11institutetext:  Department of Physics, Anhui Normal University, Wuhu, Anhui,
241000, China
11email<EMAIL_ADDRESS>22institutetext: Guangxi Key Laboratory for
Relativistic Astrophysics, School of Physical Science and Technology, Guangxi
University, Nanning 530004, China 33institutetext: Département d’Astronomie,
Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
44institutetext: Gravitational Wave Science Center (GWSC), Université de
Genève, CH-1211 Geneva, Switzerland
44email<EMAIL_ADDRESS>55institutetext: Key Laboratory of Dark
Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing, 210033, People’s Republic of China 66institutetext:
Department of Astronomy, School of Physics, Peking University, Beijing 100871,
China 77institutetext: College of Physics, Guizhou University, Guiyang city,
Guizhou Province, 550025, P.R. China 88institutetext: Max-Planck-Institut für
Gravitationsphysik (Albert-Einstein-Institut), D-30167 Hannover, Germany
99institutetext: Leibniz Universität Hannover, D-30167 Hannover, Germany

# Merging binary black holes formed through double-core evolution

Y. Qin 1122 R.-C. Hu 22 G. Meynet 3344 Y. Z. Wang 55 J.-P. Zhu 66 H. F. Song
77 X. W. Shu 11 S. C. Wu 8899

###### Abstract

Context. To date, various formation channels of merging events have been
heavily explored with the detection of nearly 100 double black hole (BH)
merger events reported by the LIGO-Virgo-KAGRA (LVK) Collaboration. We here
systematically investigate an alternative formation scenario, i.e., binary BHs
(BBHs) formed through double helium stars (hereafter double-core evolution
channel). In this scenario, the two helium stars (He-rich stars) could be the
outcome of the classical isolated binary evolution scenario involving with and
without common-envelope phase (i.e., CE channel and stable mass transfer
channel), or alternatively of massive close binaries evolving chemically
homogeneously (i.e., CHE channel).

Aims. We study the properties (i.e., the chirp masses and the effective spins)
of binary BHs (BBHs) formed through the double-core evolution, and investigate
the impact of different efficiencies of angular momentum transport within
massive He-rich stars on double-core evolution.

Methods. We perform detailed stellar structure and binary evolution
calculations that take into account internal differential rotation and mass
loss of He-rich stars, as well as tidal interactions in binaries. We
systematically study the parameter space of initial binary He-rich stars,
including initial mass and metallicity of He-rich stars, as well as initial
orbital periods. Apart from direct core collapse with mass and angular
momentum conserved, we also follow the framework in Batta & Ramirez-Ruiz
(2019) to estimate the mass and spin of the resulting BHs.

Results. We show that the radii of massive He-rich stars decrease as a
function of time, which comes mainly from mass loss and mixing in high
metallicity and from mixing in low metallicity. For double He-rich stars with
equal masses in binaries, we find that tides start to be at work on the Zero
Age Helium Main Sequence (ZAHeMS: the time when a He-rich star starts to burn
helium in the core, which is analogous to ZAMS for core hydrogen burning) for
initial orbital periods not longer than 1.0 day, depending on the initial
metallicities. Besides the stellar mass loss rate and tidal interactions in
binaries, we find that the role of the angular momentum transport efficiency
in determining the resulting BH spins, becomes stronger when considering BH
progenitors originated from a higher metal-metallicity environment. We
highlight that double-core evolution scenario does not always produce fast-
spinning BBHs and compare the properties of the BBHs reported from the LVK
with our modeling.

Conclusions. After detailed binary calculations of double-core evolution, we
have confirmed that the spin of the BH is not only determined by the interplay
of the binary’s different initial conditions (metallicity, mass and orbital
period), but also dependent on the angular momentum transport efficiency
within its progenitor. We predict that, with the sensitivity improvements to
the LVK’s next observing run (O4), the sample of merging BBHs will contain
more sources with positive but moderate (even high) $\chi_{\rm eff}$ and part
of the events are likely formed through the double-core evolution channel.

###### Key Words.:

binaries: close – stars: Wolf-Rayet – stars: black holes – stars: rotation

##  1 Introduction

The LIGO-Virgo-KAGRA (LVK) Collaboration has released the Gravitational Wave
Transient Catalog 3 (GWTC-3, The LIGO Scientific Collaboration et al., 2021b),
consisting of 69 confident binary black hole (BBH) merger events with the
detection threshold to count events with false alarm rate (FAR) ¡ 1 $yr^{-1}$.
With the targeted sample of BBHs in GWTC-3, the LVK Collaboration has also
inferred their intrinsic properties (e.g., merger rates, masses and effective
inspiral spins), among which the effective inspiral spin $\chi_{\rm eff}$
111$\chi_{\rm eff}=(M_{1}\chi_{1z}+M_{2}\chi_{2z})/(M_{1}+M_{2})$, where
$M_{1}$ and $M_{2}$ are the component masses of the two BHs, $\chi_{1z}$ and
$\chi_{2z}$ are dimensionless BH spin magnitudes aligned to the direction of
the orbital angular momentum (AM). has been widely considered as a probe to
distinguish the formation channels of merging BBH events (Abbott et al.,
2016b; Farr et al., 2017, 2018; The LIGO Scientific Collaboration et al.,
2021c; Roulet et al., 2021). The majority of the BBHs reported by the LVK
Collaboration have low $\chi_{\rm eff}$, while several BBH mergers 222GW190403
and GW190805 with high $\chi_{\rm eff}$ were reported from deeper searches in
GWTC-2.1 (The LIGO Scientific Collaboration et al., 2021a), but a low-
significance FAR threshold of 2 per day. show definitely high positive
$\chi_{\rm eff}$, e.g., $0.28_{-0.29}^{+0.26}$, $0.31_{-0.22}^{+0.20}$,
$0.33_{-0.25}^{+0.22}$, $0.37_{-0.25}^{+0.21}$, $0.52_{-0.19}^{+0.19}$, for
GW190706, GW190519, GW190620, GW170729, GW190517, respectively (Abbott et al.,
2021). We also note that these high values of $\chi_{\rm eff}$ are heavily
under debate (see e.g., Callister et al., 2022; Vitale et al., 2022,
references therein).

Substantial progress for understanding the origin of BBHs has been made in the
field over the last 7 years since the discovery of the first GW event GW150914
(Abbott et al., 2016a). However, the formation process of BBH merger events
remains an open scientific question. Leading models of BBH formation include
isolated binary evolution via either common envelope (CE, e.g., Phinney, 1991;
Tutukov & Yungelson, 1973; Belczynski et al., 2007; Ivanova et al., 2013;
Postnov & Yungelson, 2014; Belczynski et al., 2016; Vigna-Gómez et al., 2018;
Qin et al., 2018; Bavera et al., 2020; Hu et al., 2022), stable Roche-lobe
overflow (RLOF, e.g., van den Heuvel et al., 2017; Inayoshi et al., 2017;
Bavera et al., 2021; Olejak et al., 2021; Olejak & Belczynski, 2021; Gallegos-
Garcia et al., 2021; Marchant et al., 2021; Tanikawa et al., 2022; Shao & Li,
2022; van Son et al., 2022a, b), or chemical mixing (Marchant et al., 2016;
Mandel & de Mink, 2016; de Mink & Mandel, 2016; Song et al., 2016; du Buisson
et al., 2020; Riley et al., 2021), as well as dynamical assembly in globular
clusters and galactic nuclear clusters (e.g., Rodriguez et al., 2015; Antonini
et al., 2016; Safarzadeh et al., 2020; Mapelli et al., 2021; Fragione et al.,
2022), or efficient migration assisted in active galactic nuclei (AGN) disks
(Secunda et al., 2019; McKernan et al., 2020; Tagawa et al., 2020; Saavik Ford
& McKernan, 2022). Alternatively, two BHs can be the occurrence of
hierarchical stellar-mass BH mergers (Doctor et al., 2020; Kimball et al.,
2020, 2021; Gerosa & Fishbach, 2021).

Zevin et al. (2021) recently investigated multiple formation pathways
(isolated binary evolution channels and dynamical assembly channels) and found
that neither channel can contribute more than $\simeq 70\%$ of the BBHs
reported in GWTC-2. Moreover, it was pointed out in Mandel & Farmer (2022)
(also see Mapelli (2020); Mandel & Broekgaarden (2022)) that the merger rates
for BBHs can vary by orders of magnitude for different formation scenarios. So
far, it is still a challenge to quantitatively predict the properties of
merging BBHs due to uncertain physics involved in single and/or binary
evolution (Abadie et al., 2010; Dominik et al., 2015; de Mink & Belczynski,
2015; Giacobbo & Mapelli, 2018; Tang et al., 2020; Broekgaarden et al., 2022;
Belczynski et al., 2022; Peng et al., 2022).

