcome of CE
evolution crucially depends on the binding energy of the envelope as well as
on the efficiency with which energy from various sources is deposited in the
envelope (and can ultimately lead to its ejection). That is the essence of the
so-called energy formalism [477], which allows one to obtain a crude estimate
of the final (post-CE ejection) orbital separation of the binary and is
commonly used e.g. in population synthesis studies [478, 479, 480, 481, 44,
482, 189, 483, 484] (see also Section IX). In this framework, the difference
in orbital energy between the CE onset and CE ejection multiplied by an
efficiency parameter $\alpha_{CE}$ is compared against the energy needed to
unbind the envelope (sometimes additional energy sources, contributing with
different efficiencies are considered as in equation VI.3). Another parameter,
$\lambda_{CE}$ is used to characterise the binding energy of the envelope of
the donor as follows:

$E_{bind}=\frac{GM_{\rm donor}M_{\rm env}}{\lambda_{CE}R_{\rm donor}}$ (VI.8)

here $M_{\rm env}$ is the mass of donor’s the envelope. Equation VI.3 then
takes the form:

$\frac{GM_{\rm donor;ini}M_{\rm env}}{\lambda_{CE}R_{\rm
donor}}=\alpha_{CE}\left(\frac{GM_{\rm donor;fin}M_{\rm
accretor;fin}}{2a_{fin}}-\frac{GM_{\rm donor;ini}M_{\rm
accretor;ini}}{2a_{ini}}\right)\ (+\text{other energy sources})$ (VI.9)

where the indices ‘ini’ and ‘fin’ refer to the values of masses and
separations at the CE onset and after the CE ejection, respectively, and
$M_{\rm donor;fin}=M_{\rm donor;ini}-M_{\rm env}$. In the simplest
formulation, if we neglect the other energy sources term in Equation
VI.9131313In practice, if energy sources other than the orbital energy term
are included, then usually it is done in an implicit way, e.g. by increasing
the value of $\alpha_{CE}$ or decreasing the $\lambda_{CE}$ parameter., the
final separation becomes:

$a_{fin}=M_{\rm donor;fin}M_{\rm accretor;fin}\left(\frac{M_{\rm
donor;ini}M_{\rm
accretor;ini}}{a_{ini}}+\frac{2\alpha_{CE}}{\lambda_{CE}}\frac{M_{\rm
donor}M_{\rm env}}{R_{\rm donor}}\right)^{-1}$ (VI.10)

Cases when $a_{fin}$ is so small that the helium core of the donor star would
overflow its Roche lobe are interpreted as CE events leading to stellar
mergers. In the energy formalism, all the complexity of the CE process is then
hidden within the $\alpha_{CE}$ and $\lambda_{CE}$ parameters.
It can be seen that the higher the $\lambda_{CE}$, the more loosely bound the
envelope. The value of $\lambda_{CE}$ can be obtained from 1D stellar models.
It was shown that $\lambda_{CE}$ can take values spanning orders of magnitude,
depending on the envelope type (radiative or convective) and the evolutionary
stage and internal structure of the CE donor [485, 486]. Detailed calculations
show that $\lambda_{CE}$ tends to increase with the increasing fraction of
mass contained in the outer convective envelope and that this increase may be
quite drastic - $E_{bind}$ can drop by more than a factor of 10 as the
envelope of a massive giant becomes convective [448].

In order to compute $E_{bind}$ from a stellar model one needs to assume the
exact location of the boundary between the envelope (that is to be ejected)
and the core of the giant CE donor. Because most of the binding energy is
located in deep envelope layers near the core, the assumed core-envelope
boundary can have a substantial effect on the value of $E_{bind}$ (even more
than an order of magnitude difference in the case of convective-envelope
donors [487, 488, 486]). Multiple criteria for the core-envelope boundary for
CE evolution have been proposed in the literature but no consensus is reached
[433, 434]. Notably, depending on the amount of hydrogen-rich material left on
top of the helium core after the envelope is ejected, a CE phase may be
immediately followed by another phase of (stable) mass transfer [489, 490,
448, 491, 492, 493]. This would continue to affect the orbital separation of
the system and is not captured by the simple parametrisation in Equation
VI.10.
The second parameter in Eqn. VI.10, $\alpha_{CE}$, represents the efficiency
with which orbital energy is deposited inside the envelope during the
inspiral. In principle, its value should thus be limited to $\alpha_{CE}<1$.
However, any result pointing to $\alpha_{CE}>1$ is not always unphysical as it
may indicate that additional energy sources are present in the system and
contribute in ejecting the envelope (or that the $E_{bind}$ term has been
overestimated). The most likely agents for additional energy input are the
internal energy of the envelope (gas and radiation) and the energy released
upon recombination of hydrogen and helium [494, 495]. These energy sources may
not act immediately as they require the envelope to expand, decompress, cool,
and recombine. Some of the released energy is then reprocessed into kinetic
energy (and may help to remove the envelope) while the rest is likely radiated
away and lost from the system [496, 497]. The efficiency of these processes is
unclear and remains an area of active research, with some of the key aspects
being the timescale on which the internal energy is released and where in the
envelope it occurs [498, 499, 500]. It is important to realize that the
potential contribution of internal energy is often already incorporated in the
$\lambda_{CE}$ values reported in the literature (rather than in the energy
term and $\alpha_{CE}$), where it is assumed to lower the binding energy of
the envelope with maximum eficiency (i.e. neglecting radiative loses) [485,
501, 502, 486, 448]. There is no consistency in the literature in this respect
and when combining different $\lambda_{CE}$ and $\alpha_{CE}$ estimates one
should make sure that no potential energy sources (in particular the internal
energy contribution) are counted twice.
Constraining the value of $\alpha_{CE}$ has been at the forefront of CE
research for a long time. On the theoretical side, 3D hydrodynamic simulations
of the CE inspiral suggest $\alpha_{CE}<1$ (e.g. $\alpha_{CE}$ = 0.046-0.25 in
Clayton  _et al._ 503, $\alpha_{CE}\lesssim 0.5$ based on Nandez  _et al._
494, Nandez and Ivanova 498, $\alpha_{CE}\lesssim 0.4$ found by Law-Smith  _et
al._ 504, and $\alpha_{CE}$ = 0.3-0.7 across the simulations of Ondratschek
_et al._ 505, Sand  _et al._ 506, Lau  _et al._ 499 when combined with
$\lambda_{CE}$ that includes internal energy). A notable result from
simulations are cases when the inspiral slows down or ceases without ejecting
the entire envelope (e.g. just the convective part), which lowers the amount
of binding energy that needs to be overcome and, at the face value, increases
the effective $\alpha_{CE}$ value to even $\gtrsim 1-2$ [490, 499, 507]. It is
important to realize that such a variant of CE evolution, most likely only
realistic in the case of convective-envelope donors with a steep density
gradient at the bottom of the convective zone, is likely to be followed by
another phase of mass transfer that would strip the remaining H-rich layers
[489, 508, 492]. The final orbital separation (and the effective value of
$\alpha_{CE}$) may further be affected by tidal torques from the recently
ejected material [509] and is likely to depend on the mass ratio [507].
Finally, it is worth noting that the vast majority of 3D simulations of the CE
phase to date has been limited to convective-envelope donors, which are the
typical case in the low- to intermediate-mass regime, but may not be the most
common case of donors in the massive star regime [101].
Observationally, $\alpha_{CE}$ can be constrained by studying post-CE systems
(e.g. double white-dwarf binaries) and reconstructing their pre-CE orbits
[510, 511, 512, 513, 514, 515]. While this approach suffers from uncertainties
in the reconstruction of pre-CE parameters, it generally points out to
$\alpha_{CE}<1$ values, in line with most theoretical results. A notable
exception are BH low mass X-ray binary systems, the formation of which through
a CE evolution channel would possibly require $\alpha_{CE}>1$ efficiencies
[516, 517], although see [448].

In the context of BHs, the energy budget for CE evolution leading to the
formation of merging binary BH systems has been discussed extensively by [481,
486, 448, 437]. These authors conclude that in the case of massive giants (BH
progenitors) the envelope binding energies are generally too large to allow
for a successful CE ejection for $\alpha_{CE}\leq 1$, unless the CE phase is
initiated by a convective-envelope star: a red supergiant (but see also
[518]). This might be a severe limitation for evolutionary channels involving
a CE phase: whereas in the low-mass regime mass transfer from a convective
donor is expected to be common, in the high-mass regime of BH progenitors it
is limited to a narrow range of orbital periods [519, 101] and may be
completely impossible above a certain mass ($\sim 40-50\rm M_{\odot}$)
corresponding to the the Humphreys-Davidson limit [396, 84, 520].

Importantly, the conclusion that CE events from radiative-envelope donors
result in mergers are based on 1D hydrostatic stellar models and ad hoc
assumptions regarding the onset of the CE phase as well as the final state
following the inspiral [481, 486, 448, 437]. All these factors are uncertain.
Firstly, the internal envelope structure of a CE donor cannot always be
robustly predicted from 1D stellar models. For example, Klencki  _et al._
[448] showed that two red supergiant models of a similar mass, radius, and
luminosity can be characterized by significantly different envelope density
profiles and $E_{bind}$ values, an effect arising from differences in their
internal chemical abundance profiles. Recently, Renzo  _et al._ [521]
demonstrated that the envelope structure of past accretors shows systematic
structural differences compared to envelopes of single stars, leading to lower
$E_{bind}$ values. This is important in the context of binary BH merger
formation. In the classical formation scenario of such systems, the secondary
star becomes a CE donor when the primary is already a BH and after evolving
through a mass transfer episode initiated by the primary.

Secondly, the onset of the CE phase is not immediate but is preceded by a
phase of pre-CE mass transfer before the interaction becomes dynamically
unstable [435]. In the case of radiative-envelope donors, this phase can last
for hundreds of years, with a significant (but poorly constrained) fraction of
the donor’s envelope becoming stripped even before the CE inspiral begins
[433, 522, 443, 448]. On the far end of the CE phase, it is unclear where
exactly in the envelope may the inspiral slow down to the point where the
further orbital evolution is governed by secular processes such as stable mass
exchange (stripping the final envelope layers) and tidal torques between the
orbit and all the matter ejected in the process, forming a rich and complex
circumbinary medium.

Last but not least, an energy source that is rarely considered is the feedback
from accretion onto the compact object as it spirals in through the envelope.
While Eddington-limited accretion luminosity (see Section VI.3.5) would not
necessarily be a significant addition to the energy budget [486, 448], the
accretion rate may exceed this limit due to neutrino cooling or photon
trapping [523, 524, 525]. If an accretion disk forms (which is not necessarily
the case [526]) then disk winds and jets could transfer some of the super-
Eddington power into kinetic energy, thus having a massive impact in the
envelope ejection process [527, 528].

#####  VI.3.3 Impact on the donor star

As a result of the mass transfer, the donor star loses some fraction or even
the entire envelope and becomes a stripped star [529, 530, 531, 532, 533, 75].
The basic observable properties (e.g. radius, effective temperature and
surface composition) of such a star are vastly different compared to a non-
stripped star of the same mass. If the stripping process is efficient, hot
inner layers of the star are exposed and the mass-loser appears as a compact
helium-rich stripped star. This further leads to an increase in the wind mass
loss that it experiences and the amount of ionising radiation that it produces
[533, 534, 81, 535, 536, 537]. Especially if the interaction happens early
enough during the evolution of a star, it can limit the growth of the core and
reduce its compactness, thus affecting the outcome of the core collapse and
the formation of a compact remant with respect to single star evolution
predictions [531, 537, 346, 103, 538, 539]. In this context, early mass
transfer is the one that can have an impact on the properties of the helium
core of the star. Helium core mass and and the inner carbon-oxygen core
(formed as a result of core-helium burning) to a large extent determine the
subsequent evolution and the final fate of a star [540, 541]. Whatever happens
to the outer layers of a star once the carbon-oxygen core mass is established
is found to have relatively little impact on the core-collapse [542, 106,
346]. However, it may still affect the observable characteristics of the
potential supernova (in particular their classification as being envelope
stripped, based on the lack of H or He lines in the supernova spectrum [543]).
Furthermore, the mass remaining in the envelope sets the limit on the total
amount of mass that can be ejected or accreted during the formation of the
compact object.

