# Hypercritical accretion during common envelopes in triples leading to binary
black holes in the pair-instability-supernova mass gap

Enrique Moreno Méndez1 , Fabio De Colle2 , Diego López-Cámara3 Alejandro
Vigna-Gómez4 .
1Facultad de Ciencias, Universidad Nacional Autónoma de México, A. P. 70-543
04510, CDMX, México
2Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.
P. 70-543 04510 D. F. Mexico
3Cátedras CONACyT – Universidad Nacional Autónoma de México, Instituto de
Astronomía, AP 70-264, CDMX 04510, México
4Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17,
DK-2100 Copenhagen, Denmark E-mail<EMAIL_ADDRESS>0000-0002-5411-9352 0000-0002-3137-4633 0000-0001-9512-4177
0000-0003-1817-3586

(July 2022)

###### Abstract

Hydrodynamic studies of stellar-mass compact objects (COs) in a common
envelope (CE) have shown that the accretion rate onto the CO is a few orders
of magnitude below the Bondi-Hoyle-Lyttleton (BHL) estimate. This is several
orders of magnitude above the Eddington limit and above the limit for
neutrino-cooled accretion (i.e., hypercritical accretion, or HCA). Considering
that a binary system inside the CE of a third star accretes material at nearly
the same rate as a single object of the same total mass, we propose stellar-
evolution channels which form binary black hole (BBH) systems with its
component masses within the pair-instability supernova (PISN) mass gap. Our
model is based on HCA onto the BBH system engulfed into the CE of a massive
tertiary star. Furthermore, we propose a mass transfer mode which allows to
store mass lost by the binary onto a third star. Through the use of population
synthesis simulations for the evolution of BBHs and standard binary-evolution
principles for the interaction with a tertiary star, we are able to produce
BBHs masses consistent with those estimated for GW190521. We also discuss the
massive binary system Mk34 as a possible progenitor of BBHs in the PISN gap,
as well as the spin distribution of the observed mergers in the gravitational-
wave catalog.

###### keywords:

Accretion, accretion discs – stars: binaries – stars:black holes – stars:
massive – black hole mergers

##  1 Introduction

Stellar evolution predicts that low-metallicity stars with initial masses
$M_{\rm ZAMS}\lesssim 133\;\mbox{M${}_{\odot}$}$ produce black holes (BHs)
with $M_{\rm BH}\lesssim 50\;\mbox{M${}_{\odot}$}$ (Fryer, Woosley, & Heger,
2001; Heger & Woosley, 2002; Abbott et al., 2020b). If a star has $M_{\rm
ZAMS}\gtrsim 140\;\mbox{M${}_{\odot}$}$, it may experience a pair-instability
supernova (PISN, Barkat, Rakavy, & Sack, 1967; Heger & Woosley, 2002; Heger et
al., 2003). PISN occur when a massive oxygen core is sufficiently hot and
dense to efficiently create electron-positron pairs. These pairs convert
internal gas energy into rest mass, abruptly decreasing the radiation pressure
and prompting a rapid contraction of the core. This contraction ignites
oxygen, leading to a runaway thermonuclear explosion that leaves no remnant
(Heger & Woosley, 2002). The PISN results in a predicted mass gap for non-
interacting massive stars with masses between
$50\;\mbox{M${}_{\odot}$}\lesssim M_{\rm gap}\lesssim
130\;\mbox{M${}_{\odot}$}$ (see Stevenson et al., 2019, and references
therein).

The binary black hole (BBH) system progenitor of the GW source GW190521
consisted of two black holes (BHs) with masses $M_{\rm
BH,1}=85^{+21}_{-14}\mbox{M${}_{\odot}$}$ and $M_{\rm
BH,2}=66^{+17}_{-18}\mbox{M${}_{\odot}$}$ (Abbott et al., 2020a) which fall in
the theoretically predicted PISN mass gap. The merger of the BBH radiated
about $7.6^{+2.2}_{-1.9}\;\mbox{M${}_{\odot}$}$ in GWs, and resulted in a
single 142${}^{+28}_{-16}\mbox{M${}_{\odot}$}$ BH (Abbott et al., 2020a). The
inferred Kerr spin parameters of the merging BHs was $0.1\lesssim
a_{\star}\lesssim 0.9$, where $a_{\star}\equiv Jc^{2}/GM_{\rm BH}^{2}$, being
$J$ the angular momentum of the BH, and the spin parameter of the remnant BH
was $a_{\star}\simeq 0.72^{+0.09}_{-0.12}$.

Several mechanisms have been proposed to produce BBH in the PISN gap. These
include: a) multiple generations BBH mergers either in clusters (e.g.,
Fragione & Silk, 2020; Liu & Lai, 2020; Stone & Leigh, 2019), in AGN disks
(McKernan et al., 2012; Bartos et al., 2017), or in the field (Vigna-Gómez et
al., 2021); b) accretion in proto-clusters (Roupas & Kazanas, 2019a, b),
isolated binaries (van Son et al., 2020), halos (Safarzadeh & Haiman, 2020),
or onto primordial BHs (De Luca et al., 2020); c) creation of massive BBHs
that form from isolated binaries (Spera et al., 2019), the merger of two stars
below the PISN forming a star with a small core and an oversized envelope
(Renzo et al., 2020), massive BHs from population III stars (Kinugawa,
Nakamura, & Nakano, 2020; Vink et al., 2021); d) mechanisms beyond the
standard models of stellar evolution (Sakstein et al., 2020; Straight,
Sakstein, & Baxter, 2020; Ziegler & Freese, 2021).

The common envelope (CE) phase is a short lived stage during the evolution of
a binary system in which the envelope of one of the stellar components engulfs
the other component (Paczynski, 1976; Iben & Livio, 1993). Due to mass
transfer and orbital decay the orbital separation decreases up to a point in
which highly energetic and transitory phenomena may be produced, for example:
type Ia supernovae (Chevalier, 2012; Postnov & Yungelson, 2014), short GRBs
(Vigna-Gómez et al., 2020), long GRBs (Brown et al., 2007; Moreno Méndez,
2011), and GWs (Abbott et al., 2016). During the CE phase, which may occur in
several occasions during the evolution of the binary system, compact objects
may be the engulfed component (Ivanova et al., 2013) and the accretion onto
the secondary component may play an important role (Chamandy et al., 2018).

Most studies (e.g., those in the third paragraph) assume that accretion is
limited by the Eddington luminosity. However, accretion onto a single star or
a binary system may be super-Eddington or hypercritical111In super-Eddington
accretion ($\dot{M}_{\rm Edd}<\dot{M}\lesssim 3000\;\dot{M}_{\rm Edd}$) the
energy excess is lost through photons, while in hypercritical accretion (HCA)
the thermal energy is mainly lost through neutrinos, given that for
$\dot{M}\gtrsim 3000\dot{M}_{\rm Edd}$ and the accreting material has large
density and temperature (Chevalier, 1989; Brown & Weingartner, 1994). (e.g.,
Pacucci et al., 2015; Sakurai et al., 2016; Sugimura et al., 2017; Li & Cao,
2019). This is especially true for a compact object (CO) accretor. Radiation-
hydrodynamics simulations have shown that steady accretion happens once the
accretion rate is $\dot{M}\gtrsim 3000\;\dot{M}_{\rm Edd}$, with $\dot{M}_{\rm
Edd}$ being the Eddington-limited mass-accretion rate (Inayoshi et al., 2016).
In addition, HCA can also produce the masses and spins observed in some BH
binaries in high-mass, X-ray binary systems (Moreno Méndez et al., 2011;
Moreno Méndez, 2014).

In this paper, we propose that HCA can play an important role in the formation
of massive stellar BHs. We show that a BBH system with HCA during a CE phase
may produce BHs with masses within the PISN gap. We discuss two possible
scenarios: a) the evolution of a hierarchical triple system, or b) the outcome
of a BBH capturing a massive star. We propose a mechanism which allows the
system to store material, initially part of the more massive binary stars, in
the tertiary star. Both scenarios produce BBH with component masses within the
PISN mass gap through HCA during the CE phase of the massive star.

The paper is organized as follows. In Section 2 we discuss the conditions for
HCA onto a BBH during the CE phase. In Section 3 we discuss mass-transfer and
storage in triple systems. In Section 4 we describe possible evolutionary
paths of triple-systems which produce hyper accreting BHs within the CE of a
red giant or red super giant. In Section 5 and 6 we discuss our methods and
results and drive our conclusions.

##  2 Hypercritical accretion onto a BBH system during the CE phase

In this section we describe how during HCA onto a BBH system which is within
the CE of a massive star a non-negligible fraction of the massive star may be
accreted by the BBH. We also discuss how accretion and the ejection process
are the key competing mechanisms to determine the amount of hydrogen-envelope
accreted onto the inner BBH.