Alternatively, merging BBHs could be formed through the double-core evolution.
This scenario involving the CE phase has been recently investigated, focusing
on low-mass He-rich stars leading to form double NSs (Dewi et al., 2006; Hwang
et al., 2015; Vigna-Gómez et al., 2018). More massive stars with mass-ratio
close to one at low metallicities evolving from ZAMS (Zero Age Main Sequence)
in close binaries can undergo several stable mass transfer phases during core
hydrogen burning (Case A mass transfer phase) and thus form double He-rich
stars as potential progenitors of BBHs (see Figure 3 in Marchant et al.,
2016). On the other hand, two massive stars could first evolve to form a close
binary system of a He-star and a main-sequence companion star after the first
mass transfer, and subsequently the second mass transfer from MS/giant star to
the He-star leads to form massive He-rich binary stars in a short orbit.

For now most BBH systems reported by the LVK Collaboration are still
consistent with zero BH spins. Recently, by employing a variety of
complementary methods to measure the distribution of spin magnitudes and
orientations for BBH mergers, Callister et al. (2022) found that the existence
of a subpopulation of BHs with vanishing spins is not required by current
data. The fact at the moment no event necessarily requires a high spin does of
course not mean that there are not among those already detected any that may
present a high spin. High BH spins may indicate that the inefficient AM
transport mechanism within the BH progenitor is preferred (Qin et al., 2019a,
b, 2022b). This finding can be reached given the assumption that BBHs are
formed through the classical isolated binary evolution channel involving CE
phase, before which the initially more massive star collapses to form the
first-born BH. Accordingly, the progenitor of the first-born BH is in a wide
orbit in which the tides from its companion are too weak to change the spin AM
of both components. Therefore, the resultant BH spin, inherited from the AM
content of its progenitor, is exclusively determined by the AM transport
efficiency within the progenitor star during post main sequence expansion. In
case of an efficient transport, any removal of the outer layers (at the time
of CE phase) slows the whole star, even its core. In case of a less efficient
coupling, the core spins faster than the envelope and removing the envelope
will make appear a faster rotating core than in the case of the efficient AM
transport. Alternatively, it is shown in Olejak & Belczynski (2021) that fast-
spinning BHs in merging BBHs can be formed by tidal spin-up through either a
stable mass transfer phase leading to the mass ratio reversal, or the CE phase
forming equal-mass BH components. For the case of stable mass transfer (see
their Figure 1 in Olejak & Belczynski, 2021), the initially more massive star
evolves first to become a BH, and then its companion obtains enough mass via
the first RLOF to become a massive He-rich star due to losing its hydrogen
envelope onto the first-born BH in the second RLOF. The He-rich star
subsequently evolves to become a fast-spinning BH by the tides (Qin et al.,
2018). As for the other case (see their Figure 2 in Olejak & Belczynski, 2021)
the two stars initially with equal-mass instead form twin-mass He-rich stars
following the RLOF mass transfer and subsequent CE phase, after which two
fast-spinning BHs are formed via Wolf-Rayet tides. More recently, under the
assumption of the Eddington-limited accretion onto BHs and efficient AM
transport within massive stars, Zevin & Bavera (2022) investigated the
isolated binary evolution regarding forming highly-spinning BHs and concluded
that it is difficult to form systems with moderate or high spins in the
primary BH component. However, the BH can be efficiently spun up by highly
super-Eddington accretion (Bavera et al., 2021; van Son et al., 2020; Qin et
al., 2022a; Shao & Li, 2022).

The Tayler-Spruit dynamo (Spruit, 2002), produced by differential rotation in
the radiative layers, is considered as one of potential mechanisms responsible
for the efficient transport of AM between the stellar core and its radiative
envelope. In brief, the TS dynamo starts for a small radial magnetic field
component (its precise initial value has no importance since it is rapidly
enhanced by the dynamo mechanism). This component is wounded up per
differential rotation and an azimuthal component field is formed. An azimuthal
field is unstable by the Tayler instability, i.e., an nonaxisymmetric pinch
type instability, which has consequence to amplify the azimuthal field and the
radial one. The new radial component is wounded up and the instability starts
again. This amplification mechanism lasts until the growth timescale of the
magnetic field is equal to its damping timescale. Assuming that stationary
situation is reached at every time step and that the length over which the
instability can develop is small enough for allowing the excess energy in the
differential rotation to overcome the stabilizing entropy gradient and large
enough for the magnetic field to not decay too fast, it is possible to deduce
the diffusion and viscosity coeffiecients. The revised TS dynamo (Fuller et
al., 2019) is based on the fact that the damping timescale can be much longer
than the one assumed in the original Tayler-Spruit dynamo. In that case larger
magnetic fields can be reached and stronger coupling achieved (see the
discussion in Eggenberger et al., 2022).

Stellar models with the original Tayler-Spruit dynamo (TS dynamo) can well
reproduce the rotation rates for the Sun (Eggenberger et al., 2005), white
dwarfs and NSs (Heger et al., 2005; Suijs et al., 2008). However the TS dynamo
is currently challenged for explaining the slow rotation rates of cores in red
giants (Eggenberger et al., 2012; Cantiello et al., 2014). Recently, the
revised TS dynamo (Fuller et al., 2019), which was proposed to better match
lower core rotation rates for sub-giant and red giant stars in better
agreement with observed values, faces a challenge to reproduce the
observational constraints on asteroseismic data of evolved stars (Eggenberger
et al., 2019; den Hartogh et al., 2020). Applying the revised TS dynamo to
massive He-rich stars in close binary systems predicts lower BH spins when
compared with the original TS dynamo (Fuller & Lu, 2022). More recently,
Eggenberger et al. (2022) derived a new calibrated version of the original TS
dynamo to better account for the evolution of the core rotation rates along
the red giant branch stars when compared with the revised dynamo version.
There was a theoretical debate on the existence of the dynamo (Zahn et al.,
2007). Ji et al. (2022) recently performed three-dimentional
magnetohydrodynamic simulations of the Tayler instability in rotating stellar
interiors, and claimed to observe dynamo action via the amplification of
poloidal magnetic field, indicating the TS instability could be important for
magnetic field generation and AM transport in the radiative regions of
evolving stars. The detailed comparisons between different versions of TS
dynamo are beyond the scope of this work. Therefore, we are focused on the
impact of the original TS dynamo within massive He-rich stars on the spin of
resultant BH and its comparison when the TS dynamo is not included.

In this paper, we systematically investigate an alternative evolutionary
scenario to form BBHs from double He-rich stars, i.e., double-core evolution
first proposed by Brown (1995) who studied the formation of double NSss. In
Section 2, we introduce the main methods used in the stellar and binary
evolution models. We present our detailed results in Section 3. The
conclusions and discussion are summarized in Section 4.

##  2 Methods

We use release 15140 of MESA stellar evolution code (Paxton et al., 2011,
2013, 2015, 2018, 2019) to perform all of the binary evolution calculations in
this work. We adopt three different kinds of metallicities,
$\mathrm{Z}=\mathrm{Z}_{\odot},0.1\mathrm{Z}_{\odot},0.01\mathrm{Z}_{\odot}$,
where the solar metallicity is $\mathrm{Z}_{\odot}=0.0142$ (Asplund et al.,
2009). We create He-rich stars at zero-age helium main sequence following the
same method as in Qin et al. (2018); Bavera et al. (2020); Hu et al. (2022);
Fragos et al. (2022), and then relax the created He-rich stars to reach the
thermal equilibrium when the ratio of the He-burning luminosity to the total
luminosity $\geq$ 99$\%$. We model convection using the standard mixing-length
theory (Böhm-Vitense, 1958) with a parameter $\alpha=1.5$ and semiconvection
according to Langer et al. (1983) with an efficiency parameter $\alpha_{\rm
sc}=1.0$. We adopt Ledoux convection criterion to treat the boundaries of the
convective zones and consider the step overshooting as an extension given by
$\alpha_{p}=0.1H_{p}$, where $H_{p}$ is the pressure scale height at the
Ledoux boundary limit. The network of approx12.net is adopted for
nucleosynthesis.

We treat rotational mixing and AM transport as diffusive processes Heger &
Langer (2000), including the effects of Eddington-Sweet circulations, the
Goldreich–Schubert–Fricke instability, as well as secular and dynamical shear
mixing. We include diffusive element mixing from these processes with an
efficiency parameter $f_{c}=1/30$ (Chaboyer & Zahn, 1992; Heger & Langer,
2000). We use the standard efficient AM transport mechanism (e.g., Spruit,
1999, 2002). Stellar winds of He-rich stars are modeled with the standard
“Dutch” scheme, multiplied with a scaling factor of 2/3 to match the recently
updated modeling of helium stars’ winds (Higgins et al., 2021).