###### Early mass transfer maximizes the differences with respect to single
star evolution:

In view of the example shown in Figure 8, early and efficient stripping can be
expected for high metallicity BH progenitors, that are likely to become donors
when rapidly expanding on the Hertzsprung gap and get quickly stripped of
their hydrogen rich envelopes. Because the envelope loss takes place shortly
after the onset of hydrogen-shell burning, there is not enough time for the
helium core to grow in mass any further. The exposed helium core of a massive
star (a Wolf-Rayet star) experiences strong stellar winds that instead reduce
the core mass [79, 532, 216, 81]. If the collapse of its core eventually leads
to an explosion, it is likely to be observed as a stripped-envelope supernova
(i.e. classified as a hydrogen or possibly even helium poor supernova type
Ib/c). Recent models suggest that stars that encounter this kind of binary
interactions tend to form lower mass compact objects [538, 539]. If that is
the case, higher initial mass is required to produce a stellar BH, which acts
to reduce the expected number of BH forming per unit of star formation [538].
In contrast, if a star becomes a donor during the core helium burning phase or
even later, its helium core may still grow to some extent before the hydrogen
envelope is stripped. More gradual expansion during this phase allows the mass
transfer to last much longer (nuclear timescale) than in the case of mass
transfer from a Hertzsprung gap donor (thermal timescale) [395]. If the Wolf-
Rayet star is formed as a result of such a mass transfer, its lifetime (and
the associated strong wind mass loss) is thus shorter. If the mass-loser
eventually explodes as a supernova, the observable event may still show signs
of stripping but the He lines are likely to be present (i.e. it might be
classified as hydrogen-deficient supernova type IIb or Ib).
Onset of mass transfer at an even more advanced evolutionary stage (after
core-helium depletion) further reduces the importance of the interaction for
the remaining evolution of the donor. Unless the mass transfer is dynamically
unstable, there might not be enough time to strip the entire hydrogen envelope
of the star before the core-collapse. The helium core is already established
at this point of evolution.

###### Fully or partially stripped (and why it matters):

The amount of residual H-rich envelope left atop the helium core after mass
transfer was shown to depend on metallicity, with metal-poor stars retaining
more hydrogen after the interaction [544, 536, 508, 395]. The process of
envelope stripping may become even less efficient in the case of massive core-
helium burning donors (common in metal-poor populations), which were found to
interact on the long nuclear timescale and/or lead to partially stripped stars
with $\gtrsim 5-30\%$ of the initial envelope left unstripped until the end of
core-helium burning, depending on binary parameters [395]. Such stars are
predicted to be much larger and cooler than normal stripped (helium) stars,
producing far less ionizing radiation. The possibility of partial envelope
stripping is usually neglected in studies that approximate binary evolution
effects by considering naked helium stars as the starting point. Similarly, it
is often assumed that as a result of a successful CE evolution is the ejection
of the entire hydrogen-rich envelope, leaving behind a helium-star remnant.
This is likely not strictly the case, as the CE inspiral may stall before all
the hydrogen-rich layers are ejected [489, 491, 492, 493].
Understanding the exact degree of envelope stripping experienced by donors in
various types of binary interactions is important not only because of the
effect it has on their observable characteristics, but also because it may
significantly affect the subsequent evolution of the system as a whole.
Surface properties of a star determine its wind mass loss and consequently the
orbital evolution of a post-interaction binary (and likely the final
separation before the core collapse). Additionally, stars that retain even a
tiny hydrogen-rich envelope can significantly re-expand later during their
evolution, in contrast to their fully stripped counterparts that remain
compact [508], leading to another phase of RLOF.
Finally, the mass remaining in the envelope of the star at the end of its life
may affect the properties of the binary that remains after the core collapse.
In general, the formation of a compact object is thought to be accompanied by
some amount of mass ejection from the outer layers of its immediate
progenitor. If significant, such mass ejection from the collapsing star can
considerably affect the orbital parameters, or even unbind the system [545,
546]. Furthermore, a newly born compact object is thought to gain velocity
(natal kick) due to asymmetries (either in mass loss or neutrino loss [547,
548, 549]) involved in the process of its formation. While in the case of NSs
there is ample observational evidence that at least some fraction of them
forms with substantial natal kick ($\gtrsim$a few hundred km/s) [550, 551,
552, 553, 554, 555, 556, 557, 558], this is currently much less clear in the
case of BH formation [559, 560, 561, 562, 332, 563, 564, 565, 566]. In some
scenarios, natal kick velocity has been proposed to scale with the mass of the
ejected envelope (the lower the mass of the ejecta, the lower the kick
velocity [549]). If that is the case, lower mass of the envelope remaining at
the core collapse can increase the chances that the binary remains bound after
the compact object formation. It has been suggested that for sufficiently
massive progenitors, BH formation may occur via the collapse of the entire
star and the mass loss can be completely avoided [183]. In such a scenario,
higher envelope mass allows for the formation of a more massive compact
object.

#####  VI.3.4 Impact on the accretor: stellar accretors

Stars spend most of their life on the main sequence and the less massive
component of a binary is likely to still be at this stage of evolution at the
onset of the first mass transfer episode. During a mass exchange, some
fraction of the matter lost by the donor can reach and may be accreted by the
companion (thereby called the accretor). This not only affects its observable
characteristics (e.g. increased luminosity, fast rotation, envelope enriched
with the CNO-processed material from the donor), but may also significantly
affect its subsequent evolution [444, 407, 445, 408, 446, 447].

###### Rejuvenation and core growth (or lack thereof):

The accretor has to adjust its size and structure to accommodate additional
material. As noted earlier, in some cases this may lead the accretor to
filling its own Roche lobe, possibly affecting the stability and
characteristics of the mass transfer. Furthermore, as the accretor responds to
the new mass, fresh hydrogen-rich fuel from the outer layers can be mixed into
its convective core, ‘rejuvenating’ the star - its further evolution resembles
that of a younger star with a higher initial mass [407, 445, 447]. Such
response to accretion affects the structure of the (helium-rich) core-envelope
boundary of the star. That layer is where the steep rise in the density
profile happens, and hence it stores most of the binding energy of the stellar
envelope. [521] show that stars that evolved through a mass transfer phase as
accretors may have systematically lower envelope binding energies compared to
single stars of the same radius and mass. If such stars subsequently initiate
a CE, the envelope may be easier to eject and leave behind a wider binary
compared to the case with CE donors whose core-envelope boundary was not
affected by earlier accretion. Finally (inversely to the donor’s case), early
enough accretion leading to rejuvenation allows for the formation of a more
massive He core and consequently, NS/BH formation from initially less massive
stars compared to single stellar evolution predictions. Rejuvenation is
sensitive, for instance, to the composition of the accreted matter [567] and
to uncertain stellar evolution model assumptions - in particular, it may be
completely avoided if internal mixing (semiconvection) is inefficient [446].
Unless they are fully rejuvenated, accretors are bound to have different core
to envelope mass ratios than single stars (likely having undermassive helium
cores and overmassive envelopes for their mass). This was shown to affect the
radius evolution of a star [469, 568] and is a potentially important (but
poorly explored) effect for future binary interactions.
At the moment, the impact of mass transfer on the accretor star briefly
discussed above is not taken into account in most current binary evolution
population models, in which it is customary to approximate the subsequent
evolution of the accretor with a single (possibly fast-rotating) stellar
model.

###### Spin-up (followed by the likely spin-down):

The matter that is transferred to the accretor carries angular momentum and
spins up the star [569, 570, 571]. The transferred angular momentum can be
particularly significant if the accretion occurs through a disc formed around
the star, in which case the accretor may reach its critical/break-up rotation
($\omega_{crit}=(GM_{*}/R_{*}^{3})^{1/2}$ i.e. the limiting rotational
velocity at which the centrifugal force at the equator is equal to the
gravitational force) after accreting only a small fraction of its initial mass
[569]. This can possibly be avoided in very tight binaries (with orbital
periods $\lesssim$a few days), where efficient tidal interactions can prevent
the star from reaching critical rotation by locking its spin period to the
orbital period. However, if the star reaches its critical rotation, it cannot
accomodate any more angular momentum. This possibly halts further mass
accretion [572, 573], either making the mass transfer highly non-conservative
or forcing the mass to accumulate in the disk (from which it might eventually
still be accreted if the stellar rotation decreases). On the other hand, [574,
571] argue that the critically rotating star can continue accreting mass from
an accretion disk without accreting angular momentum as a result of viscous
coupling.
Increased rotation increases the wind mass loss rate [270, 234], may
significantly enhance the mixing inside the star ([275, 288, 150]; see Section
VII.1.1 for further discussion of this point) and overall, can have a profound
impact on stellar evolution [575]. However, it is unclear whether the star can
retain high rotation rate throughout its evolution. In the context of BH
properties, it is particularly important to understand whether the accretion-
enhanced rotation can be transported all the way to the core and retained
until the core-collapse, so that it can potentially lead to the formation of a
highly spinning compact object, and possibly a long gamma ray burst progenitor
[289, 576]. To this end, one needs to understand i) how much angular momentum
is lost by the star during its subsequent evolution and ii) how is the angular
momentum transported and distributed within the star (the internal rotation
does not need to be uniform, rotation of the core may decouple from the
envelope). Apart from binary effects (mass transfer and tidal interactions)
that obviously need to be considered to answer i), one has to account for
stellar winds. At low metallicity, wind mass loss is smaller and the
associated removal of angular momentum during evolution much weaker, in
principle allowing to retain higher rotation [577]. As long as the rotation of
the core and the envelope remain coupled, the loss of angular momentum from
the envelope also slows down the rotation of the core (and vice versa). In
fact, some mechanism to extract the angular momentum from the core appears
necessary to avoid overpredicting the rotation rates of young NS and WD [293,
578]. Such mechanism can be provided by magnetic torques and the Spruit-Tayler
dynamo model [301] appears sufficient to address the above issue [293, 578].
However, this mechanism alone cannot explain the asteroseismologically
observed angular momentum loss of the core of red giants and helium core-
burning low mass stars [579, 312, 580]. This led to the conclusion that more
efficient angular momentum transport may be required, as in the re-formulation
of the Spruit-Tayler dynamo model proposed by [312]. If those arguments apply
to more massive BH progenitors, the model of [312] predicts that the magnetic
torques remove most of the angular momentum from the helium core just after
the main sequence and consequently, BH natal spins are expected to be small
[313]. Possible exceptions from this rule involve BH progenitors spun up
through tidal interactions in close binaries either at very low metallicity or
at the WR stage in WR+BH binaries (see Section VII.1). However, high
observational spin estimates of BH in high mass X-ray binaries [581, 582,
583], if correct, appear challenging to reproduce with the very efficient
angular momentum transport in BH progenitor interiors (e.g. [584] but see
[585]). Alternative, more moderate angular momentum transport schemes are also
explored in the massive star models [124, 146, 586].

In summary, during the mass transfer the accretor gains (possibly a very
significant amount of) mass and angular momentum. However, this does not
necessarily mean that its further evolution can be accurately described as
that of a highly-spinning star of it’s new mass - predominantly because the
accretor’s core might not fully experience the mass and angular momentum
gained by its outer layers. Its post-interaction structure (in particular
core-envelope boundary and core to envelope mass ratio) is likely different
than predicted by any single star models, with possible (and largely
unexplored) consequences for the subsequent phases of binary interactions.

#####  VI.3.5 Impact on the accretor: BH accretors

BH in a binary can accrete mass and angular momentum from its companion when
the star evolves to fill its Roche lobe. This can obviously affect its natal
properties, however, to what degree is currently uncertain. We briefly discuss
the related problems and commonly made assumptions below.

###### How much mass can a BH accrete?

Accretion onto the compact object is often assumed to be limited by the
Eddington accretion rate ($\dot{M}_{Edd}$), defined as the maximum accretion
rate at which gravity can overcome radiation pressure of the emission driven
by spherical accretion:

$\dot{M}_{Edd}=\frac{4\pi GM_{BH}}{\eta\kappa c}\approx 2.6\times
10^{-8}\,\rm{M}_{\odot}\text{yr}^{-1}\left(\frac{M_{BH}}{\,\rm{M}_{\odot}}\right)\left(\frac{\eta}{0.1}\right)^{-1}\left(\frac{1+X}{1+0.7}\right)^{-1}$
(VI.11)

where $M_{BH}$ is the BH mass, $c$ is the speed of light, $\eta\sim 0.1$ is
the efficiency at which the BH is converting the accreted rest mass into
radiative energy ([587, 517]; the luminosity driven by accretion of mass at a
rate $\dot{M}$ is $L_{BH}=\eta\dot{M}_{acc}c$), and $\kappa=0.2(1+X)\
\text{cm}^{2}/\text{g}$ is the radiative opacity due to pure electron
scattering (a reasonable assumption for massive stars) for composition with
hydrogen mass fraction X ($\sim$0.7 for H-rich donor stars and 0 for
H-deficient donor stars).
If $\dot{M}_{Edd}$ is exceeded, the excess mass is lost from the system and is
typically assumed to carry away the specific angular momentum of the BH in its
orbital motion (i.e. $\gamma_{iso}$). It can be seen that $\dot{M}_{Edd}$
imposes a very strong limit on the BH accretion in the most massive stellar
binaries: even if a 10 $\,\rm{M}_{\odot}$ BH was to accrete at the Eddington
rate from its companion during its entire lifetime (i.e. $\approx$hydrogen
burning nuclear timescale $\lesssim$ 10 Myr for BH progenitors), the accreted
mass would be $\sim$2.6 $\,\rm{M}_{\odot}$.

