# Black Hole Ultracompact X-Ray Binaries as Galactic Low-frequency
Gravitational Wave Sources: the He Star Channel

Ke Qin School of Science, Qingdao University of Technology, Qingdao 266525,
People’s Republic of China<EMAIL_ADDRESS>School of Physics, Zhengzhou
University, Zhengzhou 450001, People’s Republic of China  Kun Xu School of
Science, Qingdao University of Technology, Qingdao 266525, People’s Republic
of China<EMAIL_ADDRESS>Dong-Dong Liu Key Laboratory for the Structure
and Evolution of Celestial Objects, Yunnan Observatories, Chinese Academy of
Sciences, Kunming 650216, People’s Republic of China  Long Jiang School of
Science, Qingdao University of Technology, Qingdao 266525, People’s Republic
of China<EMAIL_ADDRESS>School of Physics and Electrical Information,
Shangqiu Normal University, Shangqiu 476000, People’s Republic of China  Bo
Wang Key Laboratory for the Structure and Evolution of Celestial Objects,
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, People’s
Republic of China  Wen-Cong Chen School of Science, Qingdao University of
Technology, Qingdao 266525, People’s Republic of China<EMAIL_ADDRESS>School of Physics and Electrical Information, Shangqiu Normal University,
Shangqiu 476000, People’s Republic of China

###### Abstract

Black hole (BH) ultracompact X-ray binaries (UCXBs) are potential Galactic
low-frequency gravitational wave (GW) sources. As an alternative channel, BH
UCXBs can evolve from BH+He star binaries. In this work, we perform a detailed
stellar evolution model for the formation and evolution of BH UCXBs evolving
from the He star channel to diagnose their detectability as low-frequency GW
sources. Our calculations found that some nascent BH+He star binaries after
the common-envelope (CE) phase could evolve into UCXB-LISA sources with a
maximum GW frequency of $\sim 5~{}\rm mHz$, which can be detected in a
distance of 10 kpc (or 100 kpc). Once BH+He star systems become UCXBs through
mass transfer, they would emit X-ray luminosities of $\sim 10^{38}~{}\rm
erg\,s^{-1}$, making them ideal multimessenger objects. If the initial He-star
masses are $\geq 0.7M_{\odot}$, those systems are likely to experience two
Roche lobe overflows, and the X-ray luminosity can reach a maximum of
$3.5\times 10^{39}~{}\rm erg\,s^{-1}$ in the second mass-transfer stage. The
initial He-star masses and initial orbital periods of progenitors of Galactic
BH UCXB-LISA sources are in the range of 0.32-2.9 $M_{\odot}$ and 0.02-0.19
days, respectively. Nearly all BH+He star binaries in the above parameter
space can evolve into GW sources whose chirp masses can be accurately
measured. Employing a population synthesis simulation, we predict the
birthrate and detection number of Galactic BH UCXB-LISA source evolving from
the He star channel are $R=2.2\times 10^{-6}~{}\rm yr^{-1}$ and 33 for an
optimistic CE parameter, respectively.

binaries: close – galaxies: helium – stars: black holes – star: evolution –
gravitational waves

##  1 Introduction

Ultracompact X-ray binaries (UCXBs) are low-mass X-ray binaries (LMXBs) with
ultra-short orbital periods (usually less than 60 minutes), consisting of an
accreting compact object and a hydrogen-poor donor star (Nelson et al., 1986;
Nelemans & Jonker, 2010). UCXBs can help us to understand the angular momentum
loss mechanisms (Ma & Li, 2009; van Haaften et al., 2012a, b), the common-
envelope (CE) evolution (Zhu et al., 2012), and the accretion process of
compact objects (Lin & Yu, 2018), thus they are ideal laboratories for testing
stellar and binary evolutionary theory. Furthermore, UCXBs can emit continuous
low-frequency gravitational wave (GW) signals, which could be detected by the
space-borne GW detectors such as the Laser Interferometer Space Antenna (LISA;
Amaro-Seoane et al., 2017, 2023), TianQin (Luo et al., 2016; Wang et al.,
2019; Huang et al., 2020), and Taiji (Ruan et al., 2020). Therefore, UCXBs are
ideal objects pursuing multi-messenger investigations (Chen et al., 2020).

Thus far, the number of confirmed UCXBs (with an accurately measured orbital
period $\leq 80$ minutes) and candidates is around 45 (Armas Padilla et al.,
2023). Based on the accurate detection of orbital periods, 20 sources were
identified to be UCXBs in high confidence, which includes 11 persistent
sources and 9 transient sources (in’t Zand et al., 2007; Liu et al., 2007;
Heinke et al., 2013; Cartwright et al., 2013; Pietrukowicz et al., 2019; Coti
Zelati et al., 2021; Peng & Shen, 2021; Armas Padilla et al., 2023). Among
confirmed UCXBs, 19 sources were discovered to include a neutron star (NS). At
present, there exist two black hole (BH) UCXB candidates. The first one is the
luminous X-ray source X9 in globular cluster 47 Tucanae, which was thought to
be a BH accreting from a white dwarf (WD) in a close orbit (Miller-Jones et
al., 2015; Bahramian et al., 2017; Church et al., 2017; Tudor et al., 2018).
Recently, Moutard et al. (2023) found some evidences that the compact object
in UCXB 4U 0614+091 may be a BH by the simultaneous NICER and NuSTAR
observations. Chen et al. (2020) estimated that the main-sequence (MS) channel
can form 60-80 NS UCXB-LISA sources in the Galaxy. However, LISA can only
detect $\sim 4$ BH UCXBs evolving from the MS channel (Qin et al., 2023). If
the ratio of the numbers between NS and BH UCXB-LISA sources is similar to
that of the identified numbers between NS and BH UCXBs, the number of
confirmed BH UCXBs evolving from the MS channel is $\sim 1$, which is
comparable to the present observation. Therefore, it is difficult to observe
BH UCXBs.

Because of the compactness of UCXBs, their donor stars are generally thought
to be partially or completely degenerate stars such as WDs or helium (He)
stars (Rappaport et al., 1982; Podsiadlowski et al., 2002; Deloye & Bildsten,
2003). Optical spectroscopic analysis of UCXBs can help us to identify the
properties of donor stars (Nelemans et al., 2004, 2006). Employed X-ray,
ultraviolet, and optical spectroscopy, chemical elements of accretion disks in
some UCXBs may include He, C, N, O, Ne, and Si (Nelemans et al., 2010). Such a
diversity of chemical compositions implied those donor stars should evolve to
different nuclear-burning stages and interior degeneracy, which requires
different evolutionary models to account for the formation of UCXBs (Sengar et
al., 2017). It is generally thought that UCXBs in the Galactic field evolved
from the following three channels: the WD channel, the evolved MS star
channel, and the He star channel (Postnov & Yungelson, 2006; Nelemans et al.,
2010).

In the first channel, the progenitors of UCXBs are compact binaries consisting
of a NS/BH and a low-mass WD, in which GW radiation drives mass transfer
(Belczynski & Taam, 2004; van Haaften et al., 2012a; Jiang et al., 2017; Yu et
al., 2021). Sengar et al. (2017) performed complete numerical models for the
formation of UCXBs evolved from a stable mass transfer from a WD to an
accreting NS, and found that the WD channel can reproduce the observed
properties of some UCXBs with high He abundances. Using an improved mass
transfer hydrodynamics model, Bobrick et al. (2017) argued that only NS+He WD
binaries with a donor-star mass less than 0.2 $M_{\odot}$ could form UCXBs
through a stable mass transfer. van Haaften et al. (2012a) also found that the
donor-star masses have to be less than 0.4 $M_{\odot}$ to form UCXBs from
NS+CO WD binaries.

The second channel originates from a NS/BH accreting mass from a MS donor star
that fills its Roche lobe. If the MS star starts mass transfer late enough and
the orbital-angular-momentum loss by magnetic braking is efficient, the system
would evolve toward a UCXB. In this channel, UCXBs generally evolve from
BH/NS+MS binaries whose initial orbital periods are shorter than the
bifurcation period (van der Sluys et al., 2005; Sengar et al., 2017; Chen et
al., 2020; Qin et al., 2023). In the UCXB stage, the donor star is most likely
to evolve into a WD. It is noteworthy that mass transfer never ceases in the
whole evolutionary stage except for those systems with a NS and a fine-tuning
initial orbital period (Chen et al., 2020; Qin et al., 2023).