He-rich stars are modeled to reach the carbon exhaustion in the center. The
baryonic remnant mass is calculated following the “delayed” supernova
prescription as in Fryer et al. (2012). In order to calculate the mass and
spin of the BH, we follow the framework in Batta & Ramirez-Ruiz (2019), which
has been recently implemented in recent work (Bavera et al., 2020; Hu et al.,
2022). We take into account the neutrino loss as in Zevin et al. (2020). We
adopt 2.5 $M_{\odot}$ as the maximum NS mass. As a comparison (see appendix
A), we also considered BHs formed through direct core collapse without
receiving any mass loss or natal kicks (Fryer, 1999; Belczynski et al., 2008).
Very recently, it was reported on VFTS 243 that an X-ray quiet BH was born
with a negligible kick in a massive binary within the Large Magellanic Cloud
(Shenar et al., 2022).

Tidal interaction in close binary systems plays a critical role in the
evolution of the orbit and the internal AM for the two stellar components. In
this work, we use the dynamical tides model (Zahn, 1975; Hut, 1981) to
calculate the synchronization timescale ($T_{\rm sync}$), which is dependent
on the tidal coefficient $E_{2}$. The two He-rich stars are assumed to be non-
rotating at ZAHeMS. The main reason is that He-rich stars can be quickly spun
up in close orbits. For both He-rich (also H-rich) stars, Qin et al. (2018)
recently updated an approximate expression of $E_{2}$, mainly depending the
convective core radius and the star’s radius for a wide range of initial
masses and evolutionary stages at different metallicities.

In this study, we are focused on detailed investigations of a parameter space
study with various initial conditions of close double He-rich stars. We cover
the initial masses of He-rich stars from 5 - 65 $M_{\odot}$, the initial
orbital periods in a range of 0.1 - 6 days. We evolve two He-rich stars with
equal mass at different initial metallicities assuming two different AM
transport mechanisms.

##  3 Results

###  3.1 Hertzsprung-Russell diagrams of single He-rich stars

Here we present the Hertzsprung-Russell (HR) diagram of single He-rich stars
from the onset of the core helium burning (i.e., ZAHeMS) to the exhaustion of
their central carbon. All of the He-rich stars are assumed to be non-rotating
with different metallicities (1.0 $\mathrm{Z}_{\odot}$, 0.1
$\mathrm{Z}_{\odot}$ and 1.0 $\mathrm{Z}_{\odot}$), in the mass range of 5 -
60 $M_{\odot}$ at a step of 5 $M_{\odot}$. In Fig. 1, the core helium burning
phase begins on the right ends of the different curves labelled by the core
He-mass, evolution then brings the stars to the left (the effective
temperature increases). The evolution of the luminosity is different depending
on the initial metallicity, rapidly decreasing at the beginning at 1.0
$\mathrm{Z}_{\odot}$ in the high mass range, and increasing at 0.01
$\mathrm{Z}_{\odot}$ in this same mass domain. This is an effect of the
different mass loss rates at different metallicities. At high metallicities
the strong mass loss rate decreases rapidly the luminoisity. At a low
metallicity the mass is much less decreased and the main effect comes from the
fact that the mean molecular weight increases increasing the luminosity,
overcoming the effect due to the weak mass loss. These stars evolves towards
to bluer regions of the HR diagram, which is similar to H-rich stars evolving
chemically homogeneously on the main sequence. The main difference, however,
is that for more massive He-rich stars their mass decreases and as a
consequence the radius shrinks.

Figure 1: Hertzsprung-Russell diagrams of various single non-rotating He-rich
stars with different initial metallicities (Left panel: 1.0
$\mathrm{Z}_{\odot}$, middle panel: 0.1 $\mathrm{Z}_{\odot}$. bottom panel:
0.01 $\mathrm{Z}_{\odot}$.) evolving from Zero Age Helium Main Sequence
(ZAHeMS) to the central helium exhaustion. The blue dashed lines refer to
contours of constant radii.

###  3.2 Spin of BHs formed from double-core evolution

####  3.2.1 Impact of TS dynamo on BH spins

Let us first show how different efficiencies of AM transport within He-rich
stars change their rotation frequency at different evolutionary stages and
thus the resulting spin parameters of BHs. As a case study, we evolve a binary
system of two equal-mass He-rich stars, with initial mass $M_{\rm ZamsHe}$ =
39.80 $M_{\odot}$ at the initial orbital period $P_{\rm init.}$ = 0.63 days,
until the end of their central carbon depletion.

First of all, we show in Fig. 2 that the AM of the star and its core increases
rapidly at the beginning due to the tidal interaction that spun up the star.
Under the assumption that the wind mass lost is carrying the specific AM of
the mass-losing star (Jeans mass loss), the He-rich star and its inner core
will thus be slowed down. This situation, however, can be reversed for He-rich
stars in a close binary system, in which tides are efficient to spin up the
outer layers of the stars and their cores through strong coupling within the
stars. We present the impact of the TS dynamo on the evolution of the internal
rotation frequency of He-rich stars at different evolutionary stages, which
are shown in the top two panels in Fig. 2. On the top left panel, we present
models with the TS dynamo included (hereafter, TS on) given the solar
metallicity, while we leave the discussion of a lower-metallicity model for
the next section. First of all, with the TS dynamo the model in top left panel
(see blue line) shows for the He-rich star in the middle of the core helium
burning a flat distribution of a constant rotation frequency. This is because
the star evolves with TS on like a solid body during core He burning phase.
The whole star then gets spun up by the tidal interaction from its companion,
as the star evolves off its core helium burning phase, from which on the star
has rotation frequency of its outer layers slightly decreasing towards the
surface due to the occurrence of increasing chemical gradient in the late
evolutionary stage. The rotation frequency of the star continues increasing
after the middle of the core helium burning, which is due to the tidal spun-up
from its companion. In contrast, similar models without including TS dynamo
(hereafter, TS off) on the top right panel in Fig. 2, show clear differences.
The first difference is that the whole star due to less efficient coupling
(e.g., TS off) between the outer layers and the stellar core is not a solid
body from the early evolutionary stage (i.e., middle of core helium burning)
to the late stages (see yellow sold line for the model at the central carbon
ignition and red line for the central carbon depletion, respectively).
Additionally, we also note that the star with TS off has a much larger
rotational frequency throughout the whole evolutionary phase when compared
with TS-on models. This is because inefficient coupling (TS off) between the
outer layers and the stellar core allows the star to retain more AM and can
instead be spun up if tides are strong.

In this section, we present the impact of TS dynamo on the evolution of the AM
of the He-rich stars and their inner cores at different evolutionary stages.
We show in Fig. 3 that two binary evolutionary sequences of the same initial
orbital period $P_{\rm init.}$ = 0.63 days and different initial masses
$M_{\rm ZamsHe}$ = 10.00 $M_{\odot}$ (top row) and $M_{\rm ZamsHe}$ = 39.80
$M_{\odot}$ (bottom row), assuming TS on (solid lines) and off (dashed lines).
Here we only show the solar-metallicity models, and leave the discussion of
the rest in the following section. In the top left panel, we can see clear
differences of the total AM of He-rich stars with TS on (black solid line) and
off (black dashed line) starting before the middle of core helium (He) burning
stage. For the model with TS on, during the core He burning phase the total AM
of the star slowly decreases and then reaches the lower limit at the central
He depletion. Additionally, the total AM keeps almost constant during the
whole carbon (C) burning phase. Nevertheless, the model with the TS off shows
that the star’s total AM slightly decreases from the core He burning phase and
then keeps constant until the central C depletion. The AM of the carbon-oxygen
(CO) core of the He-rich star shows a similar trend, but with a shallow decay
after igniting its the central carbon. The resulting BHs calculated using the
prescription in Batta & Ramirez-Ruiz (2019) show the spins 0.08 (TS on) and
0.29 (TS off), respectively. For more massive He-rich binaries ($M_{\rm
ZamsHe}$ = 39.80 $M_{\odot}$), the bottom left panel presents a similar
finding, but with a much higher difference on the BH spin value, 0.07 (TS on)
and 0.49 (TS off). First, He-rich stars in a very close binary are
synchronised with their orbit due to strong tides, which allows more massive
star to carry more AM given the same initial orbit when compared to binary
systems of less components. On top of that, more massive He-rich stars are
expected to have less lifetime before the core-collapse, resulting in more AM
content within the progenitors and thus high resultant BH spins.