###### Is Eddington accretion the limit?

Whether accreting BHs can significantly violate the Eddington limit, and under
which conditions, is debated. Clear cases where the equation VI.11 does not
hold are the lack of spherical symmetry (e.g. in cases with strongly
collimated radiation), or highly non-steady accretion. The spherical symmetry
condition is not satisfied in many models of BH accretion. Various existing
disk accretion models allow for moderately super-Eddington (e.g. ’slim’ disks
[588, 589]) to highly super-Eddington accretion rates (e.g. ’Polish
doughnuts’-like models [590, 591]), with different rates reported with modern
numerical models (see e.g. reviews [592, 593]). Furthermore, super-Eddington
accretion has been suggested as the likely explanation for a wide range of
observed phenomena, in particular ultraluminous X-ray sources ([594, 595]; at
least some of which are known to contain accreting remnants of massive star
evolution) and the galactic binary system SS433 [459]. Highly super-Eddington
accretion could in principle allow for the BH formation within the PISN mass
gap in the course of regular binary evolution [596].

###### How much can a BH spin up?

Accretion of mass and of the associated angular momentum also affects the
natal spin of a BH. To discuss the BH spin the dimensionless spin parameter
$a_{spin}$ is often used, which is defined as follows:

$a_{spin}=\frac{cJ_{BH}}{GM_{BH}^{2}}$ (VI.12)

Here $J_{BH}$ is the BH angular momentum. In the idealised case of accretion
happening from the last stable circular orbit of a thin disk around a BH, one
can describe the $a_{spin}$ of initially non-spinning BH as follows [587, 597]

$a_{spin}=\left(\frac{2}{3}\right)^{1/2}\frac{M_{BH;ini}}{M_{BH;fin}}\left(4-\sqrt[\
]{18\frac{M_{BH;ini}}{M_{BH;fin}}^{2}-2}\right)$ (VI.13)

which holds for $M_{BH;fin}<\sqrt[\ ]{6}M_{BH;ini}$.
If the disk geometry is more complex, the disk is thick or if the accretion
does not happen from the disk at all (e.g. if the specific angular momentum of
the infalling matter is not sufficient to form a disk or if the disk is
destroyed), BH spin up is likely to be less efficient than what would be
estimated with equation VI.13 [597]. Furthermore, it has been shown that the
presence of magnetic fields within the inner part of the accretion disk may
substantially limit the spin up or even slow down the BH rotation [598, 599].
Even within the validity of the framework used to derive equation VI.13, to
spin up a non-rotating BH from $a_{spin}=0$ to nearly maximum rotation
$a_{spin}\sim 1$, the BH must accrete at least $\Delta M$ = 1.85$M_{BH;ini}$
[600]. As discussed above, if the accretion rate is Eddington limited, in
typical binaries the BH can only accrete a small fraction of that mass [600].
On the other hand, if the accretion exceeds the Eddington limit, it is
unlikely to occur through a thin disk, in which case BH spin obtained with eq.
VI.13 only yields an upper limit.

###### Can BHs accrete non-negligible mass (and spin up) during the CE?

A somewhat separate question is how much mass and angular momentum can be
realistically accreted by the binary components inspiralling during the CE. In
this case the geometry and the rate of mass transfer is distinctly different
from the stable RLOF, possibly allowing for a non-negligible accretion despite
the extremely short timescale. In particular, in case of compact object
accretors, both high $\dot{M}$ and non-spherical geometry could in principle
allow to surpass the Eddington limit. However, evolution within the CE is very
challenging to simulate and the complete self-consistent model including
accretion is not yet available (see extensive discussion of the related
challenges in [434]). To approximate the accretion during this phase, it is
common to refer to the idealized case of the capture of matter by a point mass
moving supersonically with respect to a uniform medium (the
Bondi–Hoyle–Lyttleton accretion, as described in [601, 602, 523]) and assume
some small fraction ($\sim$a few percent) of the resulting accretion rate.
This reduction initially came from the realisation that the value expected
from Bondi–Hoyle–Lyttleton accretion would lead to unrealistic level of
accretion during the CE. In particular, the highly super-Eddington accretion
onto compact objects in CE resulting from this scheme would notoriously cause
NS to accrete sufficient mass to collapse to low-mass BH, leading to
discrepancies with observations [603]. The accretion rate is also
theoretically expected to be much smaller than in the idealized case, as many
of the assumptions of the Bondi–Hoyle–Lyttleton scheme are not satisfied
during the CE (e.g. the flow is non-axisymmetric and nonuniform; see also
discussion in [434]). This is further supported by the results of 3D
hydrodynamic simulations [604, 605, 606], that suggest accretion rates an
order of magnitude below the Bondi–Hoyle–Lyttleton prediction and possibly
below the Eddington-limited rate.

### VII Other relevant binary processes

####  VII.1 Tidal forces: effect on spins and orbit

Gravitational pull by a binary companion causes deformations of the (non-
degenerate) star. These deformations are time-varying, for instance, when the
spin periods of the binary components are not synchronised or if the orbit is
eccentric. This leads to dissipation of energy through tidal friction and
allows for the exchange of angular momentum and kinetic energy between the two
stars and their orbit [607, 608]. The strength of the tidal interactions
strongly increases with the ratio of the stellar radius to the orbital
separation [607, 608]. Therefore, they become an important factor in the
evolution of close binaries, with component(s) filling significant fraction of
their Roche lobe [609].
Tidal interactions act to bring the system towards circularisation, to
synchronise the binary component spins with the orbital period (tidal locking)
and to align the spins perpendicular to the orbital plane. The timescale at
which this equilibrium state can be achieved depends on the efficiency of the
processes that are responsible for the energy dissipation (see e.g. discussion
in [609, 97]). Regardless of that mechanism, in stellar binaries with orbital
periods below a few days, the tides are expected to be so strong that the
stars are forced to rotate synchronously with the orbital motion [97]. The
timescale for circularisation is typically expected to be short enough to
bring the eccentricity to $e\approx 0$ before either of the binary components
overfills its Roche-lobe [609]. However, some observational results seem to
challenge this conclusion; e.g. [610, 611]. This does not necessarily mean
that tides are inefficient, but might indicate the presence of other processes
that are counteracting tidal circularisation and driving eccentricity in those
cases [610, 612]. Nevertheless, one should remain cautious when assuming that
the mass transfer always happens when the orbit is circularized. Even in the
absence of eccentricity-pumping processes, for highly eccentric systems the
mass exchange may proceed for some time on a non-circular orbit, which greatly
complicates the description of the orbital evolution [453, 454, 455, 456,
457]. In such cases the mass transfer rate may be strongly boosted near the
periastron (such behaviour would lead to, for instance, increasing luminosity
of the accretion-powered emitters such as X-ray binaries near periastron). In
binary population synthesis simulations, tidal circularisation is typically
assumed to happen instantaneously once certain conditions are met.
Importantly, together with this simplifying assumption comes the one about the
separation of the circularised orbit $a_{circ}$. The common choice is to
assume that the angular momentum is conserved throughout this process (only
the energy is lost from the binary as it circularizes), leading to
$a_{circ}=a_{pre;circ}(1-e^{2})$ [479, 613]. Other choices are also used, for
instance, some authors assume $a_{circ}$ equals the periastron distance of the
eccentric orbit at which the circularisation condition was met, i.e.
$a_{circ}=a_{pre;circ}(1-e)$, e.g. [49].

Two possibilities offered by strong tidal interactions in close binaries are
particularly interesting in the context of BHs and the formation of BH
mergers: i) tidally induced chemically homogeneous evolution of the binary ii)
tidal spin up of the immediate BH progenitor. We briefly discuss them below.

#####  VII.1.1 Chemically homogeneous evolution

As noted in Section VI.3.4, rapid rotation induces internal mixing. Such
mixing can operate also in the regions where convection (providing the
conventional way of re-distributing material within stars) is inefficient.
Therefore it can reduce, or - in extreme cases, completely remove the
composition gradients between the central and outer layers of the star [288,
251, 253]. In the latter case, a helium star with no hydrogen envelope is
formed at the end of the main sequence. In contrast to the conventional radial
evolution shown in Figure 8, such ‘chemically homogeneous’ stars avoid the
strong radial expansion of their envelope during the phases of core
contraction [288, 253]. Those stars can therefore evolve much closer to each
other in a binary and avoid merging with their stellar companion. This
observation led to the proposal of chemically homogeneous evolution as a
possible binary route to form BH mergers, unique in its lack of need for any
processes that can drastically tighten the birth orbit [98, 614, 291].
Furthermore, the extreme mixing allows for the formation of more massive
stellar cores (as all available fuel is used in nuclear reactions), and
therefore more massive compact objects. Chemically homogeneous evolution is
believed to be restricted to stars more massive than $\gtrsim$ 30 M⊙ (i.e. BH
progenitors) that are rapidly rotating already at the onset of the core
hydrogen burning (i.e. before the composition gradient at the core-envelope
boundary develops, as its presence may inhibit efficient rotational mixing
[253]).
One way to facilitate chemically homogeneous evolution is to form two massive
stars sufficiently close to each other so that they are forced to rotate
rapidly through strong tidal interactions. In such a case, both binary
components undergo chemically homogeneous evolution. Maintaining the tidal
locking is then crucial, which means that chemically homogeneous evolution is
expected to preferentially happen at low metallicity: strong wind mass loss
experienced by massive, high metallicity stars tends to increase the binary
separation. This can remove the tidal locking and allow the stars to
significantly spin down, inhibiting chemically homogeneous evolution [97]. If
tidal locking can be kept throughout the evolution, such a scenario can
potentially lead to the formation of rapidly spinning BHs.
Alternatively, accretion during the mass transfer can efficiently spin up the
secondary star (Section VI.3.4). If this happens early enough during its
evolution, it may allow for the chemically homogeneous evolution of the
secondary [576]. Also in this scenario low metallicity is favored, as stellar
winds extract the angular momentum from the star and spin it down.

#####  VII.1.2 Tidal spin up of the immediate BH progenitor

As discussed in Section VI.3.4, if the angular momentum is efficiently
transported within the star and can be exchanged between the core and the
envelope, the natal spins of BH are likely to be very small [615, 313]. Even
if the BH progenitor was born rapidly spinning, during its post-main sequence
expansion the initial angular momentum is mostly transported to its outer
layers. Those layers are then likely lost when it initiates the mass transfer
in a binary and/or (excluding the lowest metallicity stars) due to wind mass
loss.
However, in very close binaries, there is a window of opportunity for the
secondary to gain rotation at the very end of its life and therefore maintain
it until the collapse to a BH. This window is opened up by tides. Tidal
locking of a tight system consisting of a compact object and a helium/WR star
can lead to a strong spinup of the latter [616, 617, 618, 619, 615, 586, 620,
621]. At the same time, mass lost by the helium star through winds widens the
orbit and may eventually lead to tidal decoupling and spin down the star.
Therefore, to understand the net effect it is important to properly account
for the coupled effect of tides and stellar winds on the evolution of the
system. In general, such tidal spin up i) may only be effective for a fraction
of the (already very short) lifetime of a WR+BH/NS system and ii) for the
binary with otherwise the same properties, can operate for longer at lower
metallicity. The first point means that this scenario likely requires that the
progenitor system evolves through a CE, which allows to reach the small binary
separation at which the spinup of the secondary is efficient [450]. Such tight
WR+BH binaries are the progenitors of merging BH+BH systems and are expected
to constitute some (uncertain) fraction of the observed gravitational wave
events [586, 620, 622, 621]. The second point is somewhat counteracted by the
fact that massive low metallicity stars in binaries are likely to fill their
Roche-lobes later during their evolution than their high metallicity
counterparts (see Section VI.1). This means that if a low metallicity binary
evolves into a tight BH + WR, the remaining lifetime of the WR is likely to be
shorter than for the analogous high metallicity system. Nonetheless, the
results discussed in [620] suggest that overall, this scenario produces a
higher fraction of highly spinning immediate BH progenitors at low
metallicity.
In principle, the mechanism discussed above may produce highly spinning BHs
(even maximally spinning) from secondaries evolving in isolated binaries even
if the angular momentum transport in stellar interiors is very efficient.
However, the link between the spin of the immediate BH progenitor and the
natal BH spin (both its magnitude and orientation) may be highly non-
straightforward, as it depends on the details of the core-collapse and the
interaction of the proto-compact object with the ejected/falling back
material, e.g. [585, 623, 624].