###  2.1 Accretion onto a BBH

Previous studies have shown that accretion onto a star or compact object
orbiting inside the CE of a red giant (RG) or red super-giant (RSG) occurs at
a fraction ($\epsilon\approx 0.01-0.1$) of the Bondi-Hoyle-Littleton rate
(see, e.g., Ricker & Taam, 2012; MacLeod & Ramirez-Ruiz, 2015; Moreno Méndez
et al., 2017; Chamandy et al., 2018; López-Cámara et al., 2019, 2020). The
accretion rate onto the companion ($\dot{M}_{\rm a}$) is:

$\displaystyle\dot{M}_{\rm a}$ $\displaystyle\equiv$
$\displaystyle\epsilon\dot{M}_{\rm BHL}=\epsilon\frac{4\pi\rho
G^{2}M^{2}}{(v^{2}+c_{s}^{2})^{3/2}}$ (1) $\displaystyle\simeq$ $\displaystyle
2.8\times 10^{-5}\ \epsilon_{0.1}\ \rho_{-6}\ M_{50}^{2}\ v_{7}^{-3}\
\mbox{M${}_{\odot}$}\;{\rm s}^{-1}$

where $G$ is the Gravitational constant, $\rho_{-6}$ is the density of the
environment in units of $10^{-6}$g cm-3, $M_{50}$ the mass of the accretor in
units of 50 M⊙, $v_{7}$ the velocity of the accretor in units of $10^{7}$cm
s-1, and $\epsilon_{0.1}$ the accretion efficiency normalized to
$\epsilon=0.1$ (which depends on the structure of the envelope, in particular
on the presence of density and velocity gradients). Since the sound speed
within the envelope of a RG or RSG is $c_{s}\sim 10\;(T/10^{4}{\rm K})^{1/2}$
km s-1, and the velocity of the accreting material is of order of the
keplerian velocity ($v\sim v_{k}\gtrsim 100$ km s-1), we have
$(v^{2}+c_{s}^{2})^{3/2}\sim v^{3}$.

If the secondary is engulfed by the CE in a sudden plunge, the orbital period
of the accretor as it enters the envelope of the CE may be completely
asynchronised with respect to the rotation period of the donor. This scenario
leads, at first order approximation, to BHL accretion. The case of a gradual
plunge occurs when the donor and the accretor are fully synchronized, and
thus, the relative velocity is zero. On the other hand, if the secondary is
engulfed gradually by the CE, then approximately, Bondi accretion takes place
(case in which $v\lesssim c_{s}$ in equation 1). Density and velocity
gradients, magnetic fields, convection flows, Coriolis, centrifugal and Euler
forces, and the presence of a preexisting disk, to name some, modify the
simple picture of BHL or Bondi accretion, and thus it is uncertain, in any
case, whether a disk may or may not form around the accretor.

Meanwhile, accretion onto a BBH system depends on the orbital separation $a$
between the BHs and on the accretion radius ($r_{a}=2GM/v^{2}\sim 3\times
10^{14}~{}M_{100}~{}v_{7}^{-2}~{}{\rm cm}$; see, e.g., Edgar 2004). If the BBH
forms part of a triple system, a fraction of the envelope of the tertiary can
be accreted during a CE phase by the CE-engulfed BBH (see Section 2.2). Also,
in the case of a triple system, it must be considered that the Keplerian
velocity of the binary will never be zero, regardless of whether
synchronization is achieved or not with the envelope of the tertiary.

In order to model the effect of accretion of a binary system inside a CE of a
third star, we follow Comerford et al. (2019); Comerford & Izzard (2020).
These studies present numerical simulations of accretion onto binaries for a
wide range of orbital periods and inclinations. They find that for small
orbital separations ($a\ll r_{a}$) the accretion takes place onto a point mass
(located at the center of mass of the BBH) with a mass given by the sum of the
two individual masses ($M=M_{\rm BH_{1}}+M_{\rm BH_{2}}$). When $a\gg r_{a}$,
each BH accretes as a single object. The intermediate case ($a\sim r_{a}$) is
complex and remains to be fully studied. In the following we will focus on the
case in which $a\gg r_{a}$.

From equation 1 it is clear that the mass of the black holes ($M_{\rm BH,i}$,
with $i=1,2$) changes as $\dot{M}_{\rm BH,i}=kM^{2}_{\rm BH,i}$ (where
$k=4\pi\epsilon\rho G^{2}/v^{3}$), which by direct integration gives:

$M_{\rm BH,i}=\frac{M_{{\rm BH,{i_{0}}}}}{1-kM_{{\rm BH,{i_{0}}}}t}\;,$ (2)

where $M_{{\rm BH,{i_{0}}}}$ is the initial mass of the black hole $i$. The
black hole $i$ achieves a mass $M_{\rm BH,i}$ in a time:

$t_{\rm f,i}=\left(1-\frac{M_{{\rm BH,{i_{0}}}}}{M_{\rm
BH,i}}\right)\frac{1}{kM_{{\rm BH,{i_{0}}}}}\;.$ (3)

Since $t_{\rm f,1}=t_{\rm f,2}$, then, the following relationship between the
ratio of the initial and the final masses of the black holes ($q_{0}=M_{{\rm
BH,{2_{0}}}}/M_{{\rm BH,{1_{0}}}}$ and $q=M_{\rm BH,2}/M_{\rm BH,1}$
respectively) is obtained:

$q_{0}=\frac{qM_{\rm BH,1}/M_{{\rm BH,{1_{0}}}}}{1+q(M_{\rm BH,1}/M_{{\rm
BH,{1_{0}}}}-1)}\;.$ (4)

Using equations 2, 3, and 4 the initial masses of the BHs and the accretion
time $t_{f}$ (for the BHs to reach their final masses), are obtained as a
function of the final masses.

In Figure 1 we plot the evolution of the BBH during HCA when inside the CE of
a third star. Specifically, we plot the range of possible initial masses of
the two merging BHs (with $M_{\rm BH,1}>M_{\rm BH,2}$), the amount of mass
accreted by the binary system, and the time required for the BHs to reach the
masses of GW190521 (85M⊙ and 66M⊙). For the latter, we assume that the
accretion rate onto the BBH during the CE phase is $10\%$ of the BHL accretion
rate. The accreted mass will be a substantial fraction of the envelope of the
tertiary star in which the BBH is immerse. If the initial mass of both BHs is
large, the accretion time required to reach masses similar to those of
GW190521 is considerably smaller (a few days to over a month) and the required
mass transferred and accreted from the donor star drops to a few tens of solar
masses. For example, if the initial mass of the secondary BH is $M_{\rm
BH,2}\approx 40\;\mbox{M${}_{\odot}$}$, then the mass of the primary BH must
be $M_{\rm BH,1}\approx 45\;\mbox{M${}_{\odot}$}$ (from the solid black line)
and the accretion time for each of the BHs to reach $65\;\mbox{M${}_{\odot}$}$
and $85\;\mbox{M${}_{\odot}$}$ (respectively, and assuming 10% of the BHL
accretion rate) is $\simeq 10$ d (red solid line). The total mass transferred
and accreted from the envelope of the tertiary star needs to be $\Delta M_{\rm
CE}$ $\gtrsim 65\;\mbox{M${}_{\odot}$}$ (dashed black line). Assuming a
maximum available envelope mass of 100 M⊙ then $\gtrsim 65\%$ of the envelope
must be accreted (for further details see Section 4).

Figure 1: The solid black line indicates the initial mass of the primary BH
($M_{\rm BH,1}$) as a function of the initial mass of the secondary BH
($M_{\rm BH,2}$) in the binary system. The dashed black line represents the
mass accreted by the BBH system. The red solid line and the right $y$-axis
show the accretion time needed for the primary and secondary BHs to end up
with 85 M⊙ and 65 M⊙ respectively during the CE phase (assuming an accretion
rate given by 10% of the BHL rate). The gray shaded area is excluded as the
initial mass of the primary BH is larger than $\gtrsim$ 55 M⊙. Larger BH
masses are excluded by single stellar evolution models (see Section 1).

###  2.2 Limits on the accretion onto the BBH

The accretion onto a BBH system may be limited by several effects: a) the
radiation pressure of the accreting material, b) the orbital timescale, and c)
the amount of material ejected from the CE. In this section, we discuss these
effects and argue that they do not represent a strong limiting factor to the
accretion onto the BBH.

Firstly, we discuss the feedback due to radiation pressure and neutrino
effects. The accretion rate given by equation (1) is $\sim$10 orders of
magnitude above the Eddington mass accretion rate given by:

$\dot{M}_{\rm Edd}=\frac{L_{\rm Edd}}{c^{2}}\simeq 3.5\times 10^{-15}M_{50}\
\mbox{M${}_{\odot}$}\;{\rm s}^{-1}\;,$ (5)

where $L_{\rm Edd}$ is the Eddington luminosity and $c$ is the speed of light.
Nevertheless, the Eddington limit assumes spherical symmetry, i.e., Bondi
accretion. BHL accretion breaks spherical symmetry, leading to radiation
ejected through polar jets and material accreted via the equatorial plane.
Additionally, velocity and density gradients, as well as Coriolis and
centrifugal forces, may further break the cylindrical symmetry of the BHL
accretion (Moreno Méndez et al., 2017; López-Cámara et al., 2019, 2020).

Thus, the BHL accretion rate onto a BH which is within the CE of a massive RG
or RSG is HCA. In this case, as the accretion rate is $\dot{M}_{\rm a}\gtrsim
3\times 10^{3}\;\dot{M}_{\rm Edd}$, neutrinos are produced (Houck & Chevalier,
1991; Brown & Weingartner, 1994). Since the density and temperature of the
material in the accretion flows around the BH are well below those in core-
collapse supernovae, close to where the Eddington limit for neutrinos lies,
these neutrinos will not interact with the accretion flow. Therefore, most of
the released binding energy will be radiated away by neutrinos which leave the
CE without exerting pressure on the accreting material and allowing for
extremely large accretion rates. Hence, the radiation pressure and neutrino
emissivity will not have effect on our results.