####  3.2.2 Impact of the metallicity on BH spins

As shown in the previous section, the TS dynamo has a significant impact on
the evolution of the AM of the He-rich star and its core at later evolutionary
stages, which further determines the spin values of the BH at birth. We here
describe how the initial metallicity of He-rich stars can play a role in
determining the spins of the resulting BHs.

It is well known that the stellar winds are strongly dependent on the
metallicity of the mass-losing He-rich stars (Vink et al., 2001; Vink & de
Koter, 2005; Eldridge & Vink, 2006; Sander et al., 2020). We can see that the
He-rich star at its central carbon depletion has a much larger mass (around
37.5 $M_{\odot}$) at 0.01 $\mathrm{Z}_{\odot}$ when compared with a solar
metallicity (around 20 $M_{\odot}$). We show in the two bottom panels of Fig.
2 that, the He-rich star and its inner core have a similar and higher rotation
rate at different evolutionary stages when compared with corresponding TS-on
models at $\mathrm{Z}_{\odot}$. Additionally, the bottom right panel shows
that the He-rich star evolves deviating from a solid body, slightly decreasing
rotation rate from the stellar core to its outer layers. We also see that at
solar metallicity the TS-off models retain more AM, but the difference is much
less marked than at sub-solar metallicity (e.g., 0.01 $\mathrm{Z}_{\odot}$,
see two bottom panels in Fig. 2). This difference is caused by the effect of
the metallicity-depend wind mass loss which plays a critical in determining
the final AM content of the progenitor star.

In Fig. 3, we can see that at 0.1 $\mathrm{Z}_{\odot}$, both He-rich stars and
their cores have a higher AM at different evolutionary stages when compared
with 1.0 $\mathrm{Z}_{\odot}$ (see the top left panel in Fig. 3).
Interestingly, we note that the TS dynamo plays a small role in determining
the evolution of the AM of He-rich star and its core at a low metallicity (see
the top middle and right panel). This is expected as the wind carrying the
specific AM of the mass-losing He-rich star is weaker at lower metallicities,
which weakens the effect of AM transport within stars. Therefore, the spin
values of resultant BHs formed at lower metallicities are accordingly higher,
i.e., 0.1 $\mathrm{Z}_{\odot}$ and 0.14 at 0.01 $\mathrm{Z}_{\odot}$ for TS on
(TS off: 0.43 $\mathrm{Z}_{\odot}$ and 0.49 at 0.01 $\mathrm{Z}_{\odot}$).

Figure 2: As a function of mass coordinate, we plot the angular velocity
$\omega$ of He-rich stars at three evolutionary stages, i.e., middle of core
helium burning (blue), Central carbon ignition (yellow), and central carbon
depletion (red). The initial mass of He-rich star is $M_{\rm ZamsHe}$ = 39.80
$M_{\odot}$ and the initial orbital period $P_{\rm init.}$ = 0.63 days.
Different efficiencies of AM transport mechanism and metallicities are
assumed. Left column: TS on, right column: TS off. Top row: 1.0
$\mathrm{Z}_{\odot}$, bottom row: 0.01 $\mathrm{Z}_{\odot}$. Figure 3: AM of
the He-rich star and its Carbon-Oxygen (CO) core (black lines: $J_{\rm
total}$, blue lines: $J_{\rm CO}$) as a function of model number for two
binary sequences (Top row: two equal-mass He-rich stars with initial helium
star mass $M_{\rm ZamsHe}$ = 10.0 $M_{\odot}$, initial orbital period
$P_{init.}$ = 0.63 days; bottom row: $M_{\rm ZamsHe}$ = 39.8 $M_{\odot}$,
$P_{init.}$ = 0.63 days). Similar to Fig. 2, we assume two different
efficiencies of AM transport mechanism, i.e., solid lines: TS on, dashed
lines: TS off. Left column: 1.0 $\mathrm{Z}_{\odot}$, middle column: 0.1
$\mathrm{Z}_{\odot}$, right column: 0.01 $\mathrm{Z}_{\odot}$. Three
evolutionary stages are marked in different symbols: square: middle of core
helium burning, circle: central helium depletion, filled circle: central
carbon ignition. The spin parameters of BHs formed from He-rich stars are
presented.

####  3.2.3 Parameter space analysis

First of all, we show in the top left panel of Fig. 4, BH masses as a function
of the He-rich initial mass and initial orbital period at solar metallicity.
First, the binary systems either start to overflow their Roche lobes at the
first model for $P_{\rm init.}$ $\sim$ 0.1 days (0.2 days for $M_{\rm ZamsHe}$
$\sim$ 30 $M_{\odot}$) or undergo the second lagrangian point (L2) overflowing
for $M_{\rm ZamsHe}$ $\sim$ 38 $M_{\odot}$ and $P_{\rm init.}$ $\sim$ 0.1
days. Second, given the “delayed” supernova prescription (Fryer et al., 2012)
the lower mass limit of the He-rich star that can collapse to form a BH (given
solar metallicity) is around 12 $M_{\odot}$, below which a NS is formed
instead (The study of NS formation is not considered in this work). Third, a
He-rich star can form a BH with the maximum mass of around 26 $M_{\odot}$. At
0.1 $\mathrm{Z}_{\odot}$ (see top middle panel), we note that $\sim$ 40
$M_{\odot}$ BH can be formed. Notably, the initial orbital period starts to
have an impact on the mass of the resulting BH when its immediate progenitor
(He-rich star) has an initial mass $\gtrsim$ 40 $M_{\odot}$. This is because
He-rich stars tend to lose more masses at a higher rotation rate in a closer
binary system, which is due to the rotationally-enhanced mass loss (Langer,
1997; Maeder & Meynet, 2000). It is clearly shown at 0.01 $\mathrm{Z}_{\odot}$
that more massive BHs ($>$ 55 $M_{\odot}$ see top right panel) can be formed.
It is worth noting that the efficient AM transport within He-rich stars plays
a negligible role in determining the BH mass.

We then present in Fig. 5 the spin parameters $a_{*}$ of BHs formed from
collapsing He-rich stars in close binaries with various conditions and assumed
AM transport processes. Let us first show the spins of resultant BHs assuming
efficient AM transport within He-rich stars. We note that the tides start to
play a role when the initial orbital period $P_{\rm init.}$ is not longer than
1.0 day for all metallicities. As demonstrated in recent studies (Qin et al.,
2018; Fuller & Lu, 2022), the interplay between the tides and wind mass loss
of He-rich stars determines the AM of resultant BHs at birth and thus their
spin magnitudes. At solar metallicity, the formed BHs are found to have low
spin values (i.e., $a_{*}$ $\lesssim$ 0.4, see top left panel). This is
because the wind mass loss of He-rich stars at a high metallicity is dominant
over the tides. The spin magnitudes of BHs formed at lower metal poor
environments are shown in the middle (0.1 $\mathrm{Z}_{\odot}$) and right
panel (0.01 $\mathrm{Z}_{\odot}$). Therefore, at a given initial orbital
period ($P_{\rm init.}$ $\lesssim$ 1.0 day), high BH spins can be reached for
models at 0.1 and 0.01 $\mathrm{Z}_{\odot}$. At 0.01 $\mathrm{Z}_{\odot}$, we
clearly see in the top right panel that, the spin magnitudes continue
increasing with initial orbital period for all different initial masses of He-
rich star. This is because the progenitor of the BH has very weak winds at
0.01 $\mathrm{Z}_{\odot}$, and thus loses negligible mass and AM. Furthermore,
it is clear to see that the spins of BHs, originated from He-rich stars with
initial mass $\lesssim$ 20 $M_{\odot}$, slightly increase with initial mass.
This is because the wind of low-mass He-rich stars at very low metallicity
(0.01 $\mathrm{Z}_{\odot}$) is significantly weak. Accordingly, for initially
higher mass of He-rich stars, more infalling mass with its corresponding AM
can be accreted to the newly-formed BHs (Batta & Ramirez-Ruiz, 2019),
resulting in higher final BH spins.

Assuming inefficient AM transport within He-rich stars, we show the spins of
resultant BHs in the second row of Fig. 5. As shown clearly in the bottom left
panel, high BH spins ($>0.9$) can be reached at solar metallicity.
Additionally, the spin covers the whole range, i.e., from minimum to maximum.
For initial orbital periods $P_{\rm init.}$ $\lesssim$ 1.0 day, we note that
the BH spin gradually decreases with increasing initial mass of He-rich stars,
which is because massive He-rich stars are prone to be slowed down due to
their strong winds at high metallicity. This is in contrast to the results of
models at very low metallicity (see the top right panel), where the wind mass
loss of He-rich stars is significantly weak at 0.01 $\mathrm{Z}_{\odot}$.
Notably, we can see that He-rich stars tend to form higher-spinning BHs at
lower metallicities which correspond to weaker wind mass loss (see bottom
middle and right panel).