####  VII.2 Orbital evolution due to gravitational wave emission

Massive, accelerating objects whose motion leads to variations in the overall
mass distribution of the system (specifically, leads to changing quadrupole
moment) emit gravitational waves [625]. Binary systems are a well known
example of astrophysical sources of gravitational waves. The emitted
gravitational waves carry energy and angular momentum away from the system,
leading to a gradual decrease of the orbital separation. The accurate
measurement of such orbital decay of the binary pulsar PSR 1913+16 allowed for
the first indirect detection of gravitational waves [626, 627]. In stellar
binaries, other processes typically dominate the orbital evolution over
gravitational waves emission and the latter can be neglected. However, for
sufficiently tight systems, in which only compact objects and low
mass/stripped stars can fit, the timescale for the angular momentum loss in
gravitational waves can be high enough to significantly affect their orbital
evolution or even drive the mass transfer [628]. Some of such systems located
in our Galaxy will be detectable as gravitational waves sources with the
proposed space-based gravitational wave detectors [629]. Finally, evolution of
a double compact object (binary composed of BHs and/or NSs) that is not
affected by interactions with any other objects is fully determined by the
gravitational wave emission. gravitational wave-emission induced inspiral
eventually leads to its merger - the emitted signal is the strongest around
this phase and that is when it can be directly measured by current detectors.
For a binary whose orbital evolution is affected solely by gravitational wave
emission and whose component masses $m_{1}$ and $m_{2}$ are constant, the time
until the merger can be estimated by solving the set of equations describing
the related rate of change of the orbital separation ($a$) and eccentricity
($e$) in the lowest (quadrupolar) order [630]:

$<\frac{da}{dt}>=-\frac{64}{5}\frac{G^{3}\mu
M^{2}}{c^{5}a^{3}(1-e^{2})^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right)$
(VII.1) $<\frac{de}{dt}>=-\frac{304}{15}e\frac{G^{3}\mu
M^{2}}{c^{5}a^{4}(1-e^{2})^{5/2}}\left(1+\frac{121}{304}e^{2}\right)$ (VII.2)

where $M=m_{1}+m_{2}$ is the total mass of the binary and $\mu=m_{1}m_{2}/M$
is the reduced mass.

Figure 10:  Orbital separation decay of 30 $\,\rm{M}_{\odot}$ \+ 30
$\,\rm{M}_{\odot}$ binary for different initial separations and eccentricities
$e_{0}$ driven purely by gravitational wave emission (obtained by solving
equations VII.1 and VII.2). If the lines fall within the gray band, the merger
timescale of the corresponding binary exceeds the Hubble time.

Example solutions to the above equations are shown in Figure 10. It can be
seen that the merger timescale $\tau_{\rm mr}$ strongly depends on both the
starting separation $a_{0}$ and eccentricity $e_{0}$. In case of circular
orbits, this timescale $\tau_{\rm mr;circ}$ can be obtained directly from
equation VII.1:

$\tau_{\rm mr;circ}=\frac{5c^{5}}{256G^{3}}\frac{a_{0}^{4}}{m_{1}m_{2}M}$
(VII.3)

while the following analytical fit allows to obtain the merger time in case of
eccentric orbits [631]:

$\tau_{\rm mr}\approx\tau_{\rm
mr;circ}\left(1+0.27e_{0}^{10}+0.33e_{0}^{20}+0.2e_{0}^{1000}\right)\left(1-e_{0}^{2}\right)^{7/2}$
(VII.4)

An equal-mass m1=m2=30$\,\rm{M}_{\odot}$, circular binary with
$a_{0}$=10$\,\rm{R}_{\odot}$ will merge within only $\tau_{\rm
mr;circ}\approx$28 Myr, while for the same binary with
$a_{0}$=30$\,\rm{R}_{\odot}$ the merger timescale becomes $\tau_{\rm
mr;circ}\approx$2 Gyr. A binary on a highly eccentric orbit with e0=0.7 will
merge in only about 10% of the time that would be estimated for the same
masses and $a_{0}$ with eq. VII.3.
From eq. VII.3 one can estimate the maximum separation that a (circular)
binary can have at its formation to merge within the time shorter than the age
of the Universe ($\tau_{0}\sim 14$Gyr) due to gravitational wave emission:

$a_{\rm merging}<\left(\frac{1}{0.15}\
\frac{m_{1}m_{2}(m_{1}+m_{2})}{\,\rm{M}_{\odot}^{3}}\right)^{1/4}\left(\frac{\tau_{0}}{\rm
Gyr}\right)^{1/4}\ \,\rm{R}_{\odot}$ (VII.5)

The above equation yields $a_{\rm merging}\lesssim 50\ \,\rm{R}_{\odot}$ for
the considered m1=m2=30$\,\rm{M}_{\odot}$ binary. This is at least 10 times
smaller than the radii that typical massive stars reach during their evolution
(see Section VI.1). To end up as such a close BH+BH system, an isolated
massive star binary must either i) evolve through mass transfer phase(s) that
bring the stars close together during their evolution or ii) prevent stars in
initially close obits from expanding and merging prematurely (as in the
chemically homogeneous evolution scenario discussed in Section VII.1.1).

####  VII.3 Magnetic braking

Magnetic braking is the loss of angular momentum through stellar wind of a
magnetic star [632, 633, 634, 635, 636]. This mode of angular momentum loss
concerns primarily low mass (0.3 - 1.5 $\,\rm{M}_{\odot}$), main sequence
stars. Such objects have radiative cores and convective envelopes in which
magnetic fields can be amplified and sustained by the action of magnetic
dynamo [637, 638]. Even though the amount of mass such stars lose through
winds is often negligible, the associated angular momentum loss may be
significant because the magnetic field of the star forces the wind matter to
corotate to a large distance from the stellar surface. Magnetic braking can
completely slow down a star during its main sequence evolution, unless it is
tidally synchronised with a binary companion. In such case, tidal torques
replenish the rotational angular momentum of the star and extract it from the
orbit. Therefore, while counteracting the effect of magnetic braking, tides
lead to orbital shrinkage and possibly drive the binary to RLOF. This process
is important, for instance, in low mass X-ray binaries (and, in fact, appears
necessary to explain the observed high mass transfer rates of some of those
systems), where the coupled effects due to tides and magnetic braking need to
be considered [634, 639, 640]. The fraction of magnetic stars appears to
decrease with increasing mass: [641] estimate it to be $<$15% for
intermediate/high-mass stars. However, their wind mass-loss is no longer
negligible and if such stars experience magnetic braking, the angular momentum
loss is amplified accordingly.

### VIII Summary

The progenitors of stellar-origin BHs and NSs are very likely to exchange mass
and angular momentum with another star, and that likelihood appears to
increase with the initial stellar mass. Those exchanges happen through stable
and/or unstable mass transfer phase(s) and tides. Such interactions lead to
COs with properties that may substantially differ from those predicted within
the single star evolution theory. The properties of COs (masses, spins) that
remain in binaries after their formation can be further altered by accretion
from the companion. In particular, isolated binary progenitors of BH/NS
mergers (currently detectable GW sources) must form with relatively narrow
initial separations and likely interact multiple times during their evolution.
Binary processes are therefore expected to leave a strong imprint on their
observable characteristics (see Part V).
Good quantitative understanding of the effects of those processes is currently
hampered by uncertainties pertaining to the physics of binary interactions, in
particular criteria for mass transfer stability and the amount of mass and
angular momentum lost during various types of interactions. Models of binary
evolution also inherit uncertainties present in the detailed models of single
stars. Of particular importance in the context of binaries are those affecting
the evolution of stellar radii and the envelope structure (crucial when
determining the characteristics of the mass transfer), angular momentum
transport in stellar interiors (particularly important to understand the
effect of mass transfer on the evolution of the accretor), core-collapse
physics (determining the properties of the binary right after the formation of
the CO, in particular, deciding whether the system remains bound or not) and
stellar winds, introducing a metallicity-dependent effect on the evolution of
the orbit, stellar masses and spins (cfr. Section III). Finally, the structure
of post-interaction stars produced in binaries may be different than predicted
by any single star evolution model. The use of the latter models to
approximate the evolution of individual binary components (in particular that
of the secondary) is a common practice. However, it means that some relevant
effects may be missed by current models. This adds uncertainty to the current
binary evolution predictions that is largely unexplored.
It becomes evident that birth metallicity plays an important role in the
evolution of BH progenitors in binaries \- probably much more so than in the
case of systems with lower mass stars. There are multiple observational clues
that systems involving compact remnants of massive stars may become more
common and/or more extreme at low metallicity ( [642, 643, 644, 645, 646, 647,
648, 649, 650], (see also Part V). As highlighted throughout this section,
such link to low metallicity is also present in the current models. In the
latter case, it stems mostly from two effects: i) stellar wind mass loss rate
decreasing with metallicity and ii) later radial expansion of lower
metallicity massive stars. The first allows to retain more mass and angular
momentum within the BH progenitor (and so within the binary, limiting the
orbital expansion due to winds). This allows for the formation of more massive
and potentially - highly spinning BHs in the course of isolated binary
evolution, and opens up the possibility of chemically homogeneous evolution.
The second leads to the expectation that low metallicity BH progenitors in
binaries are more likely to initiate mass transfer during the CHeB phase of
their evolution rather than on the Hertzsprung Gap. Such interaction is much
more likely to be stable, to last longer (increasing the chances of observing
interacting BH binaries at low metallicity), and to lead to wider post-
interaction orbits than the common envelope evolution (likely encountered by
BH progenitor binaries at high metallicity). Both of those effects are
uncertain and need to be better quantified in future studies. Understanding
and characterising differences in the evolution and fates of massive stars at
different (in particular low) metallicities becomes increasingly more
important: with GW we are already detecting mergers of afterlives of massive
stars that formed anywhere in the past (in a wide range of environments and
metallicities) [651], and with the electromagnetic observations we are
beginning to uncover young galaxies in the early Universe. Their emission is
dominated by the most massive stars that are likely to be very metal-poor and
affected by binary interactions [100].

##  Part III Black Holes in binaries: formation channels - isolated
Floor Broekgaarden

### IX Introduction: The new frontier of GW paleontology

Figure 11: Depiction of GW paleontology. Observations of colliding BHs and NSs
are used as “fossils” to study how massive binary stars lived in the past.
Similarly to paleontology, this consists of learning about how the collisions
formed (backward modelling) as well as modelling theories of how stars live to
find what mergers they would leave behind (forward modelling). BHs are
represented with marbles for visualization purposes.

Probing millions of massive stars in different environments across our vast
Universe could become reality with a new emerging frontier in astrophysics
called gravitational-wave paleontology: studying the formation, lives, and
deaths of massive stars and the BHs they leave behind using the GW
observations of DCO mergers such as binary BHs as “massive star fossils”. This
is particularly exciting as GW probe massive stars and BH physics over cosmic
scales. Namely, although massive stars evolve into BHs and NSs within just a
few Myr, the time it takes for the DCO to inspiral, i.e., the time between the
formation of the DCO and the DCO merger (see Figure 12), can span a wide range
of durations from Myr to many Gyr [e.g., 652, 653]. This is because the DCO
orbital angular momentum loss by GW radiation is really small for most
systems. For a binary consisting of two COs with masses ${M}_{\rm{1}}$,
${M}_{\rm{2}}$, and semi-major axis $a$ the change in separation due to GW
radiation is given by Peters [see Equation VII.1 654] For the case of a
circular orbit consisting of two COs formed with an initial separation
${a}_{\rm{0}}$ the timescale of inspiral can be derived from Equation VII.4

Importantly, this equation reveals that the inspiral time is strongly
correlated with the initial separation ${a}_{\rm{0}}$ of the DCO system,
leading to a vast range of inspiral times even for a small range of
separations, and making typical delay times of order $\,\mathrm{Gyr}$ common
for systems with ${a}_{\rm{0}}\gtrsim 2\,\rm{R}_{\odot}$. This means that the
observed population of DCO mergers, even the current “local” detections
observed by LIGO, Virgo, and KAGRA, probe a mixture of events with progenitor
systems that could have formed throughout the entire cosmic history. This
makes GW a potential unique probe of massive stars through cosmic history. At
the same time, this mixture also makes it challenging to unravel the formation
histories of GW sources as the GW detections do not directly measure the
properties of their progenitor stars.

Figure 12: Schematic display of the different times in the formation and
evolution of a binary system that impact the time $t_{\rm{m}}$ at which a BBH
system will merge in the Universe. The inspiral times can span many Myr to Gyr
such that BH and NS mergers observed with GW detectors today probe massive
stars that formed at different cosmic times, in different environments, and
with different properties.