In the He star channel, the direct progenitors of UCXBs are BH/NS+He star
binaries. It is generally believed that BH/NS+He star binaries are the
evolutionary products of high-mass X-ray binaries, in which the hydrogen
envelope of the He-star progenitors are fully ejected in the CE stage (Quast
et al., 2019; Abdusalam et al., 2020; Götberg et al., 2020; Wang et al.,
2021). Due to the close orbit, the GW radiation dominates the orbital
evolution of the nascent BH/NS+He star system and triggers a mass transfer.
The BH/NS accretes He-rich materials from the He star that fills the Roche
lobe and the system appears as an UCXB (Savonije et al., 1986; Dewi et al.,
2002; Heinke et al., 2013; Wang et al., 2021). Employing detailed stellar
evolution models, Jiang et al. (2021, 2023) found that NS/BH+He star binaries
can evolve into double NS or BH+NS systems through stripped supernova
explosions, and these double NS and BH+NS systems would evolve toward high-
frequency GW events that can be discovered by aLIGO. Binary population
synthesis (BPS) simulations indicated that binary systems consisting of a
NS/BH and a naked He star can account for $\sim 50\%-80\%$ UCXBs (Zhu et al.,
2012). By a population-synthesis simulation, Lommen et al. (2005) proposed
there are $\sim 200$ BH+He star binaries and $\sim 540$ NS+He star binaries in
the Milky Way.

Compared with those dim WDs, He stars are close to MS stars in the
Hertzsprung-Russell (H-R) diagram (Gräfener et al., 2002; Geier et al., 2010;
Götberg et al., 2018). Therefore, BH+He star systems can be observed in the
detached stage, and provide more evolutionary details such as the mass-
transfer efficiency (Packet, 1981; de Mink et al., 2007), and the stellar wind
(Puls et al., 2008; Smith, 2014; Vink, 2017). In general, low-mass He stars
are referred to as the hot subdwarfs (Han et al., 2007), while high-mass He
stars are called Wolf-Rayet stars (Gräfener et al., 2002). At present, several
sources consisting of a compact star and a He star have been reported. For
example, LS V+22 25 (LB-1) was proposed to contain a stellar mass ($\approx
8M_{\odot}$) BH and a low-mass (0.5-1.7 $M_{\odot}$) stripped He star
(Eldridge et al., 2020; Yungelson et al., 2020). PG 1432+159, HE 0532-4503, PG
1232-136, and PG 1743+477 were also thought to be candidates containing a BH
and a He star with a very thin hydrogen envelope (Geier et al., 2010).
However, no X-ray emissions from these detached binaries were detected.
HD49798 (Mereghetti et al., 2009), M101 ULX-1 (Liu et al., 2013), IC 10 X-1,
and NGC 300 X-1 are most likely X-ray sources including He stars, in which the
X-ray emissions originated from a NS/BH accreting from the stellar winds of He
stars. Especially for the latter two sources, Tutukov & Fedorova (2016)
proposed that the accretion disk and the limited X-ray luminosity of
$10^{38}~{}\rm erg\,s^{-1}$ can be interpreted by a stellar-mass BH accreting
from a Wolf-Rayet companion through the wind-Roche lobe overflow (RLOF)
mechanism (see also El Mellah et al., 2019a, b). However, both the stellar
wind accretion and the RLOF models can explain the observations of galactic
strong X-ray sources Cygnus X-3 and SS 433 (Lommen et al., 2005), which are
also thought to consist of a compact object and a mass-transferring He star.
As a consequence, binary systems consisting of a compact object and a He star
are potential galactic strong X-ray sources. Furthermore, they are also
promising low-frequency GW sources in the Galaxy (Wang et al., 2021; Liu et
al., 2023). Therefore, it is of great significance to study the evolution of
BH binaries including He stars.

In this paper, we perform a detailed stellar evolution model for a large
number of BH+He star binaries to investigate whether or not they can evolve
toward UCXBs and low-frequency GW sources in the Milky Way. In Section 2, we
describe the binary evolution code. Some detailed simulated results are shown
in Section 3. The discussion and conclusion are presented in Sections 4, and
5, respectively.

##  2 Binary evolution model

The Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al.,
2011, 2013, 2015, 2018, 2019) is a popular code in the stellar and binary
evolution field. In this work, we use a binary updated version (r12115) of the
MESA to model the formation and evolution of BH UCXBs. We assume that the CE
stage would produce many detached BH-He star binaries, which are taken to be
the evolutionary beginning point of detailed stellar evolution. For
simplicity, the BH is considered a point mass with an initial mass of $M_{\rm
BH,i}=8~{}M_{\odot}$. The code only models the nuclear synthesis of the He
star and the orbital evolution of the binary. Therefore, the evolutionary
fates of BH-He star binaries depend on the initial He-star mass ($M_{\rm
He,i}$) and the initial orbital period ($P_{\rm orb,i}$) for a given input
physics. We build a zero age main sequence He star model with solar
metallicity (Wong & Schwab, 2019), which consists of 98% helium and 2%
metallicity (i.e. $Y=0.98,Z=0.02$). The lowest He-star mass is taken to be
0.32 $M_{\odot}$, under which the center He burning would extinguish (Iben,
1990; Han et al., 2002; Yungelson, 2008). For each binary system, we run the
MESA code until the time step reaches a minimum time-step limit or the stellar
age is greater than the Hubble time (14 Gyr).

For the wind setting of the He star, the ”Dutch” options with a scaling factor
of 0.8 are used in the schemes including $hot_{-}wind_{-}scheme$,
$cool_{-}wind_{-}RGB_{-}scheme$, and $cool_{-}wind_{-}AGB_{-}scheme$ (Glebbeek
et al., 2009). We use Type 2 opacities for extra C/O burning during and after
He burning. Furthermore, the time step options with
$mesh_{-}delta_{-}coeff=1.0$ and $varcontrol_{-}target=10^{-3}$ are adopted.
Our inlists are available at doi:10.5281/zenodo.10075413.

The fast wind of the He star is thought to carry away its specific orbital-
angular momentum. The wind-accretion efficiency of the BH via Bondi-Hoyle-like
accretion is low (Tauris et al., 2017). For example, the wind-accretion
efficiency is $\sim 0.003$ for a binary with a $8.8~{}M_{\odot}$ BH and a
$6.0~{}M_{\odot}$ He star (Jiang et al., 2023). Therefore, we ignore the wind
accretion in the whole evolutionary process. Once the He star fills its Roche
lobe, the mass transfer initiates from the donor star to the BH at a rate of
$\dot{M}_{\rm tr}$. During the mass transfer, we adopt the accretion
efficiency scheme given by Tauris & van den Heuvel (2006), i.e.
$\alpha=0,\beta=0.5$, and $\delta=0$, here $\alpha$, $\beta$, and $\delta$
represent the fractions of mass loss from the He star in the form of fast
wind, the ejected mass from the vicinity of the BH and from a circumbinary co-
planar toroid, respectively. Therefore, the accretion rate of the BH is
$\dot{M}_{\rm acc}=(1-\beta)\dot{M}_{\rm tr}=0.5~{}\dot{M}_{\rm tr}$.