Figure 4: BH mass $\mathrm{M}_{\mathrm{BH}}$ as a function of the He-rich
star’s initial mass and orbital period, are marked with the color of the
filled circles while gray squares represent that the compact objects formed
through direct core-collapse of the He-rich star are NSs. Left column: 1.0
$\mathrm{Z}_{\odot}$, middle column: 0.1 $\mathrm{Z}_{\odot}$, right column:
0.01 $\mathrm{Z}_{\odot}$. Top row: TS on; bottom row: TS off. The cross
symbols represent the systems overflowing their Roche lobes at their initial
models, while the plus symbols refer to the systems overflowing the second
Legrangian point (L2) at the initial models. Figure 5: As in Fig. 4, but the
color denotes the BH spin parameter $\mathrm{a}_{\star}$.

###  3.3 Merging timescales and comparisons with observed merging BBHs

After two BHs form from the core-collapse of He-rich stars, gravitational wave
(GW) emission shrinks the separation by removing the orbital AM, and
eventually leads to the merger the compact objects. The timescale for two
point masses to spiral in through GW emission from an initial eccentricity
being zero (circular orbit) is given by Peters (1964)

$\centering T_{\rm
merger}=\frac{5}{512}\frac{c^{5}}{G^{3}M^{3}}\frac{2q^{-2}}{1+q^{-1}}a^{4},\@add@centering$
(1)

where $M$ is the BH mass, $q$ the mass ratio of the two BHs ($q=1$ for our
case) and $a$ is the orbital separation.

We show the color bar in Fig. 6 corresponding to $T_{\rm merger}$ of merging
BBHs due to GW emission. First, comparing the two rows of different AM
transport mechanisms shows negligible impact on the merging timescale. This is
because significant differences are only expected for the AM content of the BH
progenitors, rather than the properties (two component masses and the final
separation) of the binary system just after the birth of two BHs. Second, the
parameter space of systems that are able to merge within a Hubble time is
extended in lower metallicities. This is because BH progenitors at a higher
metallicity tend to lose more mass, and the BBHs at birth thus have larger
separations ($T_{\rm merger}\propto a^{4}$). Third, given a specific initial
orbital period and metallcity, BBHs with initially higher mass have shorter
merging timescales ($T_{\rm merger}\propto M^{-3}$).

Figure 7 presents the merging timescales $T_{\rm merger}$ as a function of the
effective inspiral spin $\chi_{\rm eff}$ and the chirp mass $M_{\rm chirp}$.
69 high-confidence BBH events (false alarm rate $<$ 1 per year) officially
reported from the LVK are also shown in grey for comparison in each panel. We
present systems formed from the same initial orbital period in dashed line. We
assume that the formed BHs have spin components perfectly aligned to the
direction of the orbital AM. First of all, we note that the initial
metallicity plays an important role in forming systems with the observable
properties ($\chi_{\rm eff}$ and $M_{\rm chirp}$). i.e., lower metallicities
corresponding to formed systems with higher $\chi_{\rm eff}$ and larger
$M_{\rm chirp}$. More specifically, the $M_{\rm chirp}$ can be reached around
26 $M_{\odot}$ at solar metallicity (40 and 58 $M_{\odot}$ at 0.1 and 0.01
$\mathrm{Z}_{\odot}$, respectively). Furthermore, the magnitude of $\chi_{\rm
eff}$ can vary from 0.0 ($P_{\rm init.}$ = 1.0 day) and 1.0 ($P_{\rm init.}$ =
0.2 days). We note that so far no BBHs with $\chi_{\rm eff}$ = 1 has been
reported from the LVK collaboration. Third, the AM transport mechanism in
these observable properties of BBHs starts to play a more important role at
higher metallicities (solar metallicity, see the two left panels in Fig. 7).
Additionally, under the assumption of inefficient AM transport, double He-rich
stars can form observable BBHs with $\chi_{\rm eff}$ ¿ 0.80 (¡ 0.5 with TS on)
with initially $P_{\rm init.}$ = 0.4 days and $\chi_{\rm eff}$ ¿ 0.5 (¡ 0.25
with TS on) with initially $P_{\rm init.}$ = 0.6 days, respectively. It is
also clearly shown in the left panels of Fig. 7 that more BBHs formed from
double He-rich stars at solar metallicity will not be merged within a Hubble
time (see black triangles) when compared with low-metallicity models. We note
the trend that the observed BBHs with higher values of both $\chi_{\rm eff}$
and $M_{\rm chirp}$, can be better explained in our modeling at lower
metallicites. In particular, the fraction of the BBHs, which have $\chi_{\rm
eff}$ higher than that of GW190517 (it has the highest $\chi_{\rm eff}$
reported in the LVK), is 8.9% at 1.0 $\mathrm{Z}_{\odot}$, 20.3% at 0.1
$\mathrm{Z}_{\odot}$, and 26.9% at 0.01 $\mathrm{Z}_{\odot}$ (For TS off:
18.1% at 1.0 $\mathrm{Z}_{\odot}$, 23.2% at 0.1 $\mathrm{Z}_{\odot}$, and
28.7% at 0.01 $\mathrm{Z}_{\odot}$), respectively. GW190521 was reported from
the LVK to have the highest $M_{\rm chirp}$ (Abbott et al., 2020), which might
be a straddling binary using a population informed prior (Fishbach & Holz,
2020). This event is an outlier in our models, as the upper limit of the BH
mass in this study is assumed not to be higher than $\sim$ 65 $M_{\odot}$ due
to (pulsational) pair-instability supernovae (see discussion in the next
section). Additionally, GW190517-055101 has the largest $\chi_{\rm eff}$
reported in the second Gravitational-Wave Transient Catalog (GWTC-2) (Abbott
et al., 2021), which can be explained with our models at lower metallicities
(see the second and third rows in Fig. 7), regardless of the assumed
efficiencies of AM transport. Therefore, the BBH progenitor of this event
might have gone through the double-core evolution at low metallicities (e.g.,
$Z<$ 0.1 $\mathrm{Z}_{\odot}$.)

Figure 6: As in Fig. 5, but the color represents merger time
$\mathrm{T}_{\mathrm{merger}}$. The black dots represent the systems whose
merger times are longer than a Hubble time Figure 7: $T_{\rm merger}$ (colored
dots) as a function of $\chi_{\rm eff}$ and $M_{\rm chirp}$. The first row
including three panels corresponds to different initial metallicities (left
panel: 1.0 $\mathrm{Z}_{\odot}$, middle panel: 0.1 $\mathrm{Z}_{\odot}$, right
panel: 0.01 $\mathrm{Z}_{\odot}$) of He-rich stars. Similar to the first row,
but the second row involves inefficient AM transport mechanism, i.e., TS off.
The dashed lines refer to the specific initial orbital periods, marked with
numbers in unit of “days”. The observed high-confidence BBH events (69 BBHs
with false alarm rate $<$ 1 per year) reported from the LVK are shown with the
error bars at 90% credibility in each panel. Black triangles represent BBHs
whose $T_{\rm merger}$ is longer than a Hubble time. Two BBH events are
highlighted, square: GW190517, star: GW190521.

##  4 Conclusions and Discussion

In this work, we first present the Hertzsprung-Russel diagrams of single non-
rotating He-rich stars in a mass range of 5 - 60 $M_{\odot}$ at different
metallicities, evolving from ZAHeMS to the central helium exhaustion. We then
systematically study an alternative formation scenario of BBHs (i.e., the
double-core evolution) by modeling double He-rich stars in various parameter
spaces (metallicity and initial mass of He-rich stars, as well as the orbital
period). Furthermore, we also investigate the impact of the different AM
transport mechanism on the evolution of He-rich stars in different
evolutionary stages, the properties of resulting BBHs at birth, as well as the
merging timescale.