### X Formation pathways to GW sources

####  X.1 The separation challenge

Historically, it has been challenged whether DCO systems that merge within the
age of our Universe could exist at all. This is because all formation channels
have to overcome the so-called “separation challenge”. Using
14$\,\mathrm{Gyr}$ as a proxy for the age of the Universe, Equation VII.4
provides an estimate for the maximum possible separation for a DCO system such
that the inspiral can have taken place by today. Setting
$t_{\rm{insp}}\lesssim 14\,\mathrm{Gyr}$ gives for a binary consisting of two
1$\,\rm{M}_{\odot}$ COs that the minimum separation required is
${a}_{\rm{0}}\lesssim 3.7\,\rm{R}_{\odot}$ (see also [655]). This is extremely
small! To give some intuition, imagine scaling down everything in the Universe
by a factor of $10^{10}$. The diameter of the sun, $2\,\rm{R}_{\odot}\approx
14\cdot 10^{8}\,\rm{m}$, then becomes $14\,\rm{cm}$, or about half the size of
a cabbage. The diameters of massive stars are slightly larger: about $\sim
4\,\rm{R}_{\odot}$ for $8\,\rm{M}_{\odot}$ stars141414On the ZAMS., or about
one cabbage, and even larger for more massive stars (with the radii
approximately scaling as $M^{0.8}$). In this scaled-down Universe, two COs in
a binary system need to be a few cabbage distances apart in order to merge
within the age of the Universe. Intuitively one might thus start out with two
massive stars that are a $\sim 4\,\rm{R}_{\odot}$ (two cabbages) apart that
then eventually form the required DCO system with a separation of about $\sim
4\,\rm{R}_{\odot}$ that can merge within the age of the Universe. However,
massive stars typically expand during their lives to $100-1000$ times their
initial size, e.g., during the Hertzsprung Gap phase [380, 656]. Such an
expanding star would completely engulf the other star if the stars are only a
few solar radii apart, and the system would merge as stars, never making a DCO
system [e.g., 657]. Importantly, all formation pathways thus need to find a
way to overcome or avoid this separation challenge, such that the system can
form a tight DCO system that can merge within the age of our Universe and be
observed with GW observation in the current day.

Figure 13: Summary of proposed formation channels to form DCO mergers

####  X.2 Formation Channels

The main formation scenario leading to DCO coalescences is still under debate.
Several theories have been proposed. Figure 13 gives an overview of the most
commonly discussed channels that we discuss in more detail below. The
formation channels are classically divided into two categories: stars born in
the “field”, which evolve isolated as a binary, triple, or higher order
multiples, and dynamical channels, where the stars are born in dense stellar
environments such as globular clusters and young open star clusters and they
merge due to gravitational dynamical interactions. This Section focuses on
binaries from the isolated channel, which we will discuss in great detail
below. The other channels are also briefly mentioned below and we refer the
reader for more details to the reviews by [190, 655, 658, 656, 659, 191], and
references therein, and to Part IV.

Figure 14: Schematic depiction of the isolated formation channel. The acronyms
stand for zero-age main sequence (ZAMS), Roche lobe overflow (RLOF), supernova
(SN), common-envelope phase (CE), and double compact object merger (DCO). This
figure is created using icons from [660].

#####  X.2.1 Isolated Binary Evolution formation channel ($N=2$)

The most commonly studied channel is perhaps the isolated binary evolution
formation channel. This formation pathway starts with two ($N=2$, with N the
number of stars) massive stars formed on the ZAMS in an isolated binary with a
relatively wide orbit and overcomes the separation challenge by drastically
shrinking the orbit later on in the evolution of the binary stars. We describe
a typical example of the evolution below, which is shown schematically in
Figure 14.

  1. 1.

First, two massive stars are formed on the ZAMS with a large separation (i.e.,
with typical orbital separations of $a\gtrsim 100$–$1000$$\,\rm{R}_{\odot}$).

  2. 2.

The most massive star in the binary, the primary, evolves first off the ZAMS
when the hydrogen in the stellar core is almost completely depleted from
fusing hydrogen into helium, stopping nuclear fusion in the core. This lowers
the pressure outwards causing the core to contract, creating a (further)
density gradient, and releasing gravitational binding energy that initiates
fusion to continue in a shell surrounding the core (hydrogen shell burning).
This leads the hydrogen envelope to expand to radii of $\sim
100$–$1000\,\rm{R}_{\odot}$ for these massive stars, depending on the birth
metallicity151515Where the metallicity $Z$ is here the fraction of elements
heavier than hydrogen and helium in the star. of the stars[e.g., 656]. The
star is now in the Hertzsprung Gap phase. For a sufficiently close binary, the
star’s radius will exceed its so-called Roche Lobe radius, which defines the
equipotential surface within which material is bound to the star. From
observations, this is expected to happen for $\sim 70\%$ of all massive star
binaries [95]161616But not necessarily all as a Hertzsprung Gap star.. As a
result, a mass transfer phase is initiated during which mass can flow from the
donor star to its companion, the secondary. Under the right conditions, this
mass transfer phase can be long-lasting (thermal timescale of the donor star)
and proceed stably [see, e.g., 657, and references therein] called stable
RLOF. During this mass transfer the companion star can potentially accrete a
significant fraction of the transferred mass from the donor star, depending on
the secondary’s thermal timescale and response to the mass transfer.

  3. 3.

Eventually, the mass transfer stops, leaving behind a stripped-envelope
primary star that consists of a helium core and likely a small hydrogen
envelope layer [661], which can be observed as a WR star [e.g., 81, 662].

  4. 4.

Eventually, the primary star will end its life and likely undergo a supernova
and leave behind the first CO; a NS or BH. During the collapse into the CO,
the system might receive a significant kick of hundreds of $\rm{km}/\rm{s}$,
that disrupts the majority of binaries [e.g., 663, 664, 179].

  5. 5.

In case the binary stays bound, which might be the case for less than
$\lesssim 5\%$ of the systems [e.g., 179, 665], where disruption depends on
the magnitude and orientation of the supernova kick, the separation of the
binary and the amount of ejected mass [e.g. 666, 546]. The end result is a
binary with a CO and a star. These might be observed as dormant BHs in Gaia
[667, 668, 669] or as stellar wind-fed X-ray binaries [e.g., 670, 671, 672].

  6. 6.

The secondary will eventually also evolve off the ZAMS and expand, initiating
a mass transfer phase. This mass transfer is thought to likely be non-
conservative because the CO companion can only accrete a very limited fraction
of the donated mass (typically assumed as the Eddington limit). This leads the
binary to rapidly shrink the orbital separation. Consequently, this leads to a
runaway process as the secondary’s Roche Lobe shrinks rapidly with decreasing
separation, leading the star to even more overflow its Roche Lobe. This leads
to a dynamically unstable mass transfer phase during which the CO is engulfed
in the envelope of the secondary, a common-envelope phase [673, 674]. The CO
is thought to rapidly spiral in due to the drag forces it experiences from the
surrounding gas envelope. In some cases the orbital energy can be deposited
into the envelope, allowing it to heat up and expand and eventually be
successfully ejected, halting the inspiral. If this happens, the resulting
binary consists of the CO in a tight orbit with the stripped secondary star,
an example of a possible observed system in this phase is Cyg X-3 [675, 676].
If the envelope ejection fails, the system results in a merger of the star
with the CO, which can possibly form a Thorne– Zytkow object [471] or lead to
peculiar supernovae [e.g. 677, 678, 679].

Alternatively, the reversed mass transfer phase can proceed stably under
certain circumstances and ranges of orbital periods and mass ratios for the
binary [e.g., 680, 681, 682, 683, 684, 685, 686]. Based on this, several
authors have shown that a formation pathway with only stable mass transfer
(avoiding a common-envelope event) could even dominate the merger rate for BBH
mergers [687, 688, 689] and that the role of common-envelope phases in the
formation of DCO sources might be overestimated by population models [e.g.,
686, 690, 691].

  7. 7.

At the end of this mass transfer phase the system now consists of a binary
with a CO and the stripped helium secondary star in a tight orbit.

  8. 8.

In some cases, a second mass transfer phase can take place when the helium
star expands. This is sometimes referred to in the literature as case-BB mass
transfer from the helium star onto the primary CO [cf. 692]. This typically
occurs when the secondary is a relatively low mass helium star as they expand
to larger radii compared to more massive helium stars [e.g., 225, 692] This is
thought to be especially important and frequent in the formation of binary
neutron star systems [693, 694].

  9. 9.

Eventually the secondary also undergoes a supernova and forms the second CO.
The tight orbit leads the system to likely stay bound, but a decent fraction
of binaries might still disrupt during this second supernova.

  10. 10.

The DCO system will slowly inspiral due to the loss of orbital energy through
GW and eventually merge after millions to many billions of years [695]. If the
binary has a tight enough orbit, as described by Equation VII.4, it can merge
within the age of the Universe and be observed as a GW source by current
ground-base GW detectors.

As described in the formation channel above, there are many steps in the
evolutionary pathway that a massive binary star system needs to survive in
order to form a DCO that merges in the age of our Universe. In most cases,
however, the system will instead disrupt during one of the supernovae and/or
merge during a mass transfer event. The formation of a DCO merger is a rare
outcome, and it can thus be challenging to find what exact properties of the
pairs of stars make it survive all steps and form a DCO system that survives
eventually billions of years until it collides as a DCO system and forms a GW
source. We can write down an analytically approximate equation to calculate
the DCO merger rate that this channel can produce. For BBH mergers, this comes
down to an equation along the lines of:

$\displaystyle\mathcal{R}_{\textrm{BBH}}$ $\displaystyle=f_{\textrm{init
sep}}\times f_{\textrm{primary}}\times f_{\textrm{q}}\times f_{\textrm{survive
SN1}}\times f_{\textrm{CE}}\times f_{\textrm{survive SN2}}\times
f_{\textrm{merge}}$ $\displaystyle\sim\ 0.25\ \times\ 0.0007\ \times\
0.7\times 0.05\times 0.2\times 0.5\times 0.5\approx 3\times 10^{-7}.$ (X.1)

First, we need the binary system to be close enough to initiate a mass
transfer phase during its lifetime but not too close that it undergoes a
stellar merger at ZAMS. From studies including Schneider  _et al._ [657] it
appears that typically among the four orders of magnitude in separation, about
one order of magnitude gives the binaries the right separation for it to
undergo mass transfer without a stellar merger. This gives a factor
$f_{\rm{initsep}}=0.25$, given the uniform logarithm distribution of stars in
separation [696]. Note that this separation condition also takes into account
the fraction of massive stars that are in binaries, as binary systems with too
wide orbits are the effectively single stars in Sana  _et al._ [697], giving a
binary fraction of $\sim 100\%$ when this separation range is corrected for.
Second, we multiply by $f_{\textrm{primary}}$, the fraction of stars where the
primary is massive enough to form a BH. This is approximately for stars with
masses $20\lesssim{M}_{\rm{1}}\lesssim 200$$\,\rm{M}_{\odot}$ [see 698, for
more details]. And integrating this over a Kroupa [699] initial mass function,
gives approximately
$f_{\textrm{primary}}=(\int_{8\,\rm{M}_{\odot}}^{20\,\rm{M}_{\odot}}{M}_{\rm{1}}^{-2.3}\mathop{}\\!\mathrm{d}{M}_{\rm{1}})/(\int_{0.08\,\rm{M}_{\odot}}^{200\,\rm{M}_{\odot}}{M}_{\rm{1}}^{-2.3}\mathop{}\\!\mathrm{d}{M}_{\rm{1}}))\approx
0.0007$, where we choose 200$\,\rm{M}_{\odot}$ as the maximum mass of massive
stars [e.g., 700], and 0.08$\,\rm{M}_{\odot}$ is chosen as the approximate
minimum mass for a star below which a brown dwarf is formed instead [e.g.,
701, 702]. Third, the secondary also needs to be massive enough to form the
other CO (NS). Moreover, the first mass transfer phase in the binary is likely
unstable if the secondary is significantly less massive than the primary at
the onset of the mass transfer leading to a stellar merger [657]. For this
simple calculation, we assume a mass ratio of
$q={M}_{\rm{1}}/{M}_{\rm{2}}\gtrsim 0.3$ at the ZAMS is required for stable
mass transfer [e.g., 703, 704, 705, 706, 707, 657]. From observations the mass
ratio distribution of stars is found to be roughly consistent with a uniform
distribution [cf. 708, 709, 710, 363, 697, 711]. This gives a factor
$f_{\rm{q}}\sim 0.7$. Fourth, the binary system typically has a wide orbit by
the time of the first supernova and likely disrupts by the significant natal
kicks of hundreds of $\rm{km}/\rm{s}$ the system can receive when the NS forms
[e.g., 712, 713]. Studies have found that this likely disrupts $\gtrsim 95\%$
of the systems at this point giving a $f_{\textrm{SN,1}}\approx 0.05$ [665,
714]. Fifth, the common-envelope phase might well be one of the most uncertain
aspects of the binary’s life. What stars and what fraction of stars can
successfully eject the envelope and form a tight system is uncertain. An
example from the literature for NSNS mergers is a fraction of
$f_{\rm{CE}}\approx 0.2$ [e.g. 665]. Sixth, the system also needs to survive
the second supernova, this is more likely since the system has a much tighter
orbit after the common envelope phase. We estimate the factor with a factor
$f_{\rm{SN,2}}\approx 0.5$ [cf. 665]. Seventh, once the NSNS binary has formed
it needs to have a tight enough orbit to merge within the age of the Universe
so that it can be observable today. This likely removes another half of the
systems. Giving a factor $f_{\rm{merge}}\approx 0.5$.