Second, we consider the limit imposed on accretion due to the orbital
timescale. With $\dot{M}_{\rm a}\gtrsim 10^{9}\dot{M}_{\rm Edd}$ (see
equations 1 and 5), and most of the binding energy being radiated away by
neutrinos, the BBH could increase its mass from $M_{\rm BBH,i}\simeq
55\;\mbox{M${}_{\odot}$}+45\;\mbox{M${}_{\odot}$}=100\;\mbox{M${}_{\odot}$}$
to $M_{\rm BBH,f}\simeq
85\;\mbox{M${}_{\odot}$}+65\;\mbox{M${}_{\odot}$}=150\;\mbox{M${}_{\odot}$}$
within a few weeks of CE (see Figure 1). However, if the initial orbital
separation in the triple system is of order $a_{\rm tri,i}\gtrsim 10^{14}$ cm
(considering a $M_{\star}=150\mbox{M${}_{\odot}$}$ donor star as well as the
BBH system), the initial orbital period will be:

$T=\frac{2\pi R}{v_{k}}=2\pi\sqrt{\frac{R^{3}}{GM}}\sim 100~{}a_{\rm
tri,14}^{3/2}~{}M_{100}^{-1/2}~{}{\rm d}\;.$ (6)

As the BBH can not accrete an amount of mass larger than the stellar material
available during their orbit around the companion star, the timescale for the
accretion onto the BBH system will be the largest between $T$ and $\tau_{\rm
BHL}\sim 30\;(\epsilon/0.1)^{-1}$ d. Since the BHL accretion timescale
($\tau_{\rm{BHL}}$) is smaller than the orbital timescale, accretion is
limited by $T$. As the orbital separation decreases, so does the orbital
period, and it continues to decrease until $T\sim\tau_{\rm{BHL}}$.

Last, the ejection of stellar material from the CE could limit accretion onto
the BBH (as the material ejected from the CE and that lost by the triple
system could unbinding the envelope). The BBH mass222Here, by accretion on the
BBH we refer to the accretion onto the system which could include an accretion
disk around the BBH. The actual accretion onto the BBH will be then delayed by
the viscous time. increases as $\dot{M}_{a}=\epsilon\dot{M}_{\rm BHL}$
(equation 1), while the envelope mass drops by the same amount, i.e.
$\dot{M}_{\rm env}=-\dot{M}_{a}$. We limit the amount of accreted mass as the
minimum between the BHL accreted one and the mass available at radius larger
than the orbital separation ($M_{\star}(r>a)$), that is:

$\displaystyle M_{\rm BBH}(t)$ $\displaystyle=$ $\displaystyle M_{\rm
BBH,0}+\min\left(M_{*}(r>a),\int_{0}^{t}\dot{M}_{a}(t^{\prime})dt^{\prime}\right)\;,$
(7) $\displaystyle M_{\rm env}(t)$ $\displaystyle=$ $\displaystyle M_{\rm
env,0}-(M_{\rm BBH}(t)-M_{\rm BBH,0})\>,$ (8)

where $M_{\rm BBH,0}$, $M_{\rm env,0}$ are the initial BBH and envelope
masses.

The inspiral of the BBH will be driven by the the gravitational drag due to
the envelope material accreting onto the BBH (Livio & Soker, 1988), which
produces a change in the BBH orbital energy (around the massive star) as:

$\frac{dE_{\rm orb}}{dt}=-\dot{M}_{a}v_{k}^{2}\;,$ (9)

where $\dot{M}_{a}$ is the mass accretion rate (given by equation 1), $v_{k}$
is the Keplerian velocity ($v_{k}=\sqrt{GM_{\star}(r\leq a)/a}$, with
$M_{\star}(r\leq a)$ the enclosed mass within the orbital separation), and the
orbital energy is $E_{\rm orb}=-0.5\;GM_{\rm BBH}M_{\star}(r\leq a)/a$.
Assuming that a fraction $\alpha$ of the orbital energy is deposited (as
thermal energy) into the envelope, thus, the energy deposited ($E_{d}$) is
given by

$\frac{dE_{d}}{dt}=\alpha\dot{M}_{a}v_{k}^{2}\;,$ (10)

Taking the $\alpha\lambda$ energy formalism of van den Heuvel (1976) in which
the envelope will be ejected once the energy deposited is of order of the
binding energy (e.g., Webbink, 1984; Olejak, Belczynski, & Ivanova, 2021), and
using equations 9, 10 (as well as the binding energy of the envelope,
$E_{b}$), then the condition $E_{b}\sim E_{d}$ reads

$\frac{GM(a)M(r>a_{f})}{\alpha\lambda
a}\sim\int_{0}^{t}\dot{M}_{a}v_{k}^{2}dt\;,$ (11)

where $\lambda$ is a numerical factor which depends on the density structure
of the star (de Kool, 1990).

By numerically integrating this set of equations (7-11) (and considering a
stellar core with a mass $35\ \mbox{M${}_{\odot}$}$, an envelope density
profile, between $2\times 10^{11}$ cm and $2\times 10^{13}$ cm, $\rho\propto
r^{-n}$ (with $n=2.5-2.8$) and a total envelope mass of 65 M⊙, the BBH
inspiral time is $\sim 100-170$ days, and most the material can be accreted by
the BBH (since $E_{b}\gtrsim 10E_{d}$ at all times) for $\alpha\leq 1$. Thus,
the loss of material during the accretion phase is minor (at least, in the
framework of this simple analytical method), and the system is stable.

###  2.3 Jets as an extra energy injection source.

If an accretion disk does not form around the accreting component, HCA may
only proceed through neutrino-radiation cooling. (i.e., the accretion rate
must be at least three orders of magnitude above the Eddington limit; a
condition which we have shown to be vastly fulfilled by our models; Chevalier,
1989; Brown & Weingartner, 1994). In this case, there is no extra energy
injection into the CE other than the orbital energy, parametrised by the
$\alpha\lambda$ formalism. On the other hand, when an accretion disk forms
around an accreting BH, a collimated and relativistic jet may be launched
(Armitage & Livio, 2000; Soker, 2004). In this case, as shown by Moreno Méndez
et al. (2017), the energy deposition into the CE will dominate over the
$\alpha\lambda$ and other energy sources (e.g., recombination energy,
enthalpy, tidal heating, magnetic fields, see Ivanova et al., 2013, for
further details). Switching on this mechanism can bring the system into the
so-called grazing envelope phase (Sabach & Soker, 2015; Soker, 2015; Shiber,
Kashi, & Soker, 2017; Shiber & Soker, 2018) and may even terminate the CE
phase in extreme cases by removing the envelope (Soker, 2017). Nonetheless, as
the energy deposited by the jet will drop the accretion rate, the jet power
will also drop, then alternating periods of accretion with period of
quiescence. The “self-regulation” of the jet guarantees that a significant
percentage of the CE available material will be accreted (although it does not
guarantee that the orbital separation will decrease). Given the complexity of
the problem, future numerical works are needed to understand the full impact
of jets or outflows in the CE evolution (hereafter, CEE).

##  3 Mass-transfer and mass-storage in triple systems

One of the main processes determining the evolution of a multiple-stellar
system is mass transfer between the components. In this Section, we describe
our assumptions and the uncertainties on mass transfer mechanism, particularly
in the context of a CEE from a tertiary donor with an inner BBH accretor. We
provide a qualitative description of each mechanism and we estimate the
physical quantities (in particular, orbital separations and periods)
characterizing the system.

In the Roche-lobe overflow (RLOF) mechanism the mass is transferred to the
companion once the radius of the envelope of the donor exceeds the volume-
averaged Roche-lobe (RL) radius $r_{\rm RL}$ (Eggleton, 1983):

$\frac{r_{\rm RL}}{a}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}\;,$ (12)

where $q$ is the mass ratio of the binary components
($q=M_{\rm{d}}/M_{\rm{a}}$, with $M_{\rm{d}}$ the donor and $M_{\rm{a}}$ the
accretor).

During RLOF two important processes may allow a binary system to transfer mass
(Shore et al., 1994), these are: a) when the donor star is more massive than
the accretor the orbital separation decreases (as well as the RL of the
donor); b) the reaction of the donor star to the loss of its outer layers. In
the latter if the envelope is radiative it will contract as a response to mass
loss, while if it is convective it will expand (e.g., Soberman et al., 1997).
The envelopes of RG stars is usually radiative, however, RSG stars have
convective envelopes (Klencki et al., 2020). If the star expands, it is likely
that the star will continue to fill its RL (or that it will fill it up again,
if the orbital separation increased); this process occurs in a Kelvin-
Helmholtz (or thermal) timescale (Shore et al., 1994). In order to maintain a
stable RLOF phase, at least one of these runaway processes must be at play
(Soberman et al., 1997).

Another important mass-transfer mechanism in the evolution of multiple stellar
systems is the wind RLOF (wRLOF), in which the wind escaping the donor star is
swept by the accreting object. For close binary systems, Mohamed &
Podsiadlowski (2007) found that mass transfer through wRLOF can be highly
efficient (with up to $\sim 70\%$ of the wind being captured and accreted by
the companion), when the orbital velocity of the binary is comparable to the
wind velocity. A compact orbit is a necessary condition for a highly efficient
wRLOF mass-transfer, because it allows for a large orbital velocity as well as
a stellar wind which has not been fully accelerated. Stellar winds,
particularly in massive stars, require $\sim 10-15\ \mbox{R${}_{\odot}$}$ to
fully accelerate, hence the winds in this scenario have not reached escape
velocity and can be substantially funneled through the first Lagrange point
(L1) and focused towards the accretor (e.g., Vink et al., 2011). This
mechanism is an efficient mass transfer mechanism in the evolutionary stages
of the massive stellar systems (Vink, 2018).

An additional mass-transfer mechanism may occur in triple systems,
specifically, when the inner binary system is in a CE phase and the tertiary
is far orbiting around the binary system. In this stage, a substantial amount
of material from the outer layers of a CE may be inflated without being fully
unbound from the binary system; in fact, this material may even form a
circumbinary disk (Kashi & Soker, 2011). This material may slowly move to a
large orbit around the binary (and may even fall back towards the binary if
its velocity is below the escape velocity). So, if a tertiary star orbits the
binary it may be able to capture a large fraction of this material, see
Section 4.2 and Section 4.3 for further details. This mass-transfer mechanism
can be important in triple systems and we term it common-envelope Roche-lobe
overflow (ceRLOF).