We calculate the baryonic remnant mass following the “delayed” supernova
prescription shown in Fryer et al. (2012), and taking into account the impact
of accretion feedback onto the newly-formed BHs. The upper limit of the BH
mass is around 26, 40 and 58 $M_{\odot}$ at 1.0 $\mathrm{Z}_{\odot}$, 0.1
$\mathrm{Z}_{\odot}$ and 0.01 $\mathrm{Z}_{\odot}$, respectively. We find that
tides for double He-rich stars can only be important when the initial orbital
periods are less than 1.0 day, which is similar to previous studies of a He-
rich star accompanied by a BH/NS (Qin et al., 2018; Bavera et al., 2020;
Fuller & Lu, 2022). We note that the initial metallicity of He-rich stars
should be high for the efficient AM transport to play a significant role in
determining the spin magnitude of the newly-formed BHs, since their
progenitors (massive He-rich He-stars) are more inclined to be slowed down by
stronger winds mass-loss especially when rotating like a solid body. The
$\chi_{\rm eff}$ for BBHs formed through the double-core evolution is not
always high, but it can cover the whole range of BH spin, i.e., from minimum
(0.0) to maximum (1.0), depending on the initial orbital period of the binary
systems. The chirp mass $M_{\rm chirp}$ of BBH is strongly dependent on the
initial metallicity of He-rich stars (e.g., Belczynski et al., 2010; Stevenson
et al., 2017, 2019). More specifically, the chirp mass $M_{\rm chirp}$ of the
BBH from double-core evolution at 1.0 $\mathrm{Z}_{\odot}$ can not be larger
than 26 $M_{\odot}$, regardless of the efficiency of the AM transport within
He-rich stars.

After detailed investigations of the double-core evolution, we would expect
that this channel could predict a certain fraction of BBH populations with
high $\chi_{\rm eff}$ and $M_{\rm chirp}$. More events with the above features
are expected to be captured by the LVK with its improving sensitivity in the
upcoming fourth observing run. The quantitative merger rate from this channel
is beyond the scope of the current work. Therefore we plan to investigate the
quantitative contribution of this channel to the intrinsic BBH population with
the population synthesis study and the impact of different physical processes
on the outcomes in the near future.

The formation of massive He-rich binary stars might not involve the CE phase,
in which the criteria for its occurrence are still under development. Recent
investigations suggest that the BBHs merger rate from the CE channel might be
overestimated in rapid population synthesis studies (e.g., Pavlovskii et al.,
2017; Marchant et al., 2021; Klencki et al., 2021; Gallegos-Garcia et al.,
2021; Olejak et al., 2021). Their studies indicate that stable mass transfer
channel could be a dominant channel for the formation of merging BBHs (e.g.,
Shao & Li, 2022; Briel et al., 2022). van Son et al. (2022a) recently found
that stable mass transfer channel preferentially form BBH systems with more
massive component BH masses. Furthermore, by varying the metallicity-dependent
cosmic star formation history, van Son et al. (2022b) found the variations
affect the slope of the high mass end of the BBH mass distribution, but have a
slight impact on the CE channel. In addition, massive He-rich binary stars
could be formed through the CHE channel. In this channel, the two massive
stars initially evolve in a close orbit and thus have strong chemical mixing
due to strong tides.

Here we briefly summarise some main uncertainties in our binary modeling.
First, stellar wind mass loss is one of key uncertain physical processes in
the evolution of massive stars, which can have a significant impact on the
mass and the spin of resultant BHs. Second, it is unclear whether supernova
kicks (natal kicks) are imparted onto BHs during the core-collapse process.
BHs formed from direct core-collapse of massive stars were considered to
receive no natal kick (Belczynski et al., 2008). Nevertheless, we note that a
recent work by Farr et al. (2011); Tauris (2022) which argued that, rather
than dynamical formation, isolated binary evolution can still explain the
observed BBHs if BHs have spin-axis tossed due to the supernova kicks during
their formation process in the core collapse of massive stars. The stellar
evolution theory predicts a mass “gap” in the BH birth function caused by the
(pulsational) pair-instability supernovae (Fowler & Hoyle, 1964; Rakavy &
Shaviv, 1967; Barkat et al., 1967; Fraley, 1968; Heger et al., 2003), which is
still uncertain and thus plays a critical role in determining the upper limit
of the BH mass below the “gap” (see Woosley & Heger, 2021, and references
therein). The constraints from current observations of BBHs reported from the
LVK are still weak due to a statistically small sample. Therefore, we expect
the sample of BBH events with higher $\chi_{\rm eff}$ and $M_{\rm chirp}$ will
be significantly expanded in the upcoming fourth run, which will be used to
make stronger constraints on the supernova kicks during the formation process
of BHs from massive stars.

###### Acknowledgements.

Y.Q. acknowledges the support from the Doctoral research start-up funding of
Anhui Normal University and from Key Laboratory for Relativistic Astrophysics
in Guangxi University. This work was supported by the National Natural Science
Foundation of China (Grant Nos. 12003002, 12192220, 12192221, 11863003,
12173010) and the Natural Science Foundation of Universities in Anhui Province
(Grant No. KJ2021A0106). G.M. has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 833925, project STAREX). All figures were made
with the free Python module Matplotlib (Hunter, 2007).