Combined this gives that a fraction of about $3\times 10^{-7}$ of all stars
forms a NSNS merger, making this a rare outcome among stellar populations.
Assuming a local star formation rate of $\approx 1.5\times 10^{7}\,M_{\odot}$
Gpc-3 yr-1 [cf. Equation 15 in 715] with the average mass of stars of $\sim
0.3M_{\odot}$ (from the initial mass function; Kroupa 699), the yield of
$f_{\textrm{NSNS}}=9\times 10^{-7}$ gives a NSNS merger rate of

$\displaystyle\mathcal{R}_{\rm{NSNS}}\approx\frac{1.5\times
10^{7}\,\,\rm{M}_{\odot}\rm{Gpc}^{-3}\rm{yr}^{-1}}{0.3\,\rm{M}_{\odot}}\times
3\times 10^{-7}\approx 17\,\rm{Gpc}^{-3},$ (X.2)

which falls inside the merger rates inferred from gravitational-wave
observations, as well as typical simulation results [655].

In practice, there are many uncertainties in the physical processes underlying
the formation, evolution, and deaths of massive stars and how these are
modeled when investigating the formation pathways of DCO systems. Examples
include uncertainties in the initial conditions [716, 96, 717, 718, 719, 720],
the metallicity-dependent star formation history [721, 722, 723, 724, 725,
726, 727, 728, 729], stellar winds [730, 204, 205, 731, 732, 733], stellar
rotation [e.g., 734, 735], the evolutionary tracks [736, 65, 737, 508], the
expansion of the envelopes [e.g., 661, 738], common envelope phase (and its
modeling) [739, 740, 741, 486, 736, 742, 101, 691, 743, 742, 744, 437, 688],
mass transfer processes [e.g., 745, 746, 747, 748, 749, 750, 657, 751, 752,
753, 754, 755, 756, 757], supernovae kicks [758, 551, 553, 759, 760, 542, 555,
761, 762, 763, 764, 713, 713, 354, 765, 766],
supernova remnant masses [767, 53, 768, 769, 770, 324, 771, 772, 773, 774,
775, 776, 777, 778, 335, 779, 780, 781], black hole spin up processes [782,
783, 784, 785, 786, 787, 788, 789, 790, 791, 792], whether there is a mass gap
between black holes and neutron stars [e.g., 793, 794, 338, 795, 796, 797,
798, 799, 800, 801] and the existence of a black hole mass gap due to pair-
instability supernova [802, 355, 61, 358, 803, 804, 805, 806, 807, 808, 809,
537, 810]. These uncertainties can all impact the outcome of stellar
populations, making it challenging to model them all or rely on a limited set
of models. On the other hand, it also means that observations of DCO mergers
with ground-based gravitational-wave detectors, as well as other observations,
have the potential to test and constrain the physics of these evolutionary
phases.

####  X.3 Other formation channels

Many formation channels other than the isolated binary evolution channel have
been proposed. All of these also overcome the separation channel. We describe
them briefly and refer the interested readers to the reviews from Mapelli
[811], Gerosa and Fishbach [658], Mandel and Farmer [656], Arca Sedda  _et
al._ [659], and Spera  _et al._ [191], and references therein, and to Part IV
for more details.

  * •

Population-III stars: in this scenario, the binary consists of two massive
metal-free (or extremely metal-poor) stars in a tight orbit. Such metal-poor
stars are expected to exist in the early Universe as the first generation of
stars that have not yet been exposed to enrichment processes that would lead
to higher metallicities. These stars are thought to avoid extensive radial
expansion of the envelope and avoid the separation challenge. Stellar winds
and radial expansion are driven by the presence of heavier metals, causing
these stars to experience a drastically reduced radial expansion and self-
stripping through winds [812, 813, 814, 815, 816]. Moreover, the lack of
metals might also favor the formation of higher mass stars (a flatter initial
mass function), further boosting the potential contribution from this channel
[see 817, 818, 819, 820, 821, 822, 823, 824, 825, and references therein].

  * •

Chemically Homogeneous Evolution: in this scenario, a pair of massive metal-
poor stars starts in a close enough orbit such that the stars are tidally
locked. The low metallicity again suppresses strong winds and reduces the
radial expansion of the stellar envelopes. The tidal locking leads the stars
to spin up to a rotation close to the stellar break-up frequency that leads to
efficient mixing of the helium out of the stellar core into the envelope and
fresh hydrogen from the envelope into the stellar core. This allows the stars
to evolve chemically homogeneously and use almost the entire hydrogen for
fusion until eventually, the entire star has formed a naked helium star [826,
827, 828, 829, 830, 831]. The heavy helium stars do not expand much and can
avoid mass transfer phases during the rest of their lives, avoiding the
separation challenge. The stars will eventually collapse into BHs and form a
BBH system with two BHs of similar masses ($\sim 30\,\rm{M}_{\odot}$) that are
likely highly spinning and can contribute to the DCO population observed today
[832, 833, 834, 725, 835]. In rare cases, this formation channel can also form
BHNS mergers [836].

  * •

Isolated triples or higher order multiples: massive stars are also commonly
formed in triples or higher order multiples [e.g. 837]. In such systems, the
stars might experience similar evolutionary processes as described in the
isolated binary evolution channel, but additionally, the additional star(s)
can impact the system both dynamically and through binary-type interactions
(supernovae, mass transfer, etc.). Several studies have extensively
investigated the evolution of triples and higher order multiples and found
that they can significantly contribute to the formation of DCO systems [e.g.,
838, 839, 840, 841, 842, 843, 844, 845, 846]. Another proposed channel comes
from field binaries that might be perturbed by flybys from other stars that
can increase the eccentricity and lead to higher merger rates; a channel that
lies in between isolated and dynamical formation like triples and higher order
multiples [847, 848].

  * •

Dynamical channels: instead, DCO mergers might dominantly form from stars in
dense stellar environments such as globular clusters, young star clusters, and
nuclear star clusters in the centers of galaxies.. In all these channels the
dynamical interactions play a dominant role in the formation of the DCO
system171717Where we note that isolated triples and higher multiples also
belong to this category, other examples for this are the formation of DCO in
hierarchical 3-body systems with a SMBH, and in AGN disks (see also Part IV).

  * •

Exotic channels: other, more exotic formation channels have also been
suggested. This includes the formation of BBH mergers from primordial black
holes, or from other physics beyond the standard model [e.g., 849, 850, 851,
852]. These are also discussed in Part IV.

Each of these channels comes with its own model uncertainties, contributing to
the complexity of gravitational-wave paleontology. On the other hand, the
rapidly increasing population of BBH mergers is a probe of all these physical
processes that are underlying the formation and lives of BHs and their massive
star progenitors.

##  Part IV Black Holes in binaries: formation channels - dynamical
Carl L. Rodriguez

### XI $N=3$; Triples and Three-body Encounters

The first extension beyond binary physics is to consider the presence of a
third body. But whether this triple system is stable or not changes the long-
term dynamical evolution of the system, as well as the properties of the BBHs
it produces. Configurations of three gravitational point particles which are
known to be stable for many orbital periods are often referred to as
hierarchical triples; these can be thought of as “a binary within a binary”
(that is, there exists an inner binary with semi-major axis $a_{1}$, and an
outer tertiary that orbits the binary’s center of mass at a wider separation
$a_{2}$). Due to their separation in scales, these systems typically do not
exchange energy between their inner and outer orbits (e.g. the Earth-Moon-Sun
system). Decreasing the outer orbital separation, however, also decreases the
stability of the system; while there does not exist a good first-principles
derivation of this stability criterion, numerical experiments [853] building
off the analytic arguments of [854] have shown that triples with”

$\frac{a_{2}}{a_{1}}>\frac{2.8}{1-e_{2}}\left[\left(1+\frac{M_{t}}{M_{b}}\right)\left(\frac{1+e_{2}}{\sqrt[\
]{1-e_{2}}}\right)\right]^{2/5}\left(1-0.3\frac{I}{\pi}\right)$ (XI.1)

for a system with inner-binary mass $M_{b}$, tertiary mass $M_{t}$, inner and
outer eccentricities of $e_{1}$ and $e_{2}$, and a mutual inclination of $I$,
tend to be long-lived and stable (though see [855] for an updated study
focusing on the extreme-mass ratio limit). We will refer to systems where Eqn.
XI.1 is not met as democratic triples (in contrast to stable hierarchical
triples).

####  XI.1 Hierarchical Triples

While hierarchical triples (in the absence of tides or other dissapative
forces) do not transfer energy between the inner and outer orbits (and
therefore do not alter their semi-major axes), the key feature of these
systems is their exchange of angular momentum over many orbital periods. These
secular processes, as they are often called, have been known for many decades,
going back to the original papers by Lidov [856] and Kozai [857] for whom the
Lidov-Kozai (LK, though often shortened to “Kozai”) mechanism is named. These
hierarchical triples are of interest for a wide range of systems in
astrophysics, from planetary dynamics, to the inclination of asteroids, to GW
source production.

To lowest (quadrupolar) order, the LK mechanism involves an exchange of
angular momentum between the mutual orbit of the two binaries and the inner
binary. If we assume a system in which both the inner and outer binaries are
initially circular, and one of the inner masses is zero (i.e. the test-mass
approximation), one can show that the maximum eccentricity the inner binary
can achieve during these oscillations is:

$e_{\rm{max}}=\sqrt[\ ]{1-\frac{5}{3}\cos^{2}i_{0}}$ (XI.2)

where $i_{0}$ is the initial inclination of the inner binary with respect to
the outer binary. See [858] for a detailed review. In this regime, the inner
binary can reach arbitrarily large eccentricities which, since the timescale
for binaries to merge due to GW emission scales as $T_{\rm
GW}\propto(1-e_{0}^{2})^{7}/2$ [859, c.f. Equation 5.14, ], can rapidly drive
even wide binaries to merge within a Hubble time.

The first studies of this effect in the context of stellar-mass BHs were
performed in the early 2000s by [860] in the context of BBHs formed through
four-body interactions in GCs. It was further noted by [861] that these
systems could potentially maintain their eccentricity while passing into the
LIGO sensitivity band, providing a mechanism for distinguishing dynamically-
formed binaries from those formed from isolated evolution (although subsequent
studies have shown the percentage of mergers with $e>0.1$ in the LIGO band to
be substantially lower than the reported 30% from [861]). We note that there
exists significant additional complications one can consider to this process,
such as the presence of higher-order terms in the Kozai-Lidov expansion (which
give rise to the so-called eccentric Kozai-Lidov mechanism, e.g.,[862, 863,
864, 865, 866]), complicated multi-scale cross-terms between the three-body
and post-Newtonian expansions [e.g., 867, 868, 869, 870, 871, 872], and even
the breakdown of the Kozai-Lidov orbit approximation entirely [873, 874]
(which are responsible for producing the most eccentric of the triple-induced
mergers).

Since then, many authors have explored the potential contribution of triples
formed through in a variety of astrophysical environments. Observational
evidence suggests that massive O-stars above 16 $M_{\odot}$ (the progenitors
of BHs) have an average of 2.1 companions [875, Table 13], making CO triples a
natural product of star formation. These triples can undergo Kozai
oscillations that drive their inner binaries to merge [e.g., 876, 877, 878,
879, 880], with further studies showing that such triple-mediated mergers
exhibit unique mass ratios [e.g., 881, 882] & spin-orbit alignments [e.g.,
883, 878, 884, 885, 886] that may distinguish them from either field-formed
binaries or binaries formed through other dynamical channels (although
eccentricity remains the most reliable metric for distinguishing triple-
induced mergers, despite it’s degeneracy with other dynamical channels).