In principle, ceRLOF allows for an effective mechanism to transfer mass in a
conservative way (hereafter, we refer to conservative mass-transfer as a
process in which the mass lost from the binary system is accreted by the
tertiary star) and store a substantial fraction of the envelopes of both stars
in the inner binary in the tertiary star. During the evolution of our triple-
stellar systems, mass is transferred back and forth between the stellar
components at different stages, particularly during CE stages from the inner
binary (ceRLOF). The captured mass will later be returned to the inner BBH
after the tertiary, and last-to-evolve star, becomes a giant which fills its
RL and engulfs the BBH in a third CE stage. In this scenario, a large-mass BBH
can be formed successfully (see Section 2) without requiring extremely massive
stars ($M_{\star}>150\mbox{M${}_{\odot}$}$) or systems with more than three
stars333We assume that the two SNe forming heavy BHs lose only a small
fraction of the mass of binary (for further details see Fryer et al., 2012)..

##  4 CE-BBH scenarios

In this section, we discuss two possible scenarios which may produce a BBH
system with masses close to those in the PISN mass gap which have been
inferred for GW190521 by the Ligo-Virgo-Kagra collaboration. First, we
describe the evolution of a binary system, composed by two massive stars,
which produces a BBH (Section 4.1). Then, we consider the case of ceRLOF from
a tertiary star in the case when the tertiary is pre-existent (Section 4.2),
and the case in which a tertiary star has been captured by the BBH system
(Section 4.3).

###  4.1 Evolution of the inner massive binary in a hierarchical triple
stellar system

In order to look for a binary a binary system with component masses close to
the lower edge of the pair-instability black-hole mass gap ($M_{\rm BH,1}\sim
M_{\rm BH,2}\sim 45\mbox{M${}_{\odot}$}$) we use the population synthesis code
COMPAS (see Riley et al., 2022, and references therein).We adopt the default
choices of COMPAS parameters for binary evolution which follows the classic CE
formation scenario of BBH mergers (e.g., Belczynski et al., 2016; Stevenson et
al., 2017; Vigna-Gómez et al., 2018). The evolution of such binary is shown in
Figure 2 and discussed next (see also Table 1):

Figure 2: Evolutionary pathway for a binary system with a primary star with $M_{1}=119.2\ \mbox{M${}_{\odot}$}$, secondary with $M_{2}=79.7\ \mbox{M${}_{\odot}$}$ (both with $Z=0.0001$), and an orbital separation of $a=464.6\ \mbox{R${}_{\odot}$}$. The upper panel shows the evolution of the primary, secondary and the total masses. The bottom panel shows the evolution of the radii of the two stars ($R_{1}$ and $R_{2}$), the correspondent Roche lobe radii of each of the stars ($R_{\rm RL,1}$ and $R_{\rm RL,2}$), and the orbital separation of the binary (semi-major axis $a$ and periastron $a_{\rm p}$). Stage | Event | Time | M1 | M2 | M3 | $a_{\rm{in}}$ | $a_{\rm{out,crit}}$ | $a_{\rm{out}}$ | $R_{\rm{RL,12}}$ | $R_{\rm{RL,3}}$ | $\Delta M$
---|---|---|---|---|---|---|---|---|---|---|---
|  | [Myr] | [M⊙] | [M⊙] | [M⊙] | [R⊙] | [R⊙] | [R⊙] | [R⊙] | [R⊙] | [M⊙]
i | ZAMS | 0.0 | 119.2 | 79.7 | 35 | 464.6 | $>3a_{\rm{in}}$ | 1817 | 966 | 441 | 0.0
ii | RLOF | 3.4 | 51.9 | 112.4 | 35 | 737.1 | $>3a_{\rm{in}}$ | 2216 | 1142 | 568 | 33.4
iii | SN | 3.6 | 45.8 | 112.4 | 35 | 766.5 | $3a_{\rm{in}}$ | 2300 | 1178 | 595 | 0.0
iv | CEE | 4.6 | 45.8 | 44.1 | 103 | 6.4 | $>3a_{\rm{in}}$ | 818 | 300 | 320 | 68.0
v | SN | 4.9 | 45.8 | 39.6 | 103 | 6.7 | - | 836 | 303 | 330 | 0.0
vi | RLOF/CEE | 5.0 | 86.0 | 66.0 | 37 | $<6.7$ | - | 2074 | 1045 | 551 | (40 + 26)*
Table 1: Principal stages (and main values) of the evolution of the binary
shown in Figure 2. *Note: $40~{}\mbox{M${}_{\odot}$}$ are accreted by the most
massive BH ($M_{1}$) and $26~{}\mbox{M${}_{\odot}$}$ by the least massive BH
($M_{2}$).

  1. [i]

  2. 1.

The binary system begins at the zero-age main sequence (ZAMS) with a primary
of 119.2 M⊙ and a companion of 79.7 M⊙ (both with a metallicity of $Z=0.0001$)
in a circular orbit with a semi-major axis of 464.6 R⊙.

  3. 2.

At $\approx$ 3.4 Myr, the post-main-sequence primary fills its RL and begins a
semi-conservative episode of mass transfer with the main-sequence companion.
In this first episode, the primary donates $\approx$ 67.1 M⊙, of which,
$\approx$ 33.0 M⊙ are accreted by the main-sequence companion, and the
remaining $\approx$ 34.1 M⊙ are lost from the (inner) binary. This semi-
conservative episode where mass is lost via isotropic re-emission widens the
(inner) orbit to $\approx$ 737.1 R⊙.

  4. 3.

Shortly after, at $\approx$ 3.6 Myr, the primary collapses into a 45.8 M⊙ BH.

  5. 4.

The secondary star finishes the main sequence and enters the Hertzsprung gap.
At $\approx$ 4.6 Myr it begins a dynamically unstable mass transfer episode.
This mass transfer episode results in a common-envelope episode, a process
where $\approx$ 68 M⊙ of envelope can be removed from the (inner) binary. This
process results in an $\approx$ 44.1 M⊙ stripped star in a close binary with
orbital separation of several solar radii.

  6. 5.

Finally, the secondary collapses at $\approx$ 4.9 Myr to form an $\approx$
39.6 M⊙ BH.

We choose this low-metallicity binary system in order to neglect the effect of
stellar winds in our calculations. Moreover, lower metallicity environments
are predicted to lead to more massive BBH remnants (e.g., Neijssel et al.,
2019) and compact BBHs (e.g., Vigna-Gómez et al., 2021).

###  4.2 Hierarchical triple system

The hierarchical-triple-system (HTS) channel which we propose involves a
massive inner stellar binary (described in Section 4.1) and a dynamically-
stable tertiary star orbiting around it (e.g., Hut & Bahcall, 1983). The
tertiary can capture and store a large fraction of the mass which is ejected
from the inner binary system as it becomes a BBH system. Once the BBH is
produced the tertiary may begin its RLOF phase and with this transfer mass
back onto the BBH. Such a system may form in a massive stellar-forming regions
via gravitational capture, or as a triple system in the field (Moe & Di
Stefano, 2017).

The condition for dynamical stability of a HTS is that the orbit of the
tertiary star ($a_{\rm out}$) is always larger than three times the orbit of
the binary ($a_{\rm in}$), this is, $a_{\rm out}\gtrsim 3a_{\rm in}$ (Kozai,
1962; Lidov, 1962; Shevchenko, 2017). To simplify our model, we assume that
mass is transferred from the inner binary and onto the tertiary star during
the ceRLOF stage (and not during the RLOF stage when the primary star expands;
nonetheless this early stage could also transfer up to 33.4M⊙ onto the
tertiary). If this was not the case, the initial orbital separation of the
tertiary star, as well as the subsequent stages, would need to be larger to
prevent the HTS from becoming unstable.

Using the results from COMPAS for the binary system (see Section 4.1), and for
the tertiary star evolution either conservative mass transfer during RLOF,
CEE, or ceRLOF stages, as well as isotropic mass loss during SNe, we broadly
calculate the evolution of the HTS (see Table 1 for details). As mentioned in
Section 4.1 the maximum orbital expansion of the inner binary ($a_{\rm in}$),
which occurs after the SN explosion of the primary, is $a_{\rm
in}=766.5\mbox{R${}_{\odot}$}$. With the latter, $a_{\rm
out}=2300\mbox{R${}_{\odot}$}$ and the evolution of the HTS can be modeled.
The main stages are next listed:

  1. [i]

  2. 1.

At ZAMS, the tertiary star, of mass $M_{3}\simeq 35\mbox{M${}_{\odot}$}$ and
metallicity $z=0.0001$, is located in a circular orbit at $a_{\rm out}\sim
1817~{}\mbox{R${}_{\odot}$}$.

  3. 2.

Approximately 3.4 Myr after ZAMS, the primary star reaches RLOF and transfers
$\Delta M\sim 67.1\mbox{M${}_{\odot}$}$, the secondary accretes $\sim 33.4\
\mbox{M${}_{\odot}$}$ (the rest is lost). At this stage the tertiary could
accrete a fraction of the mass which is lost (if its orbital velocity is
comparable to the ejected mass velocity), but our model does not account for
this. In response to the mass loss the binary orbit expands to $a_{\rm
in}\simeq 737.1\ \mbox{R${}_{\odot}$}$, and the orbit of the tertiary star to
$a_{\rm out}\simeq 2216\ \mbox{R${}_{\odot}$}$.

  4. 3.

At 3.6 Myr, the primary star collapses onto a $45.8\ \mbox{M${}_{\odot}$}$ BH
(losing $6.1\ \mbox{M${}_{\odot}$}$). The inner binary orbit expands to
$a_{\rm in}=766\ \mbox{R${}_{\odot}$}$, and the orbit of the tertiary star
reaches the minimum stable orbit for the HTS with $a_{\rm out}=2300\
\mbox{R${}_{\odot}$}$.