## References

  * Abadie et al. (2010) Abadie, J., Abbott, B. P., Abbott, R., et al. 2010, Classical and Quantum Gravity, 27, 173001
  * Abbott et al. (2016a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, Phys. Rev. Lett., 116, 061102
  * Abbott et al. (2016b) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016b, ApJ, 818, L22
  * Abbott et al. (2021) Abbott, R., Abbott, T. D., Abraham, S., et al. 2021, Physical Review X, 11, 021053
  * Abbott et al. (2020) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, Phys. Rev. Lett., 125, 101102
  * Antonini et al. (2016) Antonini, F., Chatterjee, S., Rodriguez, C. L., et al. 2016, ApJ, 816, 65
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Barkat et al. (1967) Barkat, Z., Rakavy, G., & Sack, N. 1967, Phys. Rev. Lett., 18, 379
  * Batta & Ramirez-Ruiz (2019) Batta, A. & Ramirez-Ruiz, E. 2019, arXiv e-prints, arXiv:1904.04835
  * Bavera et al. (2020) Bavera, S. S., Fragos, T., Qin, Y., et al. 2020, A&A, 635, A97
  * Bavera et al. (2021) Bavera, S. S., Fragos, T., Zevin, M., et al. 2021, A&A, 647, A153
  * Belczynski et al. (2010) Belczynski, K., Dominik, M., Bulik, T., et al. 2010, ApJ, 715, L138
  * Belczynski et al. (2016) Belczynski, K., Holz, D. E., Bulik, T., & O’Shaughnessy, R. 2016, Nature, 534, 512
  * Belczynski et al. (2022) Belczynski, K., Romagnolo, A., Olejak, A., et al. 2022, ApJ, 925, 69
  * Belczynski et al. (2007) Belczynski, K., Taam, R. E., Kalogera, V., Rasio, F. A., & Bulik, T. 2007, ApJ, 662, 504
  * Belczynski et al. (2008) Belczynski, K., Taam, R. E., Rantsiou, E., & van der Sluys, M. 2008, ApJ, 682, 474
  * Böhm-Vitense (1958) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * Briel et al. (2022) Briel, M. M., Stevance, H. F., & Eldridge, J. J. 2022, arXiv e-prints, arXiv:2206.13842
  * Broekgaarden et al. (2022) Broekgaarden, F. S., Berger, E., Stevenson, S., et al. 2022, MNRAS[arXiv:2112.05763]
  * Brown (1995) Brown, G. E. 1995, ApJ, 440, 270
  * Callister et al. (2022) Callister, T. A., Miller, S. J., Chatziioannou, K., & Farr, W. M. 2022, ApJ, 937, L13
  * Cantiello et al. (2014) Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., & Paxton, B. 2014, ApJ, 788, 93
  * Chaboyer & Zahn (1992) Chaboyer, B. & Zahn, J. P. 1992, A&A, 253, 173
  * de Mink & Belczynski (2015) de Mink, S. E. & Belczynski, K. 2015, ApJ, 814, 58
  * de Mink & Mandel (2016) de Mink, S. E. & Mandel, I. 2016, MNRAS, 460, 3545
  * den Hartogh et al. (2020) den Hartogh, J. W., Eggenberger, P., & Deheuvels, S. 2020, A&A, 634, L16
  * Dewi et al. (2006) Dewi, J. D. M., Podsiadlowski, P., & Sena, A. 2006, MNRAS, 368, 1742
  * Doctor et al. (2020) Doctor, Z., Wysocki, D., O’Shaughnessy, R., Holz, D. E., & Farr, B. 2020, ApJ, 893, 35
  * Dominik et al. (2015) Dominik, M., Berti, E., O’Shaughnessy, R., et al. 2015, ApJ, 806, 263
  * du Buisson et al. (2020) du Buisson, L., Marchant, P., Podsiadlowski, P., et al. 2020, MNRAS, 499, 5941
  * Eggenberger et al. (2019) Eggenberger, P., den Hartogh, J. W., Buldgen, G., et al. 2019, A&A, 631, L6
  * Eggenberger et al. (2005) Eggenberger, P., Maeder, A., & Meynet, G. 2005, A&A, 440, L9
  * Eggenberger et al. (2012) Eggenberger, P., Montalbán, J., & Miglio, A. 2012, A&A, 544, L4
  * Eggenberger et al. (2022) Eggenberger, P., Moyano, F. D., & den Hartogh, J. W. 2022, A&A, 664, L16
  * Eldridge & Vink (2006) Eldridge, J. J. & Vink, J. S. 2006, A&A, 452, 295
  * Farr et al. (2018) Farr, B., Holz, D. E., & Farr, W. M. 2018, ApJ, 854, L9
  * Farr et al. (2011) Farr, W. M., Kremer, K., Lyutikov, M., & Kalogera, V. 2011, ApJ, 742, 81
  * Farr et al. (2017) Farr, W. M., Stevenson, S., Miller, M. C., et al. 2017, Nature, 548, 426
  * Fishbach & Holz (2020) Fishbach, M. & Holz, D. E. 2020, ApJ, 904, L26
  * Fowler & Hoyle (1964) Fowler, W. A. & Hoyle, F. 1964, ApJS, 9, 201
  * Fragione et al. (2022) Fragione, G., Kocsis, B., Rasio, F. A., & Silk, J. 2022, ApJ, 927, 231
  * Fragos et al. (2022) Fragos, T., Andrews, J. J., Bavera, S. S., et al. 2022, arXiv e-prints, arXiv:2202.05892
  * Fraley (1968) Fraley, G. S. 1968, Ap&SS, 2, 96
  * Fryer (1999) Fryer, C. L. 1999, ApJ, 522, 413
  * Fryer et al. (2012) Fryer, C. L., Belczynski, K., Wiktorowicz, G., et al. 2012, ApJ, 749, 91
  * Fuller & Lu (2022) Fuller, J. & Lu, W. 2022, MNRAS, 511, 3951
  * Fuller et al. (2019) Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485, 3661
  * Gallegos-Garcia et al. (2021) Gallegos-Garcia, M., Berry, C. P. L., Marchant, P., & Kalogera, V. 2021, ApJ, 922, 110
  * Gerosa & Fishbach (2021) Gerosa, D. & Fishbach, M. 2021, Nature Astronomy, 5, 749
  * Giacobbo & Mapelli (2018) Giacobbo, N. & Mapelli, M. 2018, MNRAS, 480, 2011
  * Heger et al. (2003) Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann, D. H. 2003, ApJ, 591, 288
  * Heger & Langer (2000) Heger, A. & Langer, N. 2000, ApJ, 544, 1016
  * Heger et al. (2005) Heger, A., Woosley, S. E., & Spruit, H. C. 2005, ApJ, 626, 350
  * Higgins et al. (2021) Higgins, E. R., Sander, A. A. C., Vink, J. S., & Hirschi, R. 2021, MNRAS, 505, 4874
  * Hu et al. (2022) Hu, R.-C., Zhu, J.-P., Qin, Y., et al. 2022, ApJ, 928, 163
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90
  * Hut (1981) Hut, P. 1981, A&A, 99, 126
  * Hwang et al. (2015) Hwang, J., Lombardi, J. C., J., Rasio, F. A., & Kalogera, V. 2015, ApJ, 806, 135
  * Inayoshi et al. (2017) Inayoshi, K., Hirai, R., Kinugawa, T., & Hotokezaka, K. 2017, MNRAS, 468, 5020
  * Ivanova et al. (2013) Ivanova, N., Justham, S., Chen, X., et al. 2013, A&A Rev., 21, 59
  * Ji et al. (2022) Ji, S., Fuller, J., & Lecoanet, D. 2022, arXiv e-prints, arXiv:2209.08104
  * Kimball et al. (2020) Kimball, C., Talbot, C., Berry, C. P. L., et al. 2020, ApJ, 900, 177
  * Kimball et al. (2021) Kimball, C., Talbot, C., Berry, C. P. L., et al. 2021, ApJ, 915, L35
  * Klencki et al. (2021) Klencki, J., Nelemans, G., Istrate, A. G., & Chruslinska, M. 2021, A&A, 645, A54
  * Langer (1997) Langer, N. 1997, in Astronomical Society of the Pacific Conference Series, Vol. 120, Luminous Blue Variables: Massive Stars in Transition, ed. A. Nota & H. Lamers, 83
  * Langer et al. (1983) Langer, N., Fricke, K. J., & Sugimoto, D. 1983, A&A, 126, 207
  * Maeder & Meynet (2000) Maeder, A. & Meynet, G. 2000, A&A, 361, 159
  * Mandel & Broekgaarden (2022) Mandel, I. & Broekgaarden, F. S. 2022, Living Reviews in Relativity, 25, 1
  * Mandel & de Mink (2016) Mandel, I. & de Mink, S. E. 2016, MNRAS, 458, 2634
  * Mandel & Farmer (2022) Mandel, I. & Farmer, A. 2022, Phys. Rep, 955, 1
  * Mapelli (2020) Mapelli, M. 2020, Frontiers in Astronomy and Space Sciences, 7, 38
  * Mapelli et al. (2021) Mapelli, M., Dall’Amico, M., Bouffanais, Y., et al. 2021, MNRAS, 505, 339
  * Marchant et al. (2016) Marchant, P., Langer, N., Podsiadlowski, P., Tauris, T. M., & Moriya, T. J. 2016, A&A, 588, A50
  * Marchant et al. (2021) Marchant, P., Pappas, K. M. W., Gallegos-Garcia, M., et al. 2021, A&A, 650, A107
  * McKernan et al. (2020) McKernan, B., Ford, K. E. S., & O’Shaughnessy, R. 2020, MNRAS, 498, 4088
  * Olejak & Belczynski (2021) Olejak, A. & Belczynski, K. 2021, ApJ, 921, L2
  * Olejak et al. (2021) Olejak, A., Belczynski, K., & Ivanova, N. 2021, A&A, 651, A100
  * Pavlovskii et al. (2017) Pavlovskii, K., Ivanova, N., Belczynski, K., & Van, K. X. 2017, MNRAS, 465, 2092
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10
  * Peng et al. (2022) Peng, W., Song, H., Meynet, G., et al. 2022, A&A, 657, A116
  * Peters (1964) Peters, P. C. 1964, Physical Review, 136, 1224
  * Phinney (1991) Phinney, E. S. 1991, ApJ, 380, L17
  * Postnov & Yungelson (2014) Postnov, K. A. & Yungelson, L. R. 2014, Living Reviews in Relativity, 17, 3
  * Qin et al. (2018) Qin, Y., Fragos, T., Meynet, G., et al. 2018, A&A, 616, A28
  * Qin et al. (2019a) Qin, Y., Fragos, T., Meynet, G., et al. 2019a, IAU Symposium, 346, 426
  * Qin et al. (2019b) Qin, Y., Marchant, P., Fragos, T., Meynet, G., & Kalogera, V. 2019b, ApJ, 870, L18
  * Qin et al. (2022a) Qin, Y., Shu, X., Yi, S., & Wang, Y.-Z. 2022a, Research in Astronomy and Astrophysics, 22, 035023
  * Qin et al. (2022b) Qin, Y., Wang, Y.-Z., Wu, D.-H., Meynet, G., & Song, H. 2022b, ApJ, 924, 129
  * Rakavy & Shaviv (1967) Rakavy, G. & Shaviv, G. 1967, ApJ, 148, 803
  * Riley et al. (2021) Riley, J., Mandel, I., Marchant, P., et al. 2021, MNRAS, 505, 663
  * Rodriguez et al. (2015) Rodriguez, C. L., Morscher, M., Pattabiraman, B., et al. 2015, Phys. Rev. Lett., 115, 051101
  * Roulet et al. (2021) Roulet, J., Chia, H. S., Olsen, S., et al. 2021, Phys. Rev. D, 104, 083010
  * Saavik Ford & McKernan (2022) Saavik Ford, K. E. & McKernan, B. 2022, MNRAS[arXiv:2109.03212]
  * Safarzadeh et al. (2020) Safarzadeh, M., Hamers, A. S., Loeb, A., & Berger, E. 2020, ApJ, 888, L3
  * Sander et al. (2020) Sander, A. A. C., Vink, J. S., & Hamann, W. R. 2020, MNRAS, 491, 4406
  * Secunda et al. (2019) Secunda, A., Bellovary, J., Mac Low, M.-M., et al. 2019, ApJ, 878, 85
  * Shao & Li (2022) Shao, Y. & Li, X.-D. 2022, ApJ, 930, 26
  * Shenar et al. (2022) Shenar, T., Sana, H., Mahy, L., et al. 2022, Nature Astronomy [arXiv:2207.07675]
  * Song et al. (2016) Song, H. F., Meynet, G., Maeder, A., Ekström, S., & Eggenberger, P. 2016, A&A, 585, A120
  * Spruit (1999) Spruit, H. C. 1999, A&A, 349, 189
  * Spruit (2002) Spruit, H. C. 2002, A&A, 381, 923
  * Stevenson et al. (2019) Stevenson, S., Sampson, M., Powell, J., et al. 2019, ApJ, 882, 121
  * Stevenson et al. (2017) Stevenson, S., Vigna-Gómez, A., Mandel, I., et al. 2017, Nature Communications, 8, 14906
  * Suijs et al. (2008) Suijs, M. P. L., Langer, N., Poelarends, A. J., et al. 2008, A&A, 481, L87
  * Tagawa et al. (2020) Tagawa, H., Haiman, Z., & Kocsis, B. 2020, ApJ, 898, 25
  * Tang et al. (2020) Tang, P. N., Eldridge, J. J., Stanway, E. R., & Bray, J. C. 2020, MNRAS, 493, L6
  * Tanikawa et al. (2022) Tanikawa, A., Yoshida, T., Kinugawa, T., et al. 2022, ApJ, 926, 83
  * Tauris (2022) Tauris, T. M. 2022, ApJ, 938, 66
  * The LIGO Scientific Collaboration et al. (2021a) The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott, R., et al. 2021a, arXiv e-prints, arXiv:2108.01045
  * The LIGO Scientific Collaboration et al. (2021b) The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration, et al. 2021b, arXiv e-prints, arXiv:2111.03606
  * The LIGO Scientific Collaboration et al. (2021c) The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration, et al. 2021c, arXiv e-prints, arXiv:2111.03634
  * Tutukov & Yungelson (1973) Tutukov, A. & Yungelson, L. 1973, Nauchnye Informatsii, 27, 70
  * van den Heuvel et al. (2017) van den Heuvel, E. P. J., Portegies Zwart, S. F., & de Mink, S. E. 2017, MNRAS, 471, 4256
  * van Son et al. (2020) van Son, L. A. C., De Mink, S. E., Broekgaarden, F. S., et al. 2020, ApJ, 897, 100
  * van Son et al. (2022a) van Son, L. A. C., de Mink, S. E., Callister, T., et al. 2022a, ApJ, 931, 17
  * van Son et al. (2022b) van Son, L. A. C., de Mink, S. E., Chruslinska, M., et al. 2022b, arXiv e-prints, arXiv:2209.03385
  * Vigna-Gómez et al. (2018) Vigna-Gómez, A., Neijssel, C. J., Stevenson, S., et al. 2018, MNRAS, 481, 4009
  * Vink & de Koter (2005) Vink, J. S. & de Koter, A. 2005, A&A, 442, 587
  * Vink et al. (2001) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574
  * Vitale et al. (2022) Vitale, S., Biscoveanu, S., & Talbot, C. 2022, arXiv e-prints, arXiv:2209.06978
  * Woosley & Heger (2021) Woosley, S. E. & Heger, A. 2021, ApJ, 912, L31
  * Zahn (1975) Zahn, J. P. 1975, A&A, 41, 329
  * Zahn et al. (2007) Zahn, J. P., Brun, A. S., & Mathis, S. 2007, A&A, 474, 145
  * Zevin & Bavera (2022) Zevin, M. & Bavera, S. S. 2022, ApJ, 933, 86
  * Zevin et al. (2021) Zevin, M., Bavera, S. S., Berry, C. P. L., et al. 2021, ApJ, 910, 152
  * Zevin et al. (2020) Zevin, M., Spera, M., Berry, C. P. L., & Kalogera, V. 2020, ApJ, 899, L1