In addition to massive stellar triples, CO triples are often formed or found
in other dynamical environments. In stellar clusters, such as OCs and GCs,
triples are a common byproduct of binary-binary encounters, in which two
binaries dynamically interact and (typically) eject one of the components of
the wider binary, leaving a hierarchical triple and a single star/CO [860,
887, 888, 889, 890, 891, 892, 893]. Note that this process becomes more
efficient in smaller clusters such as OCs; more massive clusters such as GCs
and galactic nuclei form triples with similar masses (with the rate of such
mergers in GCs being two orders of magnitude lower than those from standard
binary hardening that we describe in Section XIII, e.g. [887, 890]).

The one notable exception to this is nuclear star clusters containing a super-
massive BH. In these clusters, stellar binaries can form hierarchical triples
where the SMBH acts as the tertiary driving the stellar-mass inner binary to
merger [e.g., 894, 895, 896, 897, 898, 899, 900]. We note that, unlike triples
with similar mass tertiary companions, stellar-mass binary/SMBH triples may be
directly confirmed by the measurement of Doppler shift and other effects from
the SMBH with space-based GW detectors [e.g., 901, 902, 903].

####  XI.2 Democratic Triples

While hierarchical triples are of significant interest due to their potential
long-term stability, there exists a long and relevant literature on short-
lived democractic triples, typically formed by the strong interaction of a
single star or CO and a binary. See, e.g., Figure 15. The fate of triples
formed via these ”scattering encounters” was first studied numerically 40
years ago by Hut and Bahcall [904, 905, 906], and has since spawned an entire
industry dedicated to understanding the outcome of these encounters.

Figure 15: An example strong encounter between a binary and a single object in
which the incoming particle exchanges into the binary, ejecting one of the
original binary components out to infinity. These encounters occur frequently
between massive stars and/or BHs in dense star clusters such as OCs, GCs, and
GN, and are critical for the production of BBH mergers. This scattering was
performed with the Fewbody code Fregeau  _et al._ [907].

Because of the high densities required for such encounters to take place, they
are typically found only in the cores of dense star clusters such as OCs, GCs,
and NSCs. The one notable exception is the possibility of encounters between
stars and wide binaries in galaxies, where many of the same interactions that
occur in clusters can be replicated due to the large interaction cross
sections of ultra-wide binaries. See [908, 909]. The effect of these
encounters on BBH mergers depends on whether such encounters primarily occur
among massive stars before they form BHs (as often happens in lower-density
OCs) or among BHs and BBHs in the cores of dense clusters such a GCs and GN.
We will now describe these various pathways.

### XII $N=\mathcal{O}(10^{3}-10^{5})$; Open Clusters

As we add more particles to the systems we consideration, the number of
possible interactions increases as well. Despite the previous sections where
have assumed that most stars form as either isolated binaries or triples, we
in fact know that most star formation occurs in clusters, whether they be
unbound associations or YMCs such as OCs [910]. And while the majority of
massive stars are born with at least one companion [875], most of these
binaries and triples should actually seen as living (at least at first) within
dynamically active environments, where encounters with neighboring stars can
modify the evolutionary path the binary would have followed had it evolved in
isolation. These modifications can take many forms, such as exchange
encounters (like the one pictured in Figure 15) in which one component is
replaced with another, or resonant encounters in which the orbital period and
eccentricity of the binary are dynamically scrambled to a state that would be
inaccessible to a system evolving in isolation. And while the small escape
speeds of OCs mean that a single encounter is often enough to eject the
stellar binary from the cluster, if many, or even the majority, of binaries
formed through normal star formation undergoes a single dynamical encounter,
it can have a profound effect on the population of merging BBH.

Over the past two decades, many authors have used direct $N$-body techniques
to explore the formation, evolution, and destruction of these clusters and the
the BBHs they produce [e.g., 911, 912, 913, 914, 888, 915, 889, 916, 917, 918,
919]. By assuming that the majority of star formation occurs in these
environments, they can reproduce a significant fraction (or even potentially
all) of the BBH mergers observed by LIGO [e.g., 920, 919]. The interactions,
and particularly the dynamical exchange encounters, that occur among massive
stars in these clusters create binaries that can move along unique
evolutionary pathways, producing binaries with significantly larger masses
than those found through isolated binary evolution. Dynamical encounters can
also alter the alignment between the spin and orbital angular momenta of the
binaries [921, 922], another key observable of dynamical formation (though see
§XIII.1 where we discuss spin-orbit alignment in detail).

However, one of the most unique outcomes of stellar encounters prior to BH
formation is the potential to produce individual BHs more massive than those
that can be formed from stellar collapse alone. In single stars, it is largely
assumed that BHs cannot form with masses between $\sim 50M_{\odot}$ and
$120M_{\odot}$, due to the presence of pulsational pair instabilities and
pair-instability supernova [e.g., 923, 924]. But when two stars merge in OCs,
their merger products can potentially bypass this limit by developing unique
core-to-envelope mass ratios that cannot be realized through single or binary
stellar evolution [e.g., 925, 922]. These stars, upon their formation, can
then exchange into other stellar binaries, eventually forming BBHs with masses
that cannot be the result of isolated stellar evolution, a key indicator of
dynamical formation (though as we will see below, not a unique one, as
repeated BH mergers can also create BHs with masses in the pair-instability
mass gap). Of course, stellar mergers produce merger products with inflated
radii, that then in turn increases the possibility of successive stellar
mergers. These potential “runaway” collisions have been studied for nearly two
decades [926, 927, 928, 929, 930, 931], and represents both an important
channel for the production of both the heaviest LIGO sources as well as the
potential seeds of intermediate-mass BHs and SMBHs.

### XIII $N=\mathcal{O}(10^{4}-10^{7})$; Globular Clusters

As we consider systems with more particles (and correspondingly deeper
gravitational potentials), the primary mechanism of BBH formation shifts.
While smaller clusters largely rely on a natal population of massive binaries
to create their BBHs, the stellar binaries that eventually become merging BBHs
in OCs only undergo about one strong encounter before they are ejected from
the cluster. But as we move to larger systems, such as GCs these primordial
binaries become less relevant to the formation of BBHs for two reasons: first,
the higher mass (and correspondingly higher escape speed) of the clusters
means that significant fractions of BHs can be retained by the cluster after
formation [932]. This means that, unlike OCs, BBHs actually remain (and form
dynamically!) in the clusters that create them. And second, because of their
deeper central potentials, these binaries undergo many repeated encounters
before they are ejected from the cluster. These formally chaotic three-body
encounters erase any information about the initial state of the binary,
yielding a population of BBHs whose origin is purely a function of
gravitational dynamics.

While their formation is still an area of active research, it is easiest to
consider GCs as forming from a single burst of star formation $\sim 10-14$ Gyr
ago. Within 10 Myr, the massive stars in the cluster collapse, creating a
population of stellar-mass BHs; assuming the BHs receive low or no natal kicks
due to increased supernova fallback [933, 934] or direct-collapse [935]
scenarios, a significant fraction [anywhere from 65-90%, 936, 932] can be
initially retained by the cluster. The BHs experience a drag force arising
from dynamical friction [e.g., 937, §8.1], causing their orbits to decay, and
driving them into the cluster core where they form a subsystem of BHs. These
BH subsystems are simultaneously key to understanding the overall evolution of
GCs as well as the BBHs they produce. As the cluster collapses due to
relaxation [e.g., 937], the density of BHs in the core increases. Eventually,
the density becomes high enough to facilitate three-body interactions, where
BBHs are either formed through encounters of three single BHs [938, 939, 940,
941, 936] or three-body encounters between pre-existing binaries and single
BHs (such as those described in §XI). Once a BBH has been formed it will
continue to participate in encounters with other BHs each of which will, on
statistical average, shrink the BBH’s orbital period [942]. This process
continues until either the BBH merges due to GW emission [typically in
eccentric binaries, e.g., 943, 944], or the recoil of the binary from the
three-body encounter is sufficiently large to eject it from the cluster
entirely, where it may or may not merge later due to GW emission. As a result,
it is largely the central escape speed of the cluster that sets how compact a
BBH will be [945, 946]. GCs typically process their BHs from most to least
massive, forming binaries with the heaviest BHs first before moving on to the
lower-mass objects. This, combined with their age and low stellar metallicity,
makes GCs typically more efficient at producing heavier BBH mergers in the
local Universe; for a $30M_{\odot}+30M_{\odot}$ BBH like GW150914, the optimal
cluster to produce a merger in the local universe has a present-day mass of
$\sim 3-6\times 10^{5}M_{\odot}$ [947], corresponding roughly to the upper
quartile of GC masses in the MW [948].

Over the past three decades, many authors have used either direct-summation or
Monte Carlo-based $N$-body simulations [e.g., 949, 911, 950, 951, 952, 953,
954, 955, 956, 957, 958, 959, 946, 960, 961, 962] or semi-analytic techniques
[e.g., 941, 945, 963, 964, 965, 966, 967, 968, 969] to study BBH formation
from GCs, with most estimates of the BBH merger rate in the local universe
[going back to 938] hovering around $\sim 10\rm{\leavevmode\nobreak\
Gpc}^{-3}$. While this number is potentially sufficient to explain the full
population of BBHs detected so far [970], more sophisticated analyses suggest
that they are one of several channels contributing to the LVK GW detections
[e.g., 971].

While there are still uncertainties associated with the GC channel (mostly
related to GC formation and the evolution of massive stars and compact-object
formation), the gas-free dynamics of these environments does allow several
robust predictions to be made for the BBH population, and in many ways, GCs
represent the first environments (in terms of increasing numbers of
particles), where BBH formation is purely dynamical, and does _not_ depend on
the properties of the initial binary population. This transition marks the
introduction of several observable features that are relevant to GCs, NSCs,
and AGN, which we will now describe.

####  XIII.1 Observational Signatures of Purely Dynamical Formation

First, because BBHs in GCs are primarily created and driven to merger through
chaotic three-body encounters, the spins and orbital angular momenta of the
BBHs are essentially randomized after a single strong encounter, causing them
to be isotropically distributed. GW detectors are primarily sensitive to the
projection of the BH spins along the direction of the binary angular momenta,
making the presence of spin-orbit anti-alignment in the BBH population a key
observable [e.g., 972]. Isolated binaries are expected to form BBHs with spins
largely aligned with their orbit, while both hierarchical triples and OCs tend
to push the distribution away from aligned and towards spin-orbit misalignment
(while still maintaining a preference for spin alignment). In effect, as the
number of particles increases, the isotropy of the distrubtion increases, from
nearly aligned when $N=2$ to isotropic once $N\gtrsim 10^{5}$.

Second, the strong encounters that produce BBHs also facilitate eccentric
mergers through GW captures during democratic triples. Simulations of both
isolated scatterings and full Monte Carlo $N$-body models have suggested that
$\sim 20\%$ of BBH mergers that occur inside GCs [943, 973] are the result of
GW captures during strong encounters where the emission of GWs during close
hyperbolic encounters is sufficient to create bound binaries from previously
unbound BHs. Roughly half of these captures will reach a GW frequency of 10Hz
with $e\gtrsim 0.1$ and be detectable by LIGO. As was described in §XI,
eccentricity remains a key observable of dynamical formation for BBHs, and
while the ratio between in-cluster mergers and ejected mergers can change
depending on the central escape speed (from roughly half-and-half for typical
GCs to nearly 90% or more for very massive clusters, e.g., [974, 975]), the
fraction of in-cluster mergers with eccentricity above 0.1 remains largely
constant.

Finally, if two BHs merge inside a cluster, there is nothing to stop the new
BH they form from participating in further mergers. It is well known from
numerical and analytic relativity that merging BBHs receive a recoil kick due
to the asymmetric emission of GWs [e.g., 976, 977, 978, 979, 980], with the
kicks of highly-spinning, non-equal-mass BH mergers reaching as high as
$5000\leavevmode\nobreak\ \rm{km}/\rm{s}$ [981]. However, if BHs are born with
low spins from stellar collapse (as suggested by some studies of angular
momentum transport in massive stars, e.g., [982]), then the merger products of
near-equal mass BHs can sometimes be retained by their host clusters, where
they can potentially find new BH companions through three-body encounters and
merge again. The number of previous merger events that a BH has undergone is
referred to as its “generation”, with BHs born from stellar collapse being
“first-generation”, BH formed from a single previous BH merger being “second-
generation”, and so on (following [983]). GCs, with their relatively low
central escape speeds ($\lesssim 100\leavevmode\nobreak\ \rm{km}/\rm{s}$ for
the most massive clusters, though potentially higher at formation [975]), can
easily produce BBH mergers with second-generation BH components when the BH
spins from stellar collapse are small [e.g., 960, 984], but they rarely retain
BBH third or higher generation mergers. While OCs rely exclusively on stellar
mergers to create BHs in the pair-instability mass gap, GCs represent the
first environment where BH growth is facilitated by repeated BH mergers. But
to produce multiple successive generations of BH mergers, or even intermediate
mass BHs, we must consider even deeper potential wells.