  5. 4.

At 4.6 Myr, the secondary star expands and produces a CE with the first-born
BH. We assume that no accretion onto the BH occurs (although a few M⊙ could be
accreted); instead, $68\ \mbox{M${}_{\odot}$}$ are ejected (not necessarily
unbound) from the secondary star, which gets stripped down to $44.1\
\mbox{M${}_{\odot}$}$. The ejected material, has low velocity and thus a large
fraction of it may be captured by the tertiary star, i.e., it undergoes ceRLOF
mass transfer.

  6. 5.

During the CE phase of the binary system, the orbit of the inner binary decays
to $a_{\rm in}=6.4\ \mbox{R${}_{\odot}$}$, while the tertiary star captures up
to $68\ \mbox{M${}_{\odot}$}$ of the ejected mass. Assuming conservative mass
transfer during this ceRLOF phase, the mass of the tertiary star grows to
$M_{3}\simeq 103\mbox{M${}_{\odot}$}$ which brings the mass ratio of the
tertiary to the inner binary close to one
($q_{3}=M_{3}/(M_{1}+M_{2})=103/(45.8+44.1)=1.15$), thus reducing its orbit to
$a_{\rm out}\simeq 818\ \mbox{R${}_{\odot}$}$.

  7. 6.

At 4.9 Myr, the secondary star collapses to a $39.6\ \mbox{M${}_{\odot}$}$ BH,
losing $4.4\ \mbox{M${}_{\odot}$}$ in the process. Both orbits expand a little
after the mass loss to $a_{\rm in}\simeq 6.7\ \mbox{R${}_{\odot}$}$ and
$a_{\rm out}\simeq 836\ \mbox{R${}_{\odot}$}$. The RL radius of the tertiary
star reaches $R_{\rm RL,3}\simeq 330\ \mbox{R${}_{\odot}$}$.

  8. 7.

A few hundred kyrs later, the tertiary star ends its MS stage, expands, fills
its $R_{\rm RL,3}$ and produces a CEE with the BBH. In this CEE, the BHs
accrete several tens of solar masses, ending at $M_{\rm BH1}\simeq 86\
\mbox{M${}_{\odot}$}$ and $M_{\rm BH2}\simeq 66\ \mbox{M${}_{\odot}$}$. The
CEE further decreases the inner orbital separation. While, the loss of mass
from the tertiary onto the binary decreases the mass ratio and expands the
orbit of the tertiary to $a_{\rm out}>2074\ \mbox{R${}_{\odot}$}$.

  9. 8.

Finally, the BBH will merge via GW emission in $\sim$ Myr. This timescale is
similar to that when the tertiary star will become a BH, but a sequential BBH
merger will not occur within the age of the Universe. Depending on what
happens first, and the magnitude and direction of the recoil kick of the BBH
product, the system might be disrupted or remain bound, now as a binary.

###  4.3 Star-forming region capture and CEE - BBH captures a massive star

We consider the binary system described in Section 4.1. If it captures a
tertiary massive star companion ($M_{3}>35\ \mbox{M${}_{\odot}$}$) before the
CE phase, then we can obtain a HTS system which can also produce the desired
BBH in the PISN. The tertiary star could capture $68~{}\mbox{M${}_{\odot}$}$
of the binary system in a ceRLOF and evolve similarly to the HTS in Section
4.2. Given the CEE stage of the inner binary, the tertiary star could be
captured into a much smaller orbital separation ($a_{\rm out}\gtrsim
30~{}\mbox{R${}_{\odot}$}$) without disturbing the evolution of the BBH. If
this is the case, the CEE in which mass is transferred onto the BBH may occur
in a shorter timescale than in our HTS scenario (as the tertiary star will
RLOF at an earlier time).

Alternatively, the tertiary could be captured at a later stage (e.g. once the
BBH has formed). In this scenario the captured star must be more massive
($M_{3}>100\mbox{M${}_{\odot}$}$) given that it will not necessarily be able
to capture mass during the ceRLOF of the binary. A caveat here would be that
the ejected mass (the $68~{}\mbox{M${}_{\odot}$}$) may not have escape
velocity and fall back onto the binary (and be accreted by the BBH), and
another part may form a circumbinary disk around the tertiary star. Eventually
the captured tertiary would star evolve, expand, fills its RL and produce a
CEE with the BBH where the BHs reach their final masses before merging
($M_{1}>86~{}\mbox{M${}_{\odot}$}$ and $M_{2}>66~{}\mbox{M${}_{\odot}$}$).

The capture of a wandering star, or of a star ejected from another
binary/tertiary system may happen at different evolutionary phases of the
binary. At early stages of BBH formation (e.g., stage i in Section 4.1), the
binary increases its mass (to $\approx 200~{}\mbox{M${}_{\odot}$}$) and has a
large cross section; however, the evolutionary lifetime as a stellar binary is
relatively short ($\sim$ Myr). If the tertiary is captured during the next 200
kyr (stages ii and iii), given the decrease in total mass ($\sim
160~{}\mbox{M${}_{\odot}$}$) and the increase in orbital separation ($\lesssim
766~{}\mbox{R${}_{\odot}$}$), the BBH will be more prone to disruption.
Finally, after the CE phase (stage iv) the binary system will have a much
smaller cross section ($a_{\rm in}\lesssim 7~{}\mbox{R${}_{\odot}$}$), less
mass ($M_{1}\gtrsim 46~{}\mbox{M${}_{\odot}$}$ and $M_{2}\gtrsim
40~{}\mbox{M${}_{\odot}$}$), but will have a large amount of time available to
capture a tertiary before merging ($\tau_{\rm GW}\sim 7$ Gyr). The expected
capture rate is larger during the ZAMS phase, with twice the mass and 70 times
larger orbital separation (so that the the cross section is $\sim 10^{4}$
larger) with respect to the BBH phase even though the timescale is 2000 times
smaller. Also, the binary will be closer to the star formation region where it
was formed, thus the stellar density will be larger.

Table 2: PISN-BH candidates from Abbott et al. (2021). The columns correspond to the GW event, the mass of the BH remnant ($M_{\rm rem}$), the masses of the merging BHs ($M_{\rm BH,1}$ and $M_{\rm BH,2}$), and the effective spin ($\chi_{\rm eff}$). GW event | Mrem [M⊙] | MBH,1 [M⊙] | MBH,2 [M⊙] | $\chi_{\rm eff}$
---|---|---|---|---
GW190403_051519 | $105.2^{+29.1}_{-24.1}$ | $88.0^{+28.2}_{-32.9}$ | $22.1^{+23.8}_{-9.0}$ | $0.70^{+0.15}_{-0.27}$
GW190426_190642 | $175.0^{+39.4}_{-34.3}$ | $106.9^{+41.6}_{-25.2}$ | $76.6^{+26.2}_{-33.6}$ | $0.19^{+0.43}_{-0.40}$
GW190519_153544 | $101.0^{+12.4}_{-13.8}$ | $66.0^{+10.7}_{-12.0}$ | $40.5^{+11.0}_{-11.1}$ | $0.31^{+0.20}_{-0.22}$
GW190521 | $156.3^{+36.8}_{-22.4}$ | $95.3^{+28.7}_{-18.9}$ | $69.0^{+22.7}_{-23.1}$ | $0.03^{+0.32}_{-0.39}$
GW190602_175927 | $110.9^{+17.7}_{-14.9}$ | $69.1^{+15.7}_{-13.0}$ | $47.8^{+14.3}_{-17.4}$ | $0.07^{+0.25}_{-0.24}$
GW190620_030421 | $87.2^{+16.8}_{-12.1}$ | $57.1^{+16.0}_{-12.7}$ | $35.5^{+12.2}_{-12.3}$ | $0.33^{+0.22}_{-0.25}$
GW190701_203306 | $90.2^{+11.3}_{-8.9}$ | $53.9^{+11.8}_{-8.0}$ | $40.8^{+8.7}_{-12.0}$ | $-0.07^{+0.23}_{-0.29}$
GW190706_222641 | $99.0^{+18.3}_{-13.5}$ | $67.0^{+14.7}_{-16.2}$ | $38.2^{+14.6}_{-13.3}$ | $0.28^{+0.26}_{-0.29}$
GW190929_012149 | $101.5^{+33.6}_{-25.3}$ | $80.8^{+33.0}_{-33.2}$ | $24.1^{+19.3}_{-10.6}$ | $0.01^{+0.34}_{-0.33}$
GW191109 010717 | $107^{+18}_{-15}$ | $65^{+11}_{-11}$ | $47^{+15}_{-13}$ | $-0.29^{+0.42}_{-0.31}$
GW191127_050227 | $76^{+39}_{-21}$ | $53^{+47}_{-20}$ | $24^{+17}_{-14}$ | $0.18^{+0.34}_{-0.36}$
GW200220_061928 | $141^{+51}_{-31}$ | $87^{+40}_{-23}$ | $61^{+26}_{-25}$ | $0.06^{+0.40}_{-0.38}$

##  5 Discussion

###  5.1 Channels to produce BHs with masses in the PISN mass gap

In this paper we propose two evolutionary channels in which, through HCA onto
a BBH during the CE phase, BHs with masses falling in the theoretically
predicted PISN mass gap may be produced. Thus, the BBH system progenitor of
the GW source GW190521 may have been formed via such scenarios.

Our results are based on two key processes: i) a ceRLOF, where a fraction of
the material lost by the stellar progenitors of the BBH system is stored into
a tertiary, massive star, and ii) HCA, which allows a large amount of mass to
be accreted by the BBH, growing one of the BHs (or the two BHs) well above the
PISN mass gap. We stress that, while the Eddington limit applies for spherical
symmetry and assumes electron scattering for the momentum deposition by the
photons in the accreting material, during HCA most of the energy is lost
through neutrinos (whose mean free path is many orders of magnitude larger
than that of photons).