##  Appendix A Direct core collapse with mass and angular momentum conserved

In this section, we present the results of BBH formation through double-core
evolution channel, assuming that the mass and AM are conserved during the
formation process in the core collapse of He-rich stars. As shown earlier,
tides can be only important for tidal interaction of double He-rich stars if
the initial orbital periods are less 1 days. We first show in Fig. 8 the
evolution of three cases ($M_{\rm ZamsHe}$ = 12, 20, and 40 $M_{\odot}$ for
the same $P_{\rm init.}$ = 0.6 days) from the beginning of core helium burning
to their carbon depletion in the center. We adopt efficient (TS on) and
inefficient (TS off) AM transport within He-rich stars and three initial
metallicities (1.0 $\mathrm{Z}_{\odot}$, 0.1 $\mathrm{Z}_{\odot}$ and 0.01
$\mathrm{Z}_{\odot}$).

We show in the top left panel of Fig. 8 the evolution of BH spin as a function
of the He-rich star mass and its orbital period, under the assumption that He-
rich stars at any time can directly collapse to form BHs without losing mass
and corresponding AM. Let us begin with a case of an 40 $M_{\odot}$ double He-
rich stars, it was efficiently spun up and thus formed a fast-spinning BH at
the beginning of core helium burning (see the star symbol). The orbital
separation slightly expands during the core helium burning, which however
makes BH spin ($\sim$ 0.4 at the middle of core helium burning, see the square
symbol) gradually decrease and end up with being close to zero at the central
helium depletion. We note that there is a negligible discrepancy of BH spin
calculated at between the central helium depletion (the triangle symbol) and
the central carbon depletion (the circle symbol). The other two cases could
form lower-mass binary BHs being slowly rotating in a closer binaries due to
weaker winds mass loss. At lower metallcities, the same binaries will form
faster-spinning BHs in shorter orbits (see middle left panel for 0.1
$\mathrm{Z}_{\odot}$ and bottom left panel for 0.01 $\mathrm{Z}_{\odot}$).
When the inefficient AM transport (TS off) is adopted, we can clearly see the
formed BBHs spinning faster when compared with the same metallicity.

With the same parameter space, we also compute for each binary system the
evolution of the BH spin under different metallicities and efficiencies of AM
transport. We first present results assuming efficient AM transport. As shown
in Fig. 9, all He-rich stars with an initial mass of less than 12 $M_{\odot}$,
at 1.0 $\mathrm{Z}_{\odot}$, form NSs. He-rich stars with initial orbital
period of longer than 1.0 day end up with being non-spinning BHs. We find that
BHs can have moderate spin magnitudes with $P_{\rm init.}$ in a range of 0.3 -
1.0 days, below which fast-spinning BHs are formed. Similar to Fig. 9, we can
see for lower metallicities (see Fig. 9 and Fig. 10) that the formed BHs with
spins decreasing as the orbit slowly expands. We then show the results with
different metallicities of the inefficient AM transport in Fig. 12, Fig. 13
and Fig. 14. The mass and spin of the newly-formed BHs calculated using direct
core-collapse with mass and AM conserved are slightly larger when compared
with those by taking into account the accretion feedback during core-collapse
modeling (see details in Batta & Ramirez-Ruiz 2019).

Figure 8: Evolution of the spin parameter $\mathrm{a}_{\star}$ as a function
of the orbital period and mass of He-rich stars at different metallicities and
AM transport mechanisms. Top left panel: TS on and 1.0 $\mathrm{Z}_{\odot}$,
middle left panel: TS on and 0.1 $\mathrm{Z}_{\odot}$, bottom left panel: TS
on and 0.01 $\mathrm{Z}_{\odot}$; Top right panel: TS off and 1.0
$\mathrm{Z}_{\odot}$, middle right panel: TS off and 0.1 $\mathrm{Z}_{\odot}$,
bottom right panel: TS off and 0.01 $\mathrm{Z}_{\odot}$. The spin
$\mathrm{a}_{\star}$ at different evolutionary stages are marked with symbols,
star: beginning of core He burning, square: middle of core He burning,
triangle: central He depletion, circle: central carbon depletion. Figure 9:
Evolution of the spin parameter $\mathrm{a}_{\star}$ (the color bar) as a
function of the orbital period and mass of He-rich stars with TS included at
solar metallicity. The colored lines linking the two symbols show the
evolution of the binary. The color along the line gives BH spins
$\mathrm{a}_{\star}$ along the evolution (from the ZAHeMS to the central
carbon depletion), assuming that their progenitors (He-rich stars) directly
collapse to form BHs with mass and AM conserved. Figure 10: As in Fig . 9, but
for the metallicity $\mathrm{Z}=0.1\ \mathrm{Z}_{\odot}$. Figure 11: As in Fig
. 9, but for the metallicity $\mathrm{Z}=0.01\ \mathrm{Z}_{\odot}$. Figure 12:
As in Fig . 9, but without TS included. Figure 13: As in Fig . 12, but for the
metallicity $\mathrm{Z}=0.1\ \mathrm{Z}_{\odot}$. Figure 14: As in Fig . 12,
but for the metallicity $\mathrm{Z}=0.01\ \mathrm{Z}_{\odot}$.