# Dark Matter-Induced Low-Mass Gap Black Hole Echoing LVK Observations

Shuailiang Ge<EMAIL_ADDRESS>Center for High Energy Physics, Peking
University, Beijing 100871, China  School of Physics and State Key Laboratory
of Nuclear Physics and Technology, Peking University, Beijing 100871, China
Yuxin Liu<EMAIL_ADDRESS>International Centre for Theoretical
Physics Asia-Pacific, Beijing/Hangzhou, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Jing Shu
<EMAIL_ADDRESS>School of Physics and State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing 100871, China  Center for High
Energy Physics, Peking University, Beijing 100871, China  Beijing Laser
Acceleration Innovation Center, Huairou, Beijing, 101400, China  Yue Zhao
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA

###### Abstract

The recent detection of gravitational waves from a binary merger involving a
potential low-mass gap black hole (LMBH) by LIGO-Virgo-KAGRA (LVK)
Collaboration motivates investigations into mechanisms beyond conventional
stellar evolution theories to account for their existence. We study a
mechanism in which dark matter (DM), through its capture and accumulation
inside main sequence stars, induces the formation of black holes within the
mass range of $[3,5]M_{\odot}$. We examine the distribution of these LMBHs as
a function of galaxy halo mass, particularly when paired with neutron stars.
This gives a distinct signature that can be tested with future gravitational
wave observations. We find that a viable portion of the DM parameter space
predicts a merger rate of such binaries consistent with LVK observations.

## Introduction

The evolution of a main sequence (MS) star is complicated and varies depending
on its mass. Stars within the mass range of $[0.5,8]M_{\odot}$, including the
Sun, typically evolve into red giants after depleting their hydrogen fuel and
subsequently transform into white dwarfs (WD). MS stars with masses within
$[8,20]M_{\odot}$ possess sufficient energy to trigger supernova explosions,
resulting in the violent loss of mass. This process leads to the formation of
a neutron star (NS). According to the Equation of State, the maximum mass of a
NS cannot exceed $3M_{\odot}$ Mueller and Serot (1996); Kalogera and Baym
(1996); Godzieba  _et al._ (2021). When the mass of the progenitor star
exceeds approximately $\sim 20M_{\odot}$Belczynski  _et al._ (2012), the
supernova explosion can be suppressed, leaving enough matter behind for the
formation of an astrophysical black hole (ABH). The mass of the ABH directly
produced through this process is unlikely to be smaller than $5M_{\odot}$
Belczynski  _et al._ (2012); O’Connor and Ott (2011); Fryer  _et al._ (2012);
Janka (2012); Müller  _et al._ (2016); Ertl  _et al._ (2019). 11endnote:
1There are still debates regarding the validity of this statement,
particularly concerning the fallback of outgoing matter from supernovae, as
discussed in Mandel and Müller (2020); Antoniadis  _et al._ (2022); Sukhbold
_et al._ (2016); Ertl  _et al._ (2019). As a consequence, there is a gap in
the mass distribution of compact objects between $[3,5]M_{\odot}$, a feature
consistent with observations of Galactic X-ray binaries Bailyn  _et al._
(1998); Ozel  _et al._ (2010); Farr  _et al._ (2011); Kreidberg  _et al._
(2012).

Using traditional observation methods, searching for a low-mass gap black hole
(LMBH), with mass $[3,5]M_{\odot}$ can be challenging due to its small size
and very low luminosity in electromagnetic radiation. However, gravitational
wave (GW) radiation offers a new avenue for detection, even for objects
located very far away from our Milky Way galaxy. Recently, the LIGO-Virgo-
KAGRA (LVK) Collaboration announced the detection of a merger between a NS and
an LMBH with a mass of $3.6^{+0.8}_{-1.2}M_{\odot}$ Abbott  _et al._ (2024).
This breakthrough opens up a new window for investigating the existence and
properties of LMBHs. Particularly, the presence of a low-mass black hole
(LMBH) within the mass range of $[3,5]M_{\odot}$ may suggest phenomena beyond
our current understanding, potentially necessitating novel mechanisms for
their generation. For instance, such LMBHs may originate from former triple or
quadruple systems Fragione  _et al._ (2020); Lu  _et al._ (2020); Liu and Lai
(2021); Vynatheya and Hamers (2022); Gayathri  _et al._ (2023); Bartos  _et
al._ (2023), or through dynamical capture in star clusters Clausen  _et al._
(2013); Gupta  _et al._ (2020); Ye  _et al._ (2020); Rastello  _et al._
(2020); Sedda (2020); Arca Sedda (2021). Additionally, LMBHs may be identified
as primordial black holes Bird  _et al._ (2016); Clesse and García-Bellido
(2017); Sasaki  _et al._ (2016); Kashlinsky (2016); Clesse and Garcia-Bellido
(2022). In this paper, we take the detection of the LMBH-NS merger as the
motivation and study a novel mechanism for the LMBH production, through the
dark matter (DM) capture.