In the first channel, a binary system with $\sim 120~{}\mbox{M${}_{\odot}$}$
and $\sim 80~{}\mbox{M${}_{\odot}$}$ ZAMS stars evolves inside a HTS producing
a BBH system of $\sim 45~{}\mbox{M${}_{\odot}$}$ and $\sim
40~{}\mbox{M${}_{\odot}$}$. In this scenario, most of the mass lost by the
binary system is stored in a tertiary star which later returns it to the BBH
enhancing both BH masses. In the second scenario, a tertiary star is captured
by the binary system (which in turn evolves in a similar path as the first
proposed evolutionary channel).

In both scenarios, the tertiary star evolves out of the MS stage after the BBH
with BH masses of $M_{1}\simeq{45}~{}\mbox{M${}_{\odot}$}$ and
$M_{2}\simeq{40}~{}\mbox{M${}_{\odot}$}$ has formed. The tertiary expands and
engulfs the BBH in a CE, transferring through HCA in the CE phase $\sim
40\mbox{M${}_{\odot}$}$ to the $45-\mbox{M${}_{\odot}$}$ BH, and $\sim
25\mbox{M${}_{\odot}$}$ to the $40-\mbox{M${}_{\odot}$}$ BH, leaving behind
two BHs with masses of $M_{1}\simeq{85}~{}\mbox{M${}_{\odot}$}$ and
$M_{2}\simeq{66}~{}\mbox{M${}_{\odot}$}$. Then, a BBH system with an orbital
separation $a_{\rm BS}\lesssim 10\;\mbox{R${}_{\odot}$}$ will merge in a
timescale of $\tau_{\rm GW}\lesssim T_{H}/2$, where $T_{H}$ is the Hubble
timescale444Using $z=0.82\rm$ in Carmeli, Hartnett, & Oliveira (2006) one
obtains a light travel time $\lesssim 28/[1+\rm(1+z)^{2}]\;{\rm Gyr}=6.5$ Gyr
(with which the maximum time for the BBH to merge is $\approx 7.3$ Gyr). From
Misner et al. (2017) the maximum orbital separation is:  $\displaystyle a$
$\displaystyle=$ $\displaystyle\left(\frac{256G^{3}\eta
M^{3}\tau_{GW}}{5c^{5}}\right)^{1/4}$ (13) $\displaystyle\simeq$
$\displaystyle 55\ \mbox{R${}_{\odot}$}\left(\frac{\tau_{GW}}{7{\rm
Gyr}}\right)^{1/4}\left(\frac{M_{\rm
BBH}}{150\mbox{M${}_{\odot}$}}\right)^{3/4}\left(\frac{\eta}{0.25}\right)^{1/4}$
with $M_{\rm BBH}=M_{\rm BH,1}+M_{\rm BH,2}=85\ \mbox{M${}_{\odot}$}+65\
\mbox{M${}_{\odot}$}=150\ \mbox{M${}_{\odot}$}$, $\eta=(M_{\rm BH,1}M_{\rm
BH,2})/M_{\rm BBH}^{2}\simeq 0.25$. This corresponds to an orbital period of
3.88 d. in a galaxy which is $5.3$ Gpc ($z=0.82$) from us.

###  5.2 Possible progenitors

MelnicK 34 (Mk34) is a spectroscopic binary, X-ray colliding-wind system, with
two hydrogen rich WR stars (Crowther & Walborn, 2011; Hainich et al., 2014).
Mk34 has a $155.1\pm 1$ d orbital period, and a projected distance of 2 pc
from R136 (the stellar cluster with the most massive stars in 30 Doradus,
Pollock et al. 2018). Its current age is $\sim 0.6\pm 0.3$ Myr, and the masses
of the WRs are $139^{+21}_{-18}~{}\mbox{M${}_{\odot}$}$ for the primary and
$127^{+17}_{-17}~{}\mbox{M${}_{\odot}$}$ for the secondary, corresponding to
ZAMS masses of $144^{+22}_{-17}~{}\mbox{M${}_{\odot}$}$ and
$131^{+18}_{-16}~{}\mbox{M${}_{\odot}$}$ respectively (Tehrani et al., 2019).
Thus, this system represents a possible progenitor of the BBH studied in this
paper.

Modeling the future evolution of Mk34, the primary star will have a maximum
radius of $R_{\rm max}\sim 78-126~{}\mbox{R${}_{\odot}$}$ during MS, with a
periastron orbital separation of $254~{}\mbox{R${}_{\odot}$}$ (Tehrani et al.,
2019). Also, the primary star in Mk34 will fill its Roche lobe (with
$400-500~{}\mbox{R${}_{\odot}$}$) and begin a mass transfer phase before it
forms a BBH system (with masses $<45~{}\mbox{M${}_{\odot}$}$) in 2 to 2.5 Myr
(Yusof et al., 2013). A more recent study using COMPAS finds that the future
evolution of Mk34 is uncertain Belczynski et al. (2022), allowing for a system
like the one predicted in Section 4.1. Thus, it is possible that even at
metallicities as large as those of LMC these kind of BBH mergers may occur. If
Mk34 was accompanied by a $35~{}\mbox{M${}_{\odot}$}$ tertiary or if the
binary system captures a tertiary star (probably another runaway star from
R136) within the mass range $50~{}\mbox{M${}_{\odot}$}\lesssim M_{3}\lesssim
100~{}\mbox{M${}_{\odot}$}$ and orbital separation as large as $\sim$ 10 AU,
then the system would follow a path similar to that described in Section 4.
HCA in the CE phase could lead to BBH system with masses very close to those
from GW190521.

GW190521 is also the first BBH merger which may be associated to an
electromagnetic (EM) counterpart. The optical flare (with $E\sim 10^{51}$ erg)
could be interpreted as evidence of interaction between the BBH and the
accretion disk of an active galactic nucleus (Graham et al., 2020). However,
GW190521 was at a redshift $\simeq 0.82^{+0.28}_{-0.34}$ (Abbott et al.,
2020b), and the EM event was detected $\sim 34$ d after the GW signal. Thus,
the association between GW190521 and the EM radiation is weak and remains to
be confirmed. Nonetheless, we must note that, in our scenarios, if the BBH
were to merge during the the CE of the tertiary star, it is likely that an
energetic EM counterpart (GRB-like, in terms of energy) would be observed at
the time of the merger (similar to the scenarios discussed by Soker, 2021b).
This would be particularly the case if the BBH merger occurs in the core of
the star. The BBH merger, unlike a binary neutron star merger, produces its
jets from accretion disks formed from external material, whereas binary
neutron star may produce their own disk from neutron star material (Lattimer &
Schramm, 1976; Rosswog et al., 1999; Metzger et al., 2011). The amount of
angular momentum available for the infalling material is high, so, most likely
a pair of extremely energetic jets would form and produce an energetic
electromagnetic transient counterpart to the GW signal. Fallback from the
exploded star could also help to produce a late EM signal (a hypernova-like
transient).

CEE of a BBH in a tertiary star could result in mergers within the CE if the
drag is strong. This can be prevented if the BHs accrete at BHL rates.
However, a merger of the BBH inside the core of the triple star can also
account for a GW signal accompanied by a GRB.

###  5.3 BH spins

Table 2 shows a list of GW events which have at least one of the merging BHs
in the PISN mass gap (Abbott et al., 2021). These BHs could be produced by the
HCA onto the BBH system during the CE of the tertiary star (see Section 4 for
further details). Apart from the formation channels which may produce massive
BHs, we analyse the spin that such BHs would have.

Assuming that during the formation of the BH (or BHs) in the PISN mass gap
there are weak SN kicks, mass transfer stages, and tidal synchronization
episodes, the expectation is that massive BHs will be formed with low spins
which would tend to be aligned with the orbital angular momentum (see, e.g.,
discussion in Moreno Méndez et al. 2008; Moreno Méndez 2011; unless something
such as SN kicks may alter them Tauris 2022). We highlight the fact that the
effective spins $\chi_{\rm eff}$ are distributed over a large range of values
(from $-0.29$ to $0.70$) and that there is no strong trend. Values of
$\chi_{\rm eff}\approx 0.70$ are expected in BBHs which experience multiple
sequential mergers. Focusing in our triple system evolutionary channels, we
enumerate a few mechanisms that will likely increase and/or misalign the spins
of the BHs (and which may be taken into account when predicting the
characteristics of the resulting BHs since massive stars are preferentially in
multiple stellar systems, see e.g. Moe & Di Stefano 2017):

  1. 1.

Strong SN kicks can misalign as well as spin up BHs.

  2. 2.

HCA during CE phases, especially when the BH mass doubles, can turn a spinless
BH to a Kerr BH (see Figure 6 in Brown et al. 2000) and align the BH spin to
the orbital angular momentum of the binary system.

  3. 3.

HCA in a CE of the tertiary star where the orbits are misaligned could produce
almost anything, from aligned small spins to completely misaligned large
spins.

  4. 4.

Tertiary-star orbital plane misaligned with binary orbital plane (particularly
likely in captures) can misalign the BH spins (Kozai-Lidov).