The mass-growth rate of the BH is limited by the Eddington accretion rate as

$\dot{M}_{\rm Edd}=\frac{4\pi GM_{\rm BH}}{\kappa c\eta},$ (1)

where $G$ is the gravitational constant, $c$ is the speed of light in vacuo,
$M_{\rm BH}$ is the mass of the BH, $\kappa=0.2(1+X)$ is the Thompson-
scattering opacity of electrons ($X$ is the hydrogen abundance of the
transferred material, and $X=0$ for a He donor star); $\eta=1-\sqrt{1-(M_{\rm
BH}/3M_{\rm BH,0})^{2}}$ (for $M_{\rm BH}\leq\sqrt{6}M_{\rm BH,0}$, $M_{\rm
BH,0}$ is the initial mass of the BH) is the energy conversion efficiency of
the accreting BH (Podsiadlowski et al., 2003). Therefore, the mass-growth rate
of the accreting BH is $\dot{M}_{\rm BH}={\rm min}(0.5~{}\dot{M}_{\rm
tr},\dot{M}_{\rm Edd})$. The excess materials in unit time ($\dot{M}_{\rm
tr}-\dot{M}_{\rm BH}$) are thought to be ejected at the vicinity of the BH in
the form of isotropic winds, carrying away the specific orbital angular
momentum of the BH. Thus, the angular-momentum-loss rate due to isotropic
winds can be written as

$\dot{J}_{\rm iso}=-\frac{2\pi a^{2}M_{\rm He}^{2}(\dot{M}_{\rm
tr}-\dot{M}_{\rm BH})}{(M_{\rm BH}+M_{\rm He})^{2}P_{\rm orb}},$ (2)

where $a$ is the orbital separation of the binary, $M_{\rm He}$ is the mass of
the He star, and $P_{\rm orb}$ is the orbital period. In our model, orbital
angular momentum loss via GW radiation and mass loss (fast wind and isotropic
wind) are included.

During the inspiral of two components in BH binaries, the change of the mass
quadrupole produces low-frequency GW signals with a frequency of $f_{\rm
gw}=2/P_{\rm orb}$. When the systems evolve into a close orbit, the emitting
GW signals may be detected by space-borne GW detectors such as LISA. For a 4
years LISA mission, the GW characteristic strain of our simulated BH binaries
is given by (Chen, 2020)

$h_{\rm c}\approx 3.75\times 10^{-20}\left(\frac{f_{\rm gw}}{0.001~{}\rm
Hz}\right)^{7/6}\left(\frac{\mathcal{M}}{1~{}M_{\odot}}\right)^{5/3}\left(\frac{10~{}\rm
kpc}{d}\right).$ (3)

$d$ is the distance of the GW sources. For simplicity, the chirp mass
$\mathcal{M}$ can be expressed as

$\mathcal{M}=\frac{(M_{\rm BH}M_{\rm He})^{3/5}}{(M_{\rm BH}+M_{\rm
He})^{1/5}}.$ (4)

In the numerical calculation, the corresponding binaries are thought to be
LISA sources if the calculated characteristic strain exceeds the LISA
sensitivity curve given by Robson et al. (2019).

It is worth noting that the chirp mass in Equation (4) should be applied in a
detached system that its orbital decay is entirely caused by GW radiation. For
BH UCXBs, mass transfer is inevitable to influence their orbital evolution.
However, the mass transfer in compact BH binaries is driven by GW radiation
and goes along a timescale close to that of GW radiation (van Haaften et al.,
2012a, b). Therefore, the estimation of the chirp mass in Equation (4) remains
reliable in semi-detached BH binaries.

##  3 Simulation result

For a fixed initial BH mass and a given input physics, the evolutionary fates
of BH+He star binaries depend on $P_{\rm orb,i}$ and $M_{\rm He,i}$. As
evolutionary examples, we model the evolution of 10 BH+He star binaries with
different $M_{\rm He,i}$ and $P_{\rm orb,i}$, which divide into three groups
as follows: (1) group 1 with $M_{\rm He,i}$ = 0.6 $M_{\odot}$, $P_{\rm orb,i}$
= 0.03, 0.06, 0.09, 0.11 days; (2) group 2 with $M_{\rm He,i}$ = 0.32, 0.4,
0.6, 0.8 $M_{\odot}$, $P_{\rm orb,i}$ = 0.06 days. (3) group 3 with $M_{\rm
He,i}=1.2,1.8,2.8~{}M_{\odot}$, $P_{\rm orb,i}=0.06$ days. Groups 1 and 2 only
experience one RLOF, while there exist two or more RLOFs for group 3. Some
main evolutionary parameters of three groups are listed in Table 1.

###  3.1 Orbital evolution

Figure 1 shows the evolution of orbital periods with the stellar age for BH+He
star binaries in three groups. Because of small donor-star masses, the donor
stars in groups 1 and 2 fill their Roche lobes until their orbital periods are
in the range of 13 to 48 minutes, thus these systems appear as ultracompact
detached binaries in a long timescale. In group 1, those systems with $P_{\rm
orb,i}=0.03,0.06,0.09~{}\rm days$ can firstly be detected by LISA as low-
frequency GW sources at a distance of 10 kpc, and 100 kpc (these two distances
are the typical distances of the sources in the Galaxy and the Large/Small
Magellanic Cloud)(the system with $P_{\rm orb,i}=0.03~{}\rm days$ is visible
by LISA at a distance of 10 kpc at the beginning of binary evolution), then
experience RLOF and become UCXBs. The system with $P_{\rm orb,i}=0.11~{}\rm
days$ initiates the mass transfer after it is visible by LISA at a distance of
10 kpc, then it appears a UCXB that can be detected by LISA at a distance of
10 kpc, and 100 kpc (i.e. UCXB-LISA source).

In group 2, the system with $M_{\rm He,i}=0.8~{}M_{\odot}$ is visible by LISA
in a distance of 10 kpc at the beginning of binary evolution, then it fills
its Roche lobe and appears as a UCXB-LISA source. Subsequently, continuous
orbital shrinkage makes it a strong GW source that can be detected by LISA at
a distance of 100 kpc. The other three systems with low donor-star masses
first appear as low-frequency GW sources that can be detected by LISA at a
distance of 10 kpc and 100 kpc, then evolve into UCXB-LISA sources. Because of
the short orbital periods, the GW radiation drives the orbital periods of the
systems in groups 1 and 2 to continuously decrease to a minimum of
$6-12.4~{}\rm minutes$, which are very similar to the minimum orbital periods
($8-10~{}\rm minutes$) obtained by Wang et al. (2021) in NS+He star systems.
Most evolutionary tracks of groups 1 and 2 emerge ”knee” features, which are
consistent with the positions that the RLOF starts. This phenomenon originates
from the orbital expansion caused by the mass transfer from the less massive
He star to the more massive BH, which dilutes the orbital decay due to GW
radiation.

In group 3, three systems first appear as low-frequency GW sources that can be
detected by LISA at a distance of 10 kpc, then begin case BA mass transfer at
the first solid circles. The orbital period of the system with $M_{\rm
He,i}=1.2~{}M_{\odot}$ continuously decreases after the mass transfer.
However, the orbits of the other two systems with $M_{\rm He,i}=1.8$, and
$2.8~{}M_{\odot}$ show a widening tendency. In theory, the orbit should widen
when the mass is transferred from the less massive donor star to the more
massive BH. In contrast, GW radiation causes the orbit to shrink. The
evolutionary fate of the orbit would depend on the competition between the
mass transfer and GW radiation. The orbital shrinkage of the system with
$M_{\rm He,i}=1.2~{}M_{\odot}$ originates from a low mass-transfer rate (see
also Figure 2), which can not conquer the orbital decay caused by GW
radiation. After the case BA mass transfer ceases, three systems start a case
BB mass transfer soon (the solid circles are approximately in the same
positions as those open circles, and the timescales between these two circles
are 0.346, 0.389 and 0.340 Myr for $M_{\rm He,i}=1.2,1.8$, and 2.8
$M_{\odot}$, respectively). Subsequently, two systems $M_{\rm He,i}=1.8$, and
$2.8~{}M_{\odot}$ can not be detected by LISA at a distance of 10 kpc due to a
further expansion of the orbits. Once case BB mass transfer ends, the
continuous orbital shrinkage by GW radiation causes the system with $M_{\rm
He,i}=1.8~{}M_{\odot}$ to evolve toward LISA sources that can be detected at a
distance of 10 kpc, and 100 kpc. The evolution of the systems with $M_{\rm
He,i}=1,2$, and $2.8~{}M_{\odot}$ stops due to a numerical difficulty after
the case BB mass transfer ceases.

Figure 1: Evolution of BH+He star binaries in the orbital period vs. stellar
age diagram for groups 1, 2, and 3. The solid circles and open circles denote
the onset and end of mass transfer, respectively. The solid triangles and
stars represent the onset that BH binaries can be detected by LISA at
distances of $d=10$ and 100 kpc, respectively. BH binaries can not be detected
by LISA at a distance of 10 kpc at the open triangles. The horizontal dashed
lines represent the threshold period (90 minutes) of UCXBs.