The existence of DM is widely accepted, yet its properties remain in mystery.
Numerous efforts are dedicated to studying the interaction between DM
particles and ordinary matter, such as nucleons and electrons, especially
through DM direct detection experiments Abrams  _et al._ (2002); Aprile  _et
al._ (2017); Cui  _et al._ (2017); Kavanagh (2018); Digman  _et al._ (2019);
Clark  _et al._ (2020); Carney  _et al._ (2023). The general interpretation of
the null results in these experiments suggests a very weak interaction. On the
other hand, if the interaction between DM particles and ordinary matter is too
strong, the DM may not be able to freely penetrate the atmosphere and reach
the experimental devices as expected. This leads to the untested extreme of DM
particles with strong interactions, not explored by these experiments. In this
study, we consider the DM mass in the range of
$m_{\chi}\in[10^{4},10^{9}]\mathrm{GeV}$ and cross-sections in the range of
$\sigma_{\chi\mathrm{H}}\in[10^{-28},10^{-22}]\mathrm{cm^{2}}$. This overlaps
with the parameter space for strongly interacting DM models which are still
consistent with various experimental constraints Kavanagh (2018); Albuquerque
and Baudis (2003); Caplan  _et al._ (2024); Bhoonah  _et al._ (2021).

Although DM particles in this strongly coupled regime may evade detection in
terrestrial experiments, they could readily be captured and accumulate inside
an MS star. The continuous accumulation of DM particles may lead to the
formation of a mini BH at the stellar center, potentially altering the
ultimate fate of the star. Specifically, the presence of such a mini BH may
cause a star that would otherwise become a WD or a NS to instead become a
LMBH.

In this paper, we first examine the criteria for efficient DM capture and
subsequent collapse to form a small BH capable of surviving Hawking radiation.
We then explore how the presence of such a mini BH alters the final state of a
star across various mass regimes and calculate the probability of a star being
converted to a LMBH within a given halo mass. We estimate the probability
distribution of detecting a LMBH-NS merger as a function of halo mass. This is
a unique prediction of the DM-induced LMBH formation mechanism. With the
future expansion of the GW network and enhancements in GW detector
sensitivities, such a distribution serves as a discriminator to differentiate
this mechanism from others. At last, we show that a substantial portion of
parameter space in our DM model is capable of yielding a merger rate of LMBH-
NS binaries consistent with the numbers reported by LVK.

## DM accumulated inside a star

In this section, we calculate the rate of DM being captured by an MS star with
a mass ranging within $[3,5]M_{\odot}$. As it passes through the star’s
interior, a DM particle interacts with stellar material, resulting in energy
loss. When this energy loss is sufficiently large, the particle becomes
gravitationally bound to the star and ultimately falls toward its core.

A detailed study of such a capture process can be found in Gould (1987);
Jungman  _et al._ (1996); Garani and Palomares-Ruiz (2017); Acevedo  _et al._
(2021); Ray (2023); Leane and Smirnov (2023). In this study, we simplify the
interactions between DM and stellar matter as one-dimensional head-on
collisions. This simplification provides a reasonable estimation of the orders
of magnitude, as demonstrated in Bramante  _et al._ (2018, 2020); Acevedo  _et
al._ (2021).

We first estimate the typical energy loss for each collision between a DM
particle, denoted as $\chi$, and hydrogen, the predominant component within an
MS star. Assuming the initial velocity of the dark matter particle as
$v_{\text{i}}$, the final velocity of the DM after a single scattering event
with a hydrogen can be expressed as
$v_{\text{f}}=v_{\text{i}}\sqrt{1-z\beta}$, where $\beta$ is defined as
$\beta\equiv 4m_{\rm H}m_{\chi}/(m_{\rm H}+m_{\chi})^{2}$ and $z$ is a
geometric factor averaged to $z=1/2$ for isotropic spin-independent collision
angles. For the parameter space we are interested in, DM mass is always much
larger than the mass of hydrogen, which leads to $\beta\approx 4\,{\rm
GeV}/m_{\chi}$.

When the DM particle passes through the star’s interior, the average number of
collisions can be estimated as

$N(\theta)=\int_{0}^{L}n_{\rm H}(r)\sigma_{\chi\rm H}\mathrm{d}L.$ (1)

Here, $\theta$ represents the angle between the velocity of the DM particle
and the radial direction $\hat{r}$ of the star upon entry. Neglecting any
change in the DM propagation direction, the distance traveled by the DM
particle