### XIV $N=\mathcal{O}(10^{5}-10^{8})$; Nuclear Star Clusters

The final frontier of dense star clusters, both in terms of the number of
particles, and computational difficulty, remains the NSCs that populate the
centers of many galaxies. These environments, which can span from $N=10^{5}$
(similar to low-mass GCs) to $N\gtrsim 10^{8}$ numbers of stars and binaries,
represent the highest density stellar regions known, with some NSCs (such as
the nuclei in M32 and the MW) exceeding $\sim 10^{7}M_{\odot}/\rm{pc}^{3}$ (on
scales of the central $\sim 0.1\rm{pc}$ and $\sim 0.01\rm{pc}$ respectively,
[985, 986]). Observations of nearby NSCs [987] showing a clear trend between
the mass of the host galaxy and the mass and density profile of their NSCs
with more massive galaxies hosting more massive NSCs with flatter profiles.
See [988] for a detailed review of our current observational understanding of
NSCs, their hosts, and their potential evolutionary paths.

Unlike GCs, which are relatively simple stellar environments (typically
spherical and forming in a single burst of star formation lasting $\lesssim
10\rm{Myr}$), NSC sadmit many complicated structures (e.g. disks,
triaxiality), formation histories (possibly forming from infalling GCs [e.g.,
989], in situ star formation [e.g., 990], or a combination of the two; see §7
of [988]), with the formation history and other properties being closely
related to the evolutionary history and morphology of their host galaxies.
Their internal dynamics is also complicated by the presence of SMBHs, which
cause a significant increase in stellar density and velocity dispersion within
the central regions of the NSC (via the formation of a “Bahcall-Wolf Cusp”
[991, 992]).

Given the extreme computational difficulties of modeling NSCs, most studies of
BBH formation in these environments have relied upon analytic models of
varying degrees of sophistication, with assumptions that change dramatically
depending on whether or not an SMBH is present. We now explore both of these
scenarios.

####  XIV.1 GN without SMBHs

While the actual occupation fraction of SMBHs inNSCsis highly uncertain, there
are GN, such as the nucleus of M33 [993], that are known to lack SMBHs. In
many ways,NSCswithout central-massive BHs can be modeled (to lowest order) as
ultra-massive GCs with more complicated star-formation histories. In these
systems, the predominant mechanism for forming and merging BHs is three-body
binary formation and dynamical hardening, and many of their observable
properties, such as the BH spin orientations, are similar to that of GCs (see
§XIII.1). But, because of their substantially higher central densities (and
correspondingly higher central escape velocities), they can both retain
lighter-mass BHs following their supernova [e.g. 994] and more easily retain
multiple generations of BH merger products, easily producing BHs in the pair-
instability mass gap or in the intermediate-mass regime [e.g., 974, 995, 994].

Because all but the least massive NSCs remain beyond the capabilities of
direct $N$-body integrators, most studies of BH mergers in nuclei have relied
on a combination of three-body scatterings [e.g., 996, 997, 998] and semi-
analytic techniques [e.g., 999, 995, 994, 1000, 1001, 1002]. Because of their
higher escape speeds, a greater fraction of NSC BH mergers occur within the
cluster (as compared to GCs), which in turn produces roughly four times as
many highly eccentric mergers in the LIGO band (with $\sim 20\%$ passing 10Hz
with $e>0.1$, [1002]).

####  XIV.2 GN with SMBHs

In NSCs with SMBHs, the central regions of the NSC become completely dominated
by the Keplerian potential of the central BH which, in combination with two-
body relaxation puhsing the system towards dynamical equilibrium, drives the
system towards a high density, high velocity dispersion configuration known as
a Bahcall-Wolf Cusp [991, 992]. As with GCs, BHs tend to mass segregate into
the centers of SMBH cusps, increasing their density towards the central
regions of the GN. However, the increased velocity dispersion in these central
regions also tends to suppress binary formation via three-body encounters,
where the rate of binary formation scales as $1/v^{9}$, [1003]. Instead, the
formation of CO mergers in SMBH cusps is largely powered by two-body GW
capture, similar to eccentric BBH production in GCs. This idea has been
invoked for decades [e.g., 1004, 1005, 1006] to explain formation of hard
binaries in dense stellar systems and the long-term evolution of GN.

Because the velocity distribution and mass segregation of BHs in cusps can be
well understood from collisional theory [e.g., 1007, 1008], the rates, mass,
and eccentricity distributions of BHs merging in NSC cusps can be predicted
analytically (for a given SMBH mass and stellar/BH population). The first
detailed treatment of this came from [1009], which showed that the production
of mergers is dominated by BHs closest to the SMBH. More recent work by [1010,
1011] has shown that this channel produces a strong correlation between the
mass of the merging BHs and their eccentricity (as more massive BHs segregate
deeper into the cusp where the Keplerian velocities are higher, requiring a
closer encounter pericenter for GW capture to be effective). While easily
analytically tractable, the overall merger rate of BBHs in NSC cusps is likely
lower than that induced by the Kozai-Lidov process about the SMBH in the
absence of gas.

In the presence of gas, however, the dynamics changes considerably. In
particular, in active galactic nuclei (AGN), the presence of a gas disk can
significantly alter the dynamics of the NSC. Once a disk forms in an NSC, BHs
find themselves either ground down into the disk from isotropically-
distributed orbits in the NSC [e.g. 1012, 1013, 1014], with the gas drag from
repeated plunges through the disk causes the inclination of stellar orbits to
align with the disk, or they form in situ in the fragmenting disk itself
[e.g., 1015]. Once embedded within the disk, the BHs begin to experience
torques due to both gas drag and gravitational torques from the disk itself,
causing the BHs to migrate throughout the disk (in a fashion very similar to
planetary migration within proto-planetary disks, [1016, 1017]).

While the rates of BH mergers in AGN disks are highly uncertain (spanning
several orders of magnitude [e.g., 1018]), the effective reduction of the
dynamics into a 2-dimensional disk alters several of the key features of
dynamical formation in BBH mergers. First, by focusing three-body encounters
into a co-planar environment, it has been shown that the number of highly-
eccentric mergers increases dramatically compared to 3-dimensional
environments, with $\gtrsim 30\%$ of mergers retaining significant
eccentricity into the LIGO band [1019, 1020], while interactions with the
gaseous disk itself may also produce eccentric mergers through evection
resonances [1021] (though we note that 3D hydrodynamical simulations of BBHs
in AGN disks have come to less definitive results [1022]). At the same time,
the proximity of these mergers to the central-massive BH means that any AGN-
produced BH mergers occur in environments with extremely large escape speeds,
potentially facilitating many generations of hierarchical BH mergers [1023,
1024, 1025]. While such an efficiency of hierarchical BH production may in
fact _overproduce_ massive BHs compared to the LIGO/Virgo population [e.g.,
1026], the finite lifetime of AGN disks places a natural upper-limit on the
production efficiency of such high-mass mergers.

Of course, the most tantalizing potential observable of the AGN channel is the
potential production of a bright optical transient associated with the merger
of a BH embedded in the disk [e.g., 1027, 1028]. A joint observation of both
an electro-magnetic (EM) transient and GWs from a BH merger would not only
definitively confirm the AGN channel as a potential source of BBH mergers, but
would allow for a detailed probe of the interior of AGN environments [1027],
and even independent standard-siren measurements of the Hubble constant [1029,
1030] (at redshifts significantly higher than those of binary neutron star
mergers). While there were initial claims that the high-mass BBH GW190521 may
have had an associated EM flair detected by the Zwicky Transient Factory
[1031], to date no EM counterpart has been confidently associated with a GW
event [1032, 1033], and current surveys suggest the contribution from the
brightest AGN may be limited [1034]. However, in spite of the significant
uncertainties and difficulties modeling these environments, the potential of
producing an EM counterpart makes the AGN channel one of the most compelling
dynamical environments and an area of significant active research.

##  Part V Black holes in binaries through observations: X-ray, detached,
lensing, LIGO
Tana D. Joseph and Sara Saracino

Please note: This is a rapidly evolving field, especially as regards
observations, so we apologize if some references are missing, especially very
recent ones. We have included all the studies that, to the best of our
knowledge, have provided the most significant impact to the field.

### XV Introduction

Previous Chapters have extensively discussed the importance of stellar-mass
black holes (BHs), their formation channels and their abundance in the
Universe. The main goal of this Chapter is instead to provide the reader with
clear tools to understand how to reveal these elusive objects and how their
properties can be constrained. We start this journey by dividing stellar-mass
BHs into three main categories: 1) isolated BHs, 2) BHs in binary systems with
luminous companions (stars) and 3) BHs in binaries with compact objects, such
as another BH or a neutron star (NS). Each of these categories must be treated
separately, as the techniques and observational instruments we use to detect
them are mostly different. For the sake of simplicity, in describing the
different methodologies we will not consider triples or multiple systems.

### XVI Isolated Black Holes

Being the product of the evolution of very massive stars (M $>$
20$M_{\odot}$), BHs are expected to constitute 0.1% of all objects populating
the Universe [1035, 1036, 1037, 1038] and once produced, we are unaware of any
mechanism that can destroy them. They can have very different properties
depending on how they form and what kind of interactions they experience
during their lives. A fraction of these objects are isolated BHs, meaning that
they are not part of a binary system, hence they cannot be detected thanks to
interactions with their companions or any motions around them (see the
following Sections for a detailed description of BHs in binaries). There are
three main reasons why a large fraction of isolated BHs exists: 1) Not all
massive stars are born in binaries, a fraction of them are born and evolve as
single stars [96]. 2) If two stars are in a tight binary orbit, they can
experience a common envelope phase that can bring the two stars to merge and
form a massive star which would then explode as supernovae, leaving behind an
isolated BH. 3) In a wide binary composed of two stars, one of which is quite
massive, it could happen that the BH produced after the supernovae explosion
receive enough “natal kick” to become unbound [545].

If a BH lives isolated in the Universe, it has very little chance of
interacting with any other object so that the effect of this interaction would
allow its detection. In addition, the accretion rate of material from the
surrounding is expected to be so low that it does not emit any detectable
high-frequency radiation (for example X-rays). Thus, discovering an isolated
BH is very challenging and there is only one specific phenomenon that appears
to be promising in this regard: the gravitational microlensing.

A microlensing event occurs when either a star or a compact object (called
lens hereafter) passes almost exactly in front of a background star (called
source hereafter). As predicted by general relativity [1039], the lens
magnifies the image of the source, producing an apparent amplification of its
brightness. Moreover, it slightly shifts the apparent position of the source
[1040, 1041, 1042]. Figure 16 visually illustrates the concept of
gravitational microlensing, when the lens is a BH. When the BH passes nearly
in front of a background star, it bends light from the star towards Earth,
which temporarily amplifies the star’s observed brightness. BHs are able to
warp space-time so much that it noticeably alters the source’s apparent
location in the sky. The two images caused by lensing are very close to each
other to be spatially resolved, but the change in brightness of the two images
produces a shift in the position of the centroid of the source that can be
detected via high-precision astrometry. By exploiting photometric microlensing
to study how the luminosity of the source varies over time and astrometric
microlensing to measure the amount of relativistic deflection of the apparent
position of the source as the lens passes in front of it, the mass of the lens
can be measured directly and its real nature inferred. Having both
information, in fact, is very important as it allows us to break the
degeneracy that exists between the mass of the lens and the relative source-
lens proper motion. Unfortunately this is not the case for most detectable
microlensing events, for which we only have the photometric information.

The mathematical formalism that physicists use to describe a microlensing
event is rather complicated but a brief description of the main parameters
involved can help better understand the phenomenon and their intrinsic
limitations. The perfect alignment of the source, the lens and the observer,
which tends to occur very rarely, produces a special type of microlensing
event, where the reflected image is not observed as a single point, but as a
ring of light around the lens. This ring is called the Einstein ring with an
angular radius $\theta_{E}$ known as the angular Einstein radius. The latter
is an important quantity used to describe the characteristic scale of a
gravitational microlensing event, regardless of the exact observer/lens/source
configuration, and it is defined by the equation:

$\theta_{E}=\sqrt[\ ]{\frac{4GM_{L}}{c^{2}}\frac{\pi_{LS}}{1au}}$ (XVI.1)

where $M_{L}$ is the mass of the lens and $\pi_{LS}$ is the relative lens-
source parallax. The latter can also be written as:

$\pi_{LS}=\pi_{L}-\pi_{S}=1au\left(\frac{1}{D_{L}}-\frac{1}{D_{S}}\right)$
(XVI.2)

with $D_{L}$ and $D_{S}$ as distances between observer and lens and between