Figure 2: Evolution of mass-transfer rate of BH binaries in groups 1, 2, and 3
in the mass-transfer rate vs. mass-transfer timescale diagram. $t$, and
$t_{\rm rlof}$ represent the stellar age and the time when the donor star
begins RLOF, respectively.

###  3.2 Evolution of mass-transfer rates

Figure 2 plots the evolution of the mass-transfer rates of BH X-ray binaries
in three groups. At $t_{\rm rlof}\approx 0.007-54.9~{}\rm Myr$ (see also Table
1), the He stars fill their Roche lobes. The mass transfer begins early for
those massive He stars. Because of a positive correlation between the mass and
the radius ($R_{\rm He}\propto M_{\rm He}^{0.654}$, see also Tauris & van den
Heuvel, 2006) for zero-age MS He stars, the more massive He star is easy to
fill its Roche lobe.

For low-mass He stars with $M_{\rm He,i}\leq 0.8~{}M_{\odot}$, GW radiation
dominates the orbital evolution in the early mass-transfer stage. Due to the
shrinkage of the orbit, the GW-radiation timescale decreases, thus the mass-
transfer rate slowly enhances. After the mass transfer dominates the orbital
evolution, the orbital period achieves a minimum, at which the mass-transfer
rate emerges a maximum ($10^{-7}-10^{-6}~{}M_{\odot}\,\rm yr^{-1}$, Yungelson,
2008). Subsequently, the He star evolves to more degenerate, and the
correlation between the mass and the radius is negative, resulting in a
decreasing mass-transfer rate (Avila-Reese, 1993; Yungelson, 2008). For high-
mass He stars with $M_{\rm He,i}\geq 1.2~{}M_{\odot}$, mass transfer dominates
the orbital evolution due to high mass-transfer rates. The orbital periods
increase or slowly decrease, producing decreasing mass-transfer rates.

Three BH+He star binaries in group 3 experience mass transfer twice. The first
mass transfer last a relatively long timescale ($1.6-5.8~{}\rm Myr$) at a rate
of $\sim 10^{-7}~{}\dot{M}_{\odot}\rm\,yr^{-1}$, which increases with the
increase of $M_{\rm He,i}$. As the He abundance in the core drops below 0.1,
it develops a carbon-oxygen (CO) core. Due to the contraction of the He star,
the binary gradually becomes a detached system, and the first mass transfer
ceases. After the core He is exhausted, the He star begins the He-shell
burning. With the continuous growth of CO core, the He star begins to expand
and initiates the second mass transfer. The mass-transfer rate in the second
stage is significantly higher than that of the first mass-transfer stage, and
the duration is shorter. A high donor-star mass tends to produce a high mass-
transfer rate and a short mass-transfer duration. For $M_{\rm He,i}$ = 1.8 and
2.8 $M_{\odot}$, the second mass-transfer rate can exceed
$10^{-6}~{}\dot{M}_{\odot}\rm\,yr^{-1}$. The orbital-expand effect caused by
such a high $\dot{M}_{\rm tr}$ conquers the orbital-shrinkage effect caused by
GW radiation, and the binary orbits begin to widen (see also Figure 1), then
the mass-transfer rate continuously decreases. The whole burning He-shell is
almost stripped in the second mass-transfer stage, and the remaining CO core
becomes highly degenerate. Because of a negative correlation between the mass
and the radius of a degenerate star, the binary detaches again. In fact, the
BH+He star systems in group 3 may undergo a third RLOF, in which the CO WD
fills the Roche lobe and triggers a mass transfer. Due to the limitation of
the minimum time step, the duration of the third mass transfer is too short
($\leq$ 1 yr) to show in Figure 2.

It is noteworthy that the BH+He star system with $M_{\rm
He,i}=2.8~{}M_{\odot}$ emerges a transient ultra-high mass-transfer rate
($\sim 10^{-5}~{}\dot{M}_{\odot}\rm\,yr^{-1}$) in the first mass-transfer
stage. The mass transfer proceeds in a thermal time-scale (Ergma & Fedorova,
1990; Yungelson, 2008), thus the mass-transfer rate is very high in this
stage, which is much higher than $\dot{M}_{\rm Edd}$. With a rapid decrease of
the convective-core mass, the He star rapidly contracts until it re-reaches
thermodynamic equilibrium (Dewi et al., 2002). Subsequently, the mass transfer
will continue steadily on the nuclear timescale.

###  3.3 Evolution of X-ray luminosities

Compared with detached binaries, BH UCXBs are ideal multimessenger sources for
the detection in both GW and electromagnetic wave bands. X-ray observations of
these GW sources could reveal more information about the history of binary
evolution (Madhusudhan et al., 2008; Patruno & Zampieri, 2008), and provide
some constraints on the nature of the companion, the structure of accretion
disk, the relatively accurate position (Liu et al., 2013; Motch et al., 2014;
Yao & Feng, 2019), and so on. In the mass transfer phase, the X-ray luminosity
of the accretion disk surrounding a BH can be calculated by (Köding et al.,
2002)

$L_{X}=\left\\{\begin{array}[]{cl}\epsilon\dot{M}_{\rm
acc}c^{2},\quad\dot{M}_{\rm crit}<\dot{M}_{\rm acc}\leq\dot{M}_{\rm Edd}\\\
\epsilon\left(\frac{\dot{M}_{\rm acc}}{\dot{M}_{\rm crit}}\right)\dot{M}_{\rm
acc}c^{2},\quad\dot{M}_{\rm acc}<\dot{M}_{\rm crit}\\\ \end{array}\right.$ (5)

where $\epsilon$ is the radiation efficiency of the accretion disk,
$\dot{M}_{\rm crit}$ is the critical accretion rate depending on the
transition between the low/hard and high/soft states. In the calculation, we
take $\epsilon=0.1$, and $\dot{M}_{\rm crit}=10^{-9}~{}M_{\odot}\,\rm yr^{-1}$
(Narayan & Yi, 1995).

Figure 3: Evolution of X-ray luminosities of BH binaries in groups 1, 2, and
3. $t$, and $t_{\rm rlof}$ represent the stellar age and the time when the
donor star fills its Roche lobe, respectively. BH UCXBs will not be detected
by LISA at a distance of 10 kpc after the open triangles.

Figure 3 depicts the evolution of X-ray luminosities of BH UCXBs. It is
noteworthy that BH UCXBs evolved from the He star channel produce relatively
high X-ray luminosities of $\sim 10^{38-39}~{}\rm erg\,s^{-1}$, which is $2-5$
orders of magnitude higher than those in the MS channel ($\sim
10^{33-36}~{}\rm erg\,s^{-1}$, see also Qin et al., 2023). However, the
timescales of BH UCXBs evolved from the He star channel are $\sim 1-10~{}\rm
Myr$, which is much shorter than those ($\sim 50-900~{}\rm Myr$) in the MS
channel. All systems in groups 1 and 2 are visible LISA sources at a distance
of 10 kpc or 100 kpc during the UCXB stage. Two systems with high donor-star
masses ($M_{\rm He,i}$ = 1.8 and 2.8 $M_{\odot}$) in group 3 can not be
detected by LISA in the later stage of the second mass-transfer phase because
of a rapid orbital expansion. When the binaries evolve to the minimum orbital
period, the mass-transfer rates reach the maxima, and the X-ray luminosities
emerge peaks ($\sim 10^{39}~{}\rm erg\,s^{-1}$). For a high $M_{\rm He,i}$, BH
UCXBs tend to become so-called ultraluminous X-ray sources (ULXs, $L_{X}\geq
10^{39}~{}\rm erg\,s^{-1}$) (Feng & Soria, 2011; Kaaret et al., 2017), in
which the maximum X-ray luminosity is $3.5\times 10^{39}~{}\rm erg\,s^{-1}$.
Recently, Zhou et al. (2023) found the first smoking gun evidence for the
existence of He donor star in ULXs by Very Large Telescope Multi Unit
Spectroscopic Explorer observations. Similarly, the population synthesis study
also indicated that NS X-ray binaries containing the He stars can account for
a large part of ULXs in Milky Way-like galaxies (Shao et al., 2019).

###  3.4 Detection of GW signals

Since the Laser Interferometer Gravitational-Wave Observatory (LIGO) first
detected the high-frequency GW signal from the double BH merger event GW150914
(Abbott et al., 2016), the detection of GW opens a new window to understand
the distant universe. GW signals provide us with more useful information about
stellar and binary evolution. Figure 4 plots the evolution of BH+He star
systems in the characteristic stain versus GW frequency diagram. Because of
the same chirp mass, the evolutionary tracks of the four systems in group 1
overlap before RLOF. Due to short initial orbital periods, the nascent BH+He
star system with $P_{\rm orb,i}=0.03,$ and 0.06 days can be instantly detected
by LISA and Taiji at a distance of 10 kpc after the CE stage. It is impossible
to form detached BH+WD systems via isolated binary evolution (Qin et al.,
2023). Therefore, those sources with chirp masses similar to those of BH+He
star systems should be the post-CE systems, and provide indirect evidence of
CE evolutionary stages. When the mass transfer starts, these low-frequency GW
sources also appear as UCXBs, which are the ideal multimessenger detection
sources. For a long detection distance of 100 kpc, those binaries have to
spend a longer evolutionary timescale to evolve toward a shorter orbital
period, thus appearing as GW sources only in a few Myr. BH X-ray binaries in
Group 3 experience a rapid orbital expansion, thus two systems with $M_{\rm
He,i}=1.8$, and $2.8~{}M_{\odot}$ are not potential GW sources in the later
stage of the second mass transfer. However, the GW radiation will drive the
orbit of the system with $M_{\rm He,i}=1.8~{}M_{\odot}$ to continuously shrink
after the mass transfer ceases, resulting in a low-frequency GW source that is
detectable in a long timescale. Since the chirp masses are constant, the
evolutionary tracks of detached systems are lines with a slope of $7/6$
($h_{\rm c}\propto f_{\rm gw}^{7/6}$ according to equation 3). In groups 1 and
2, the maximum GW frequency of BH UCXBs is 5.6 mHz, while the maximum
frequency can reach about 63.5 mHz for a detached BH binary in group 3 (see
also Table 1).

Figure 4: Evolution of BH+He star binaries in groups 1, 2, and 3 in the
characteristic strain vs. GW frequency diagram. The detection distances of the
black and green curve groups are 10, and 100 kpc, respectively. To avoid
overlap, the dashed, dotted, and dashed-dotted curves are slightly moved up
and down in parallel in the top panel (actually, these three curves overlap
with the solid curve). The blue, purple, and red curves denote the sensitivity
curve of LISA (Robson et al., 2019), TianQin (Wang et al., 2019), and Taiji
(Ruan et al., 2020) based on the numerical calculation of 4-year observations,
respectively.

To understand the progenitor properties of BH UCXBs, we model the evolution of
a large number of BH+He star binaries. Figure 5 summarizes the initial contour
of the progenitors of BH UCXBs in the $P_{\rm orb,i}-M_{\rm He,i}$ diagram.
All BH+He star binaries with initial parameters located between two solid
curves can evolve toward UCXBs (with durations of $\geq$ 0.1 Myr) that can be
detected by LISA at a distance of 10 kpc. The initial He-star masses and
initial orbital periods of progenitors of Galactic BH UCXB-LISA sources are in
the range of $0.32-2.9~{}M_{\odot}$ and $0.02-0.19$ days, respectively. Such
an initial parameter space is slightly wider than that ($0.32-1.2~{}M_{\odot}$
and $0.01-0.1$ days) for the progenitors of NS UCXB-LISA sources evolving from
the He star channel (Wang et al., 2021). It is clear that the systems with
relatively long initial orbital periods ($P_{\rm orb,i}\geq 0.05~{}\rm days$)
and high donor-star masses ($M_{\rm He,i}\geq 0.7~{}M_{\odot}$) may experience
two RLOF stages. For massive He stars with $M_{\rm He,i}\geq 1.2M_{\odot}$,
all BH binaries will experience two RLOFs. Some systems marked by crosses can
also evolve into low-frequency GW sources, however, they can not become valid
UCXBs with a duration of $\geq$ 0.1 Myr. Those BH+He star binaries that
experienced two RLOF stages could produce a high GW frequency in the final
evolutionary stage (see also $M_{\rm He,i}=1.8~{}M_{\odot}$ in Figure 4),
while these sources are not UCXBs because of the absence of mass transfer. For
a detection distance of 10 kpc, these binaries are most likely as
multimessenger sources in the first mass-transfer stage.

Table 1: Some Important Evolutionary Parameters of BH+He Star Binaries in
Groups 1, 2, and 3.

$M_{\rm He,i}$ | $P_{\rm orb,i}$ | $t_{\rm rlof}$ | $P_{\rm orb,min}$ | $t_{\rm orb,min}$ | $f_{\rm gw,max}$ | $L_{\rm X\\_10,max}$ | $L_{\rm X\\_100,max}$ | $\bigtriangleup t_{\rm LISA,10}$ | $\bigtriangleup t_{\rm LISA,100}$
---|---|---|---|---|---|---|---|---|---
($M_{\odot}$) | (days) | (Myr) | (min) | (Myr) | (mHz) | ($10^{38}~{}\rm erg\,s^{-1}$) | ($10^{38}~{}\rm erg\,s^{-1}$) | (Myr) | (Myr)
0.6 | 0.03 | 0.852 | 12.3 | 4.58 | 2.72 | 7.67 | 7.67 | 5.42 | 5.42
0.6 | 0.06 | 10.0 | 12.1 | 14.8 | 2.75 | 6.97 | 6.97 | 15.4 | 6.17
0.6 | 0.09 | 31.4 | 11.1 | 38.9 | 3.0 | 6.48 | 6.48 | 20.3 | 8.13
0.6 | 0.11 | 54.9 | 6.01 | 58.1 | 5.55 | 34.7 | 34.7 | 15.8 | 3.43
0.32 | 0.06 | 20.5 | 8.12 | 20.6 | 4.11 | 25.6 | 25.6 | 15.9 | 2.26
0.4 | 0.06 | 16.2 | 12.4 | 16.9 | 2.69 | 8.97 | 8.97 | 17.4 | 4.47
0.8 | 0.06 | 6.69 | 11.5 | 17.7 | 2.9 | 6.66 | 6.66 | 17.7 | 11.1
1.2 | 0.06 | 3.13 | 48.3 | 9.67 | 0.691 | 33.9 | $-$ | 21.5 | $-$
1.8 | 0.06 | 0.962 | 0.525 | 260 | 63.5 | 34.6 | $-$ | 17.5 | $1.82^{*}$
2.8 | 0.06 | 0.007 | 86.3 | 0.007 | 0.386 | 35.1 | $-$ | 2.16 | $-$

††footnotetext:

Note. The columns list (in order): the initial He-star mass, the initial
orbital period, the stellar age at the beginning of RLOF, the minimum orbital
period, the stellar age at the minimum orbital period, the maximum frequency
of GW, the maximum X-ray luminosity when binaries are LISA sources at a
distance of 10 kpc and 100 kpc, the detection timescale that the system can be
detected by LISA at a distance of 10 and 100 kpc.

$*$ The system can be detected by LISA at a distance of 100 kpc as a detached
system.

Figure 5: Parameter space distribution of BH+He star binaries with different
evolutionary fates in the initial orbital period vs. initial He-star mass
diagram. The BHs are taken to be a constant mass of 8 $M_{\odot}$. The solid
curves represent the boundaries that can evolve into UCXBs (with durations
$\geq$ 0.1 Myr) that can be detected by LISA at a distance of 10 kpc. The
solid and open circles denote the progenitors of BH UCXB-LISA sources that
experience one RLOF, and two RLOF, respectively. The crosses stand for BH+He
star systems that cannot evolve into valid UCXBs (with durations $\leq$ 0.1
Myr). The open triangles correspond to BH+He star systems that He stars have
already filled their Roche lobes at the beginning of evolution.