###  5.4 Other considerations

The amount of mass accreted during the CE stage, estimated in Section 4 should
be taken as an upper limit. The error bars of the BH masses in GW190521 are
$\sim 15-20$% at $1\sigma$ (Abbott et al., 2020a). So, it is possible that the
BHs may have lower masses, thus reducing the amount of mass accreted. The PISN
mass gap for BHs ($M_{\rm GAP}$) is limited between $55\lesssim M_{\rm
GAP}/\mbox{M${}_{\odot}$}\lesssim 135$ (Fryer, Woosley, & Heger, 2001) but the
lower limit may be higher. The latter may increase to $65\
\mbox{M${}_{\odot}$}$ considering low mass-loss during the evolution of
massive ZAMS stars (Woosley, 2017; Woosley & Heger, 2021). Other studies
propose that it could be of order 80 or 90 M⊙ (Belczynski, 2020; Belczynski et
al., 2010). Thus, by taking an upper BH mass limit $\sim 65\
\mbox{M${}_{\odot}$}$, our models can form a BBH with M${}_{\rm BBH,1}\simeq
65\ \mbox{M${}_{\odot}$}$ and M${}_{\rm BBH,2}\simeq 60\
\mbox{M${}_{\odot}$}$, hence reducing the mass accreted by $\sim$ a half (to
$\sim 25$ M⊙) in order to form the observed BBH in GW190521.

Stellar models which attempt to form BHs with masses within the PISN mass gap,
assume that the hydrogen envelope collapses onto the BHs (e.g., Renzo et al.,
2020), and drives them directly into the PISN; after that, they still need to
find another BH which goes through a similar process to merge with. In our
model we get BHs with masses above the PISN mass gap via the ceRLOF mechanism
by transferring the hydrogen envelope onto the tertiary star and returning it
to the BBH at a later stage.

Another mechanism which could lead a stellar binary to evolve into a
GW190521-like system is when a star with $M_{\star}\gtrsim
100~{}\mbox{M${}_{\odot}$}$ passes close to the BBH and is tidally disrupted
(or a Mk34-like system with $\sim 50-\mbox{M${}_{\odot}$}$ BHs is produced).
In this case the BHs accrete several tens of solar masses of disrupted-star
material, bringing them into the mass range of GW190521 as well as closer
together. Mass loss in pulsational PISNe may also lead to mass transfer onto
the tertiary star, especially for material which does not acquire escape
velocity or with a velocity $v\sim v_{\rm orb}$. This material is usually lost
during single stellar evolution, but a fraction of it can be recovered in
binary- or triple-system evolution.

The presence of strong jets and/or large $\alpha\lambda$ values could prevent
efficient mass transfer during ceRLOF. The formation of jets/winds/outflows
from accretion disks around the accreting BHs can remove the envelope of a
massive star, thus they can limit the mass growth of the BHs. A small orbital
separation between the BBH components can limit the formation of large
accretion disks. Thus the orbital separation of the BBH could be an important
factor for the amount of mass accreted by the BHs. Another relevant factor is
that the ore massive the BHs the larger the BHL radius, thus it is likely that
substantial mass growth by HCA during a CE will also depend on the initial
mass of the BHs as they undergo CEE.

Klencki et al. (2021) argue that unless the CEE occurs when the envelopes are
mostly convective, these cannot be ejected. Then, assuming that accretion is
negligible, the CE mass which is not ejected produces an orbital drag, and the
binary merges. In our case, the mass which is not ejected ends up being
rapidly accreted by the BHs, thus, preventing a merger. Also, in a triple
system, mass can be removed from the inner binary by reaching the Roche lobe,
without having to reach the escape velocity. It is then transferred onto the
(tertiary) companion. Hence, the closer the tertiary star is, the more
efficient the ceRLOF phase.

##  6 Conclusion

In this paper we propose an evolutionary channel in which a BBH system, via
HCA during the CE phase, produces BHs with masses within the PISN mass gap
(e.g., the BBH system which produced the GW190521 event). We have considered
two main scenarios. In the first scenario, the evolution of a HTS with an
inner binary system with two very-massive stars. During the evolution of these
two stars, part of the material is shed away and it is stored in a tertiary
companion (our ceRLOF mass transfer mode as described in Section 3). The
massive binary produces a BBH system with masses below the PISN gap limit
(e.g., $\sim$ 45 M☉ and $\sim$ 40 M☉). Later, the tertiary star leaves the MS,
expands and engulfs the BBH in its envelope, the BBH accretes back the
previously lost mass and results in a BBH with masses in the PISN mass gap. In
the second scenario, a massive binary captures a tertiary star. If the star is
captured during the MS of the binary, it may capture by ceRLOF mass from the
inner binary and proceed similarly to the first model. Alternatively, the
captured star may be a very-massive star, captured after the BBH has formed.
As the tertiary evolves out of its MS stage it produces a CE with the BBH and
transfers mass onto it, leaving, again a BBH with masses in the PISN mass gap.

The main bottleneck for these scenarios may come from whether the BBHs are
faster at accreting the CE material than the CE at absorbing energy from the
system and unbounding it. If these processes are nearly equally efficient, the
accretion during a CE stage of a star with $M_{\star}\gtrsim
100\mbox{M${}_{\odot}$}$ by the BBH could account for the masses of the
GW190521 event.

We also discuss possible progenitors of BBH systems with massive components.
Mk34 represents a possible progenitor of the BBH studied in this paper. BBHs
which are born in multiple systems will produces massive BHs with a disperse
distribution of $\chi_{\rm eff}$ values.

Our paper takes simulations of binaries from a population-synthesis code
(COMPAS) and includes them in the simplified analytical evolution of a HTS.
Future work should study the hydrodynamic evolution of the ceRLOF, the role of
HCA in BBHs, and incorporate them into a triple-star population-synthesis
software. Such population study would determine rates and configurations of
BBHs in the PISN mass gap via the ceRLOF+HCA formation mechanism described in
this manuscript. The rates and configurations could be then compared with the
current GW catalog and the forthcoming detections of current and future GW
observatories.

## Acknowledgements

F.D.C and D.L.C. acknowledge support from the UNAM-PAPIIT grant AG100820.
D.L.C. is supported by Cátedras CONACyT at the Instituto de Astronomía (UNAM).
A.V.G. acknowledges support by the Danish National Research Foundation
(DNRF132). This research has made use of NASA’s Astrophysics Data System as
well as arXiv.