The measurement of chirp mass is very significant in constraining the masses
of two components. For a detached binary system, the chirp mass can be derived
by

$\mathcal{M}=\frac{c^{3}}{G}\left(\frac{5\pi^{-8/3}}{96}f_{\rm
gw}^{-11/3}\dot{f}_{\rm gw}\right)^{3/5},$ (6)

where $\dot{f}_{\rm gw}$ is the GW-frequency derivative. Therefore, the
measurement of chirp mass depends on the accuracy of the $\dot{f}_{\rm gw}$.
The space GW detectors are only able to detect $\dot{f}_{\rm gw}$ for those
ultra-compact binaries with a large signal-to-noise ratio (SNR) and a small
orbital period close to the minimum orbital period (Tauris, 2018). The minimum
$\dot{f}$ that LISA can measure is given by (Takahashi & Seto, 2002; Tauris,
2018):

$\dot{f}_{\rm gw,min}\approx 2.5\times
10^{-17}\left(\frac{10}{S/N}\right)\left(\frac{4~{}\rm yr}{T}\right)^{2}~{}\rm
Hz\,s^{-1},$ (7)

where $S/N$ is the SNR of GW signals, and $T$ is the mission duration of LISA.

Figure 6 illustrates the evolution of $\dot{f}_{\rm gw}$ with the GW
frequencies for our simulated three groups. Taking $S/N=10$ and $T=4$ yr, we
have the detection limitation of GW-frequency derivative as $\dot{f}_{\rm
gw,min}=2.5\times 10^{-17}~{}\rm Hz\,s^{-1}$, which is plotted by the
horizontal dashed lines in Figure 6. In the orbital shrinkage stages, the
angular momentum loss rates by GW radiation continuously increase, resulting
in a rapid increase of $\dot{f}_{\rm gw}$. The maximum $\dot{f}_{\rm gw}$ is
close to the peak GW frequency, however, $\dot{f}_{\rm gw}$ sharply decreases
to be 0 at the maximum GW frequency, then its sign turns into negative due to
an orbital expansion. Except for the orbital expansion stage of two systems in
group 3, other BH UCXBs could provide detectable $\dot{f}_{\rm gw}$ in a
timescale of $1~{}\rm Myr$. All evolutionary curves of $\dot{f}_{\rm gw}$ in
the climbing stage are approximate lines with a slope of $n=11/3$, which
implies a relation $\dot{f}_{\rm gw}\propto f_{\rm gw}^{11/3}$. This is
consistent with that the GW radiation dominates the orbital evolution of the
binaries. According to equation (6), $\dot{f}_{\rm gw}\propto f_{\rm
gw}^{11/3}$ for a constant chirp mass in detached binaries, in which the
angular-momentum loss is fully contributed by GW radiation (Webbink & Han,
1998; Piro, 2019). Therefore, if a braking index as $\dot{f}_{\rm gw}\propto
f_{\rm gw}^{n}$ is defined, its measurement can diagnose whether the binary
emitting low-frequency GW signals is a detached system.

Figure 7 presents the parameter space of BH+He star systems whose chirp masses
can be accurately measured. For BH UCXB-LISA sources evolving from the He star
channel, only several systems evolved from the progenitors with $M_{\rm
He,i}=0.7-1.5~{}M_{\odot}$ and $2.4-2.6M_{\odot}$ and relatively long $P_{\rm
orb,i}$ are difficult to detect $\dot{f}_{\rm gw}$. For the MS channel, only
BH UCXBs with initial orbital periods very near the bifurcation period have
detectable $\dot{f}_{\rm gw}$ (Qin et al., 2023). Therefore, BH UCXB-LISA
sources evolved from the He star channel are most likely to measure their
chirp masses.

Figure 6: Evolution of GW frequency derivative of BH UCXBs in groups 1, 2, and
3 in the $|\dot{f}_{\rm gw}|$ vs. $f_{\rm gw}$ diagram. The horizontal dashed
line represents the minimum $\dot{f}_{\rm gw}$ that can be detected by LISA
for $S/N=10$ and $T=4~{}\rm yr$ (see also equation 7). Figure 7: Parameter
space distribution that describes the detectability of $\dot{f}$ for BH+He
star systems in the initial orbital period vs. initial He-star mass diagram.
The solid diamonds represent BH+He star systems whose $\dot{f}$ can be
detected by LISA during the UCXB stage. The solid curves represent the
boundary that can evolve toward the valid UCXB stage (durations $\geq$ 0.1
Myr). The dashed curves represent the boundary that $\dot{f}$ can be detected
(the lower boundary coincides with the solid curve).

##  4 Discussion

###  4.1 Origin of BH+He star binaries

Similar to NS+He star systems, BH+He star systems could also be descendants of
high-mass X-ray binaries, in which the companion of the BH loses its hydrogen
envelope through stellar wind or a mass-transfer stage (Bhattacharya & van den
Heuvel, 1991; Dewi et al., 2002). Cyg X-3 was proposed to include a
$2-4.5~{}M_{\odot}$ BH (or NS) and a $7.5-14.2~{}M_{\odot}$ Wolf-Rayet donor
star (Zdziarski et al., 2013), and it may be the progenitor of a Galactic
double BH or BH-NS (Belczynski et al., 2013). In addition, massive He stars
might be formed through quasi-chemically homogeneous evolution (Woosley &
Heger, 2006; Yoon et al., 2006; Cantiello et al., 2007). However, our
simulations find that BH binaries with massive He stars ($M_{\rm
He,i}>3.0~{}M_{\odot}$) are hard to evolve into detectable UCXBs with a
duration longer than 0.1 Myr. Jiang et al. (2023) performed 1D model of post-
CE BH binaries with short orbital period ($\leq 0.2$ days) consisting of a BH
and a massive He star ($M_{\rm He,i}\geq 3.3~{}M_{\odot}$) that experiences
stable mass transfer, and found that their mass-transfer timescales are
shorter than 0.1 Myr, which is consistent with our results.

NS/BH+He star binaries could evolve from NS/BH+MS binaries through Case B or
Case C mass transfer (Dewi et al., 2002). For those systems with a large mass
ratio, the mass transfer is dynamically unstable. Subsequently, the binary
systems enter a CE phase (Ivanova et al., 2013), and the evolutionary products
of those systems that experienced Case B mass transfer in the CE phase are
binaries with an unevolved He star and a NS/BH. Those NS/BH+He star binaries
produced from Case B are so close that they initiate a Case BA mass transfer
in the core He burning stage. It is clear that our simulated systems in the
parameter space of the BH UCXB-LISA source are formed from Case B mass
transfer. Case C mass transfer would produce an evolved He star and a NS/BH
after the CE phase, subsequent mass transfer would start after core He is
exhausted.

###  4.2 He star Channel and MS channel

There exists a bifurcation period for the formation of UCXBs in the MS
channel, which is defined as the maximum initial orbital period forming UCXBs
within a Hubble time (van der Sluys et al., 2005; Sengar et al., 2017; Chen et
al., 2020; Qin et al., 2023). However, in the He star channel there is not a
critical period similar to the MS channel because of short initial orbital
periods. In the He star channel, the initial donor-star masses of the BH+He
star system that can evolve into UCXB-LISA sources are in a wide range from
0.32 to $2.9~{}M_{\odot}$. However, this range is shortened to be
$0.4-1.6~{}M_{\odot}$ in the MS channel because the stars with radiative
envelope will not experience magnetic braking (Qin et al., 2023). Certainly,
the initial orbital period range of the He star channel is much narrower than
that of the MS channel. In the MS channel, the maximum GW frequency emitting
by BH UCXBs is $\sim 3~{}\rm mHz$ (Qin et al., 2023), which is slightly
smaller than that ($5.6~{}\rm mHz$) in the He star channel. Because of the
high compactness of He stars, BH+He star systems can evolve into LISA sources
that can be detected at a distance of 100 kpc, while this phenomenon is
impossible for the MS channel.

To understand the final evolutionary fates of the He stars, we plot the
evolution of five BH-He star binaries in an H-R diagram in Figure 8. The final
luminosities and the effective temperatures of five He stars are ${\rm
log}(L/L_{\odot})\sim-1$ to $-3$, and $6300-31000~{}\rm K$, respectively. Such
luminosity and effective temperature ranges are comparable to those of WDs. It
is clear that those BH binaries with a massive donor star
($1.2-1.8~{}M_{\odot}$) had already evolved into detached systems before the
donor stars evolved into WDs by a contraction and cooling stage. Therefore,
the He star channel could form detached BH-WD systems. It is noteworthy that
the MS channel can only form mass-transferring BH-WD systems rather than
detached BH-WD systems (Qin et al., 2023). Similar to the MS channel, BH
binaries with a low-mass ($0.32-0.6~{}M_{\odot}$) He star can only evolve into
semi-detached BH-WD binaries rather than detached systems. Because those WDs
evolving from the He star channel lack a thin H envelope, their effective
temperatures are much higher than those of WD evolving from the MS channel.

###  4.3 He star Channel and Dynamic Process Channel

In globular clusters and young dense clusters, compact BH-WD binaries might
assembled through BHs capturing WDs in a dynamic process such as tidal
captures. Subsequently, GW radiation causes their orbits to continuously
shrink, and detached BH-WD binaries appear as low-frequency GW sources that
can be detected by space-borne GW detectors. Once WDs enter the tidal radius
of BHs, they will be disrupted by BHs, and these systems become UCXBs. For a
semi-detached BH-WD binary with $M_{\rm BH}=8~{}M_{\odot}$ and $M_{\rm
WD}=0.2~{}M_{\odot}$, the estimated minimum GW frequency is $6.4~{}\rm mHz$
(Qin et al., 2023). Because the tidal radius of the BH satisfies (Hopman et
al., 2004)

$R_{\rm t}=\left(\frac{M_{\rm BH}}{M_{\rm WD}}\right)^{1/3}R_{\rm WD}\propto
M_{\rm WD}^{-2/3},$ (8)

the tidal radius will be smaller if the WD captured by the BH is more massive.
According to the Keplerian third law, it would yield a relatively high minimum
GW frequency. However, the maximum GW frequency is $5.6~{}\rm mHz$ for BH
UCXBs evolving from the He star channel. Therefore, the GW frequency could be
a probe to test the formation channel of BH UCXBs.

Figure 8: Evolutionary tracks of several BH+He star binaries with different
initial donor-star masses and an initial orbital period of 0.06 days in the
H-R diagram. The solid and open circles represent the onset and end of mass
transfer, respectively

###  4.4 Influence of tidal effects

In the detailed binary evolution models, we ignore the tidal effects. In
principle, the tidal coupling between the orbit and the He donor star can
influence the orbital evolution of BH X-ray binaries. During the shrinkage
stage of the orbit, the donor star spins up to corotate with the orbital
rotation due to tidal coupling. This spin-up indirectly consumes the orbital
angular momentum, extracting orbital angular momentum from the binary system
at a rate of

$\dot{J}_{\rm t}=-I\dot{\Omega},$ (9)

where $I$ is the momentum of inertia of the He star, $\dot{\Omega}$ is the
derivative of the orbital angular velocity. Ignoring the influence of the mass
transfer, the change rate of the orbital angular velocity $\Omega$ satisfies

$\frac{\dot{\Omega}}{\Omega}\sim-3\frac{\dot{J}_{\rm gw}}{J},$ (10)

where $J$ and $\dot{J}_{\rm gw}$ are the total angular momentum of the system
and its loss rate via gravitational radiation. Therefore, the ratio between
$\dot{J}_{\rm t}$ and $\dot{J}_{\rm gw}$ is

$\frac{\dot{J}_{\rm t}}{\dot{J}_{\rm gw}}\sim 3\frac{I\Omega}{J}=3\frac{I}{\mu
a^{2}},$ (11)

where $\mu=M_{\rm BH}M_{\rm He}/(M_{\rm BH}+M_{\rm He})$ is the reduced mass
of the binary system.

Considering a BH binary with $M_{\rm BH}=8~{}M_{\odot},M_{\rm
He}=1~{}M_{\odot}$, and $P_{\rm orb}=0.1~{}\rm days$, we have $\mu
a^{2}=3.0\times 10^{55}~{}\rm g\,cm^{2}$. Since the radius of the He star
$R_{\rm He}=0.212R_{\odot}(M_{\rm He}/M_{\odot})^{0.654}$ (Tauris & van den
Heuvel, 2006), the momentum of inertia of the He star can be estimated to be
$I=0.4M_{\rm He}R_{\rm He}^{2}=1.7\times 10^{53}~{}\rm g\,cm^{2}$. According
to equation (11), the rate of angular momentum loss via the tidal effects is
approximately two orders of magnitude smaller than that via gravitational
radiation. Therefore, the influence of the tidal effects on the orbital
evolution is trivial.

###  4.5 Influence of BH Masses on the Parameter Space

The binary population synthesis simulations found that the newborn BHs have a
mass range of $5-16~{}M_{\odot}$ in BH binaries with normal-star companions,
in which the BH-masses distribution emerges a peak at $\sim 7-8~{}M_{\odot}$
in the Model A of Shao et al. (2019). In BH-He star X-ray binaries, the BH
masses range from 5 to $20~{}M_{\odot}$, and are most likely to gather at
$\sim 7-8~{}M_{\odot}$ in the Model A (Shao & Li, 2020). Based on these
statistical results, we adopt a constant initial BH mass of $8~{}M_{\odot}$ in
the detailed binary evolution models.

For a same He star, a high BH mass naturally results in a high chirp mass, a
high characteristic strain, and a long detection distance of BH UCXB-LISA
sources. When $M_{\rm He,i}=0.6~{}M_{\odot}$ and $P_{\rm orb,i}=0.06~{}\rm
days$, our calculations found that the maximum GW frequencies in the UCXB
stage are 2.55, 2.75, and 2.91 mHz for $M_{\rm BH,i}=15,8$, and
$5~{}M_{\odot}$, respectively. During the evolution BH binaries, a high BH
mass would produce an efficient angular momentum loss via GW radiation, and a
high mass transfer rate. The rapid mass transfer from the less massive He star
to the more massive BH drives the orbits of the systems to widen, resulting a
relatively small maximum GW frequency. Therefore, there exist inverse
correlation between the initial BH mass and the maximum GW frequency in BH
UCXBs evolved from the He star channel.

When $M_{\rm BH,i}=5~{}M_{\odot}$, and $M_{\rm He,i}=1~{}M_{\odot}$, our
simulations show that the initial orbital periods of BH+He star binaries that
can evolve into BH UCXB-LISA sources are in the range of $0.04-0.08$ days,
which is slightly narrower than that ($0.04-0.09$ days, see also Figure 5) in
$M_{\rm BH,i}=8~{}M_{\odot}$. If the BH has a high initial mass of
$15~{}M_{\odot}$, the range of initial orbital periods turns into $0.04-0.1$
days. Therefore, a high/low initial BH mass tends to widen/reduce the initial
orbital period range. This tendency originates from the gravitational
radiation is the dominant mechanism driving the orbit of BH+He star binaries
to shrink. A high mass BH naturally produces a high rate of angular momentum
loss, driving the BH+He star binaries with a slightly long period to evolve
into BH UCXB-LISA sources. Furthermore, the initial He-star masses that can
evolve into BH UCXB-LISA sources are in the range of $0.32-3.0~{}M_{\odot}$
and $0.32-2.9~{}M_{\odot}$ for $M_{\rm BH,i}=5$, and $15~{}M_{\odot}$,
respectively. Therefore, the influence of the initial BH masses on the initial
parameter space is trivial.

###  4.6 Detectability of BH UCXBs as LISA sources

We employ a rapid binary evolution code developed by Hurley et al. (2000,
2002) to study the birthrate of BH UCXB-LISA sources in the Galaxy. Based on
the binary population synthesis (BPS) approach, the primordial binary samples
are produced in the way of Monte Carlo simulations. Subsequently, a sample of
$1\times 10^{7}$ primordial binaries are evolved until the formation of BH+He
star systems through the rapid binary evolution code. Similar to Chen et al.
(2020), the initial input parameters and basic assumptions in the BPS
simulation are as follows: (1) All primordial stars are thought to be members
of binary systems orbiting in circular orbits. (2) The primordial primary mass
distribution arises from the initial mass function derived by Miller & Scalo
(1979), and it produces the secondary mass distribution by adopting a constant
mass ratio ($0<q\leq 1$) distribution $n(q)=1$. (3) The initial separations
distribution is taken to be constant in ${\rm log}a$ for wide binaries with
orbital periods longer than 100 yr, then changes into a uniform distribution
for close binaries (Eggleton et al., 1989). (4) The standard energy
prescription proposed by is used to tackle the CE ejection process (Webbink,
1984), in which the efficiency $\alpha_{\rm CE}$ of ejecting the envelope and
the parameter $\lambda$ describing the stellar mass-density distribution are
merged as a degenerate parameter $\alpha_{\rm CE}\lambda$. BH UCXB-LISA
sources are thought to be produced if the parameters of the BH+He star systems
are consistent with those of the progenitors of BH UCXB-LISA sources in Figure
5. The influence of the initial BH mass on the initial parameter space is
trivial, thus its affect on the population synthesis simulations can be
ignored. As a consequence, the criterion determined compact objects in the BPS
simulation is just a BH, no matter whether their masses are $8~{}M_{\odot}$.

In Figure 9, we plot the evolution of the birthrates of BH UCXB-LISA sources
evolving from the He star channel as a function of time when we take a
constant star formation rate (SFR) of $5~{}M_{\odot}\,\rm yr^{-1}$ for
Population I. The Monte Carlo simulations predict the birthrates of BH UCXB-
LISA sources in the Galaxy to be $R=2.2\times 10^{-6}$, and $3.6\times
10^{-7}~{}\rm yr^{-1}$ when the degenerate CE parameter $\alpha_{\rm
CE}\lambda=1.5$, and 0.5 ($\alpha_{\rm CE}$ is the CE ejection efficiency, and
$\lambda$ is the stellar structure parameter, Webbink, 1984), respectively.
These two birthrates are $1-2$ orders of magnitude higher than that ($\sim
3.9\times 10^{-8}~{}\rm yr^{-1}$) estimated in the MS channel (Qin et al.,
2023), however, are one order of magnitude lower than those ($3.1-11.9\times
10^{-6}~{}\rm yr^{-1}$) predicted for NS UCXBs evolving from the He star
channel (Wang et al., 2021).

According to Table 1, the mean detection timescale $\bigtriangleup t_{\rm
LISA,10}\approx 15~{}\rm Myr$ for BH UCXB-LISA sources evolving from the He-
star channel for a detection distance of 10 kpc. Therefore, we can estimate
the detection number of BH UCXB-LISA sources formed by the He-star channel in
the Galaxy to be $N=R\bigtriangleup t_{\rm LISA,10}\approx 33$, and 5 as
$\alpha_{\rm CE}\lambda=1.5$, and 0.5, respectively. For a low degenerate CE
parameter $\alpha_{\rm CE}\lambda=0.5$, the detection number of BH UCXB-LISA
sources evolving from the He-star channel is similar to that from the MS
channel ($\approx 4,$ Qin et al., 2023). However, the detection number of BH
UCXB-LISA source evolving from the He star channel is one order of magnitude
higher than that from the MS channel for a conventional degenerate CE
parameter $\alpha_{\rm CE}\lambda=1.5$.

Figure 9: Evolution of the birthrates of BH UCXB-LISA sources evolving from
the He star channel as a function of time when we take a constant SFR of
$5~{}M_{\odot}\,\rm yr^{-1}$ for Population I. The solid, and dashed curves
represent the evolutionary tracks when the degenerate CE parameter
$\alpha_{\rm CE}\lambda=0.5$, and 1.5, respectively.

##  5 Conclusion

BH UCXBs can emit both low-frequency GW signals and X-ray emission, making
them intriguing multi-messenger detection sources. In this paper, we employ
the MESA code to simulate the formation and evolution of BH UCXBs evolving
from the He star channel and diagnose their detectability in both X-ray and GW
bands. Taking an initial BH mass of $M_{\rm BH,i}=8~{}M_{\odot}$, our main
conclusions are summarized as follows:

  1. 1.

The mass transfer of BH+He star binaries is sensitive to the masses of He
stars. When $M_{\rm He,i}\geq 0.7~{}M_{\odot}$, those systems may experience
two RLOFs. The mass-transfer rates in second RLOF stages can reach $\sim
10^{-6}~{}M_{\odot}\,\rm yr^{-1}$. Such a high mass-transfer rate causes their
orbits to rapidly expand, thus some systems can not appear as UCXB-LISA
sources at a distance of 10 kpc in the later stage of the second mass-transfer
phase.

  2. 2.

The mass-transfer rates of BH UCXBs evolving from the He star channel are much
higher than those from the MS channel, producing X-ray luminosities that are
$2-5$ orders of magnitude higher than those from the MS channel. Due to the
short initial orbital periods, the BH+He star systems become UCXBs at the
beginning of the mass transfer and emit an X-ray luminosity of $\sim
10^{38}~{}\rm erg\,s^{-1}$. The maximum X-ray luminosity in the first mass-
transfer stage can exceed $10^{39}~{}\rm erg\,s^{-1}$. Those BH binaries with
massive He stars experience a second RLOF, and produce a maximum X-ray
luminosity of $3.5\times 10^{39}~{}\rm erg\,s^{-1}$, which exceeds the
threshold luminosity of ULXs.

  3. 3.

The shortest orbital period in the BH UCXB stage is 6 minutes, which
corresponds to a GW frequency of 5.6 mHz. Because of short initial orbital
periods, our simulated BH+He star binaries are already BH UCXB-LISA sources
within a distance of 10 kpc (or 100 kpc) at the onset of the first mass
transfer. BH X-ray binaries with massive He stars ($M_{\rm He,i}=1.8$, and
$2.8~{}M_{\odot}$) will not be detected by LISA at a distance of 10 kpc in the
later stage of the second mass-transfer phase due to a rapid orbital
expansion. However, the continuous orbital shrinkage due to GW radiation
causes the system with $M_{\rm He,i}=1.8~{}M_{\odot}$ to evolve toward LISA
sources after the second mass transfer ceases, emitting GW signals with a
frequency of up to 63.5 mHz.

  4. 4.

Compared with NS UCXBs from the He star channel (Wang et al., 2021), the
progenitors of BH UCXB-LISA sources have a slightly wide initial parameter
space. The initial He-star masses and initial orbital periods of the
progenitors of Galactic BH UCXB-LISA sources are in the range of
$0.32-2.9~{}M_{\odot}$ and $0.02-0.19$ days, respectively. Meanwhile, nearly
all systems in this parameter space can evolve into BH UCXB-LISA sources whose
chirp masses can be accurately measured. However, only a tiny fraction of BH
UCXB-LISA sources possess measured chirp masses for the MS channel (Qin et
al., 2023).

  5. 5.

In the He star channel, those BH binaries with a massive He star
($1.2-1.8~{}M_{\odot}$ ) can evolve into detached BH-WD binaries, which can
not be achieved in the MS channel. Meanwhile, the effective temperatures of
those WDs evolving from the He star channel are much higher than those of WD
evolving from the MS channel because of the absence of a thin H envelope.

  6. 6.

The Monte Carlo BPS simulations predict the birthrates of Galactic BH UCXB-
LISA sources evolving from the He star channel to be $R=2.2\times 10^{-6}$,
and $3.6\times 10^{-7}~{}\rm yr^{-1}$ when $\alpha_{\rm CE}\lambda=1.5$, and
0.5, respectively. In a 4-year LISA mission, the detection numbers of Galactic
BH UCXB-LISA sources for the He-star channel are 33, and 5 for $\alpha_{\rm
CE}\lambda=1.5$, and 0.5, respectively.

  7. 7.

Compared with the MS channel, BH UCXB-LISA sources from the He star channel
possess high X-ray luminosities and high possibility of detecting the chirp
masses. Therefore, BH UCXB-LISA sources evolving from the He star channel are
ideal multimessenger objects that deserve to be pursued in the X-ray and GW
community.

We cordially thank the anonymous referee for the detailed and insightful
comments that improved this manuscript. This work was partly supported by the
National Natural Science Foundation of China (under grant Nos. 12273014,
12225304, 12273105, 12373044, and 12203051), the Western Light Project of CAS
(No. XBZG-ZDSYS-202117), the Youth Innovation Promotion Association CAS (No.
2021058), and the Natural Science Foundation (under grant No. ZR2021MA013) of
Shandong Province.

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