## References

  * Abbott et al. (2016) Abbott B. P., Abbott R., Abbott T. D., et al., 2016, PhRvL, 116, 241103.
  * Abbott et al. (2020a) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, Phys. Rev. Lett., 125, 101102.
  * Abbott et al. (2020b) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, ApJ, 900, L13.
  * Abbott et al. (2021) Abbott R., Abbott T. D., Abraham S., Acernese F., Ackley K., Adams A., Adams C., et al., 2021, PhRvX, 11, 021053.
  * Armitage & Livio (2000) Armitage P. J., Livio M., 2000, ApJ, 532, 540.
  * Barkat, Rakavy, & Sack (1967) Barkat Z., Rakavy G., Sack N., 1967, PhRvL, 18, 379.
  * Bartos et al. (2017) Bartos I., Kocsis B., Haiman Z., et al., 2017, ApJ, 835, 165.
  * Belczynski et al. (2010) Belczynski K., Bulik T., Fryer C. L., et al., 2010, ApJ, 714, 1217.
  * Belczynski et al. (2016) Belczynski K., Holz D. E., Bulik T., et al., 2016, Natur, 534, 512.
  * Belczynski (2020) Belczynski K., 2020, ApJL, 905, L15.
  * Belczynski et al. (2022) Belczynski K., Romagnolo A., Olejak A., Klencki J., Chattopadhyay D., Stevenson S., Coleman Miller M., et al., 2022, ApJ, 925, 69.
  * Brown & Weingartner (1994) Brown G. E., Weingartner J. C., 1994, ApJ, 436, 843.
  * Brown et al. (2000) Brown G. E., Lee C.-H., Wijers R. A. M. J., Lee H. K., Israelian G., Bethe H. A., 2000, NewA, 5, 191.
  * Brown et al. (2007) Brown G. E., Lee C.-H., Moreno Méndez E., 2007, ApJL, 671, L41
  * Carmeli, Hartnett, & Oliveira (2006) Carmeli M., Hartnett J. G., Oliveira F. J., 2006, FoPhL, 19, 277.
  * Chamandy et al. (2018) Chamandy, L., Frank, A., Blackman, E. G., et al. 2018, MNRAS, 480, 1898.
  * Chevalier (1989) Chevalier R. A., 1989, ApJ, 346, 847.
  * Chevalier (2012) Chevalier R. A., 2012, ApJL, 752, L2
  * Comerford et al. (2019) Comerford, T. A. F., Izzard, R. G., Booth, R. A., et al. 2019, MNRAS, 490, 5196.
  * Comerford & Izzard (2020) Comerford, T. A. F. & Izzard, R. G. 2020, MNRAS, 498, 2957.
  * Crowther & Walborn (2011) Crowther P. A., Walborn N. R., 2011, MNRAS, 416, 1311.
  * De Luca et al. (2020) De Luca V., Franciolini G., Pani P., et al., 2020, PhRvD, 102, 043505.
  * de Kool (1990) de Kool M., 1990, ApJ, 358, 189.
  * Edgar (2004) Edgar, R. 2004, New Astron. Rev., 48, 843.
  * Eggleton (1983) Eggleton P. P., 1983, ApJ, 268, 368.
  * Fragione & Silk (2020) Fragione, G. & Silk, J. 2020, MNRAS, 498, 4591.
  * Fryer, Woosley, & Heger (2001) Fryer C. L., Woosley S. E., Heger A., 2001, ApJ, 550, 372.
  * Fryer et al. (2012) Fryer C. L., Belczynski K., Wiktorowicz G., et al., 2012, ApJ, 749, 91.
  * Graham et al. (2020) Graham, M. J., Ford, K. E. S., McKernan, B., et al. 2020, Phys. Rev. Lett., 124, 251102.
  * Hainich et al. (2014) Hainich R., Rühling U., Todt H., Oskinova L. M., Liermann A., Gräfener G., Foellmi C., et al., 2014, A&A, 565, A27.
  * Heger & Woosley (2002) Heger A., Woosley S. E., 2002, ApJ, 567, 532.
  * Heger et al. (2003) Heger A., Fryer C. L., Woosley S. E., et al., 2003, ApJ, 591, 288.
  * Houck & Chevalier (1991) Houck J. C., Chevalier R. A., 1991, ApJ, 376, 234.
  * Hut & Bahcall (1983) Hut P., Bahcall J. N., 1983, ApJ, 268, 319.
  * Iben & Livio (1993) Iben I., Livio M., 1993, PASP, 105, 1373.
  * Inayoshi et al. (2016) Inayoshi, K., Haiman, Z., & Ostriker, J. P. 2016, MNRAS, 459, 3738.
  * Ivanova et al. (2013) Ivanova, N., Justham, S., Chen, X., et al. 2013, A&ARv, 21, 59.
  * Kashi & Soker (2011) Kashi A., Soker N., 2011, MNRAS, 417, 1466.
  * Kinugawa, Nakamura, & Nakano (2020) Kinugawa T., Nakamura T., Nakano H., 2020, MNRAS, 498, 3946.
  * Klencki et al. (2020) Klencki, J., Nelemans, G., Istrate, A. G., et al. 2020, A&A, 638, A55.
  * Klencki et al. (2021) Klencki, J., Nelemans G., Istrate A. G., Chruslinska M., 2021, A&A, 645, A54.
  * Kozai (1962) Kozai Y., 1962, AJ, 67, 591.
  * Lattimer & Schramm (1976) Lattimer J. M., Schramm D. N., 1976, ApJ, 210, 549.
  * Li & Cao (2019) Li, J. & Cao, X. 2019, ApJ, 886, 92.
  * Lidov (1962) Lidov M. L., 1962, P&SS, 9, 719.
  * Liu & Lai (2020) Liu B., Lai D., 2020, PhRvD, 102, 023020.
  * Livio & Soker (1988) Livio, M. & Soker, N. 1988, ApJ, 329, 764.
  * López-Cámara et al. (2019) López-Cámara, D., De Colle, F., & Moreno Méndez, E. 2019, MNRAS, 482, 3646.
  * López-Cámara et al. (2020) López-Cámara, D., Moreno Méndez, E., & De Colle, F. 2020, MNRAS, 497, 2057.
  * MacLeod & Ramirez-Ruiz (2015) MacLeod, M. & Ramirez-Ruiz, E. 2015, ApJ, 803, 41.
  * McKernan et al. (2012) McKernan B., Ford K. E. S., Lyra W., et al., 2012, MNRAS, 425, 460.
  * Metzger et al. (2011) Metzger B. D., Giannios D., Thompson T. A., Bucciantini N., Quataert E., 2011, MNRAS, 413, 2031.
  * Misner et al. (2017) Misner, C. W., Thorne, K. S., & Wheeler, J. A. 2017, Gravitation, by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. ISBN: 978-0-691-17779-3. Princeton NJ: Princeton University Press, 2017.
  * Moe & Di Stefano (2017) Moe M., Di Stefano R., 2017, ApJS, 230, 15.
  * Mohamed & Podsiadlowski (2007) Mohamed, S. & Podsiadlowski, P. 2007, 15th European Workshop on White Dwarfs, 372, 397
  * Moreno Méndez et al. (2008) Moreno Méndez E., Brown G. E., Lee C.-H., et al., 2008, ApJL, 689, L9.
  * Moreno Méndez (2011) Moreno Méndez E., 2011, MNRAS, 413, 183.
  * Moreno Méndez et al. (2011) Moreno Méndez E., Brown G. E., Lee C.-H., et al., 2011, ApJ, 727, 29.
  * Moreno Méndez (2014) Moreno Méndez E., 2014, ApJ, 781, 3.
  * Moreno Méndez et al. (2017) Moreno Méndez, E., López-Cámara, D., & De Colle, F. 2017, MNRAS, 470, 2929.
  * Neijssel et al. (2019) Neijssel C. J., Vigna-Gómez A., Stevenson S., et al., 2019, MNRAS, 490, 3740.
  * Olejak, Belczynski, & Ivanova (2021) Olejak A., Belczynski K., Ivanova N., 2021, A&A, 651, A100.
  * Pacucci et al. (2015) Pacucci, F., Volonteri, M., & Ferrara, A. 2015, MNRAS, 452, 1922.
  * Paczynski (1976) Paczynski B., 1976, IAUS, 73, 75
  * Pollock et al. (2018) Pollock A. M. T., Crowther P. A., Tehrani K., Broos P. S., Townsley L. K., 2018, MNRAS, 474, 3228.
  * Postnov & Yungelson (2014) Postnov K. A., Yungelson L. R., 2014, LRR, 17, 3
  * Renzo et al. (2020) Renzo M., Cantiello M., Metzger B. D., et al., 2020, ApJL, 904, L13.
  * Ricker & Taam (2012) Ricker, P. M. & Taam, R. E. 2012, ApJ, 746, 74.
  * Riley et al. (2022) Riley J., Agrawal P., Barrett J. W., Boyett K. N. K., Broekgaarden F. S., Chattopadhyay D., Gaebel S. M., et al., 2022, ApJS, 258, 34.
  * Rosswog et al. (1999) Rosswog S., Liebendörfer M., Thielemann F.-K., Davies M. B., Benz W., Piran T., 1999, A&A, 341, 499
  * Roupas & Kazanas (2019a) Roupas Z., Kazanas D., 2019, A&A, 621, L1.
  * Roupas & Kazanas (2019b) Roupas Z., Kazanas D., 2019, A&A, 632, L8.
  * Sabach & Soker (2015) Sabach E., Soker N., 2015, MNRAS, 450, 1716.
  * Safarzadeh & Haiman (2020) Safarzadeh M., Haiman Z., 2020, ApJL, 903, L21.
  * Sakstein et al. (2020) Sakstein J., Croon D., McDermott S. D., et al., 2020, PhRvL, 125, 261105.
  * Sakurai et al. (2016) Sakurai, Y., Inayoshi, K., & Haiman, Z. 2016, MNRAS, 461, 4496.
  * Shevchenko (2017) Shevchenko I. I., 2017, ASSL.
  * Shiber, Kashi, & Soker (2017) Shiber S., Kashi A., Soker N., 2017, MNRAS, 465, L54.
  * Shiber & Soker (2018) Shiber S., Soker N., 2018, MNRAS, 477, 2584.
  * Shore et al. (1994) Shore S. N., Livio M., van den Heuvel E. P. J., et al., 1994, inbi.conf
  * Soberman et al. (1997) Soberman, G. E., Phinney, E. S., & van den Heuvel, E. P. J. 1997, A&A, 327, 620
  * Soker (2004) Soker N., 2004, NewA, 9, 399.
  * Soker (2015) Soker N., 2015, ApJ, 800, 114.
  * Soker (2017) Soker N., 2017, MNRAS, 470, L102.
  * Soker (2021b) Soker N., 2021, MNRAS, 506, 2445.
  * Spera et al. (2019) Spera M., Mapelli M., Giacobbo N., et al., 2019, MNRAS, 485, 889.
  * Stevenson et al. (2017) Stevenson S., Vigna-Gómez A., Mandel I., et al., 2017, NatCo, 8, 14906.
  * Stevenson et al. (2019) Stevenson S., Sampson M., Powell J., et al., 2019, ApJ, 882, 121.
  * Stone & Leigh (2019) Stone N. C., Leigh N. W. C., 2019, Nature, 576, 406.
  * Straight, Sakstein, & Baxter (2020) Straight M. C., Sakstein J., Baxter E. J., 2020, PhRvD, 102, 124018.
  * Sugimura et al. (2017) Sugimura, K., Hosokawa, T., Yajima, H., et al. 2017, MNRAS, 469, 62.
  * Tauris (2022) Tauris T., 2022, arXiv, arXiv:2205.02541
  * Tehrani et al. (2019) Tehrani K. A., Crowther P. A., Bestenlehner J. M., et al., 2019, MNRAS, 484, 2692.
  * van den Heuvel (1976) van den Heuvel E. P. J., 1976, IAUS, 73, 35
  * van Son et al. (2020) van Son L. A. C., De Mink S. E., Broekgaarden F. S., et al., 2020, ApJ, 897, 100.
  * Vigna-Gómez et al. (2018) Vigna-Gómez A., Neijssel C. J., Stevenson S., et al., 2018, MNRAS, 481, 4009.
  * Vigna-Gómez et al. (2020) Vigna-Gómez A., MacLeod M., Neijssel C. J., et al., 2020, PASA, 37, e038.
  * Vigna-Gómez et al. (2021) Vigna-Gómez A., Toonen S., Ramirez-Ruiz E., et al., 2021, ApJL, 907, L19.
  * Vink et al. (2011) Vink, J. S., Muijres, L. E., Anthonisse, B., et al. 2011, A&A, 531, A132.
  * Vink (2018) Vink J. S., 2018, A&A, 615, A119.
  * Vink et al. (2021) Vink J. S., Higgins E. R., Sander A. A. C., et al., 2021, MNRAS, 504, 146.
  * Webbink (1984) Webbink, R. F. 1984, ApJ, 277, 355.
  * Woosley (2017) Woosley S. E., 2017, ApJ, 836, 244.
  * Woosley & Heger (2021) Woosley S. E., Heger A., 2021, ApJL, 912, L31.
  * Yusof et al. (2013) Yusof, N., Hirschi, R., Meynet, G., et al. 2013, MNRAS, 433, 1114.
  * Ziegler & Freese (2021) Ziegler J., Freese K., 2021, PhRvD, 104, 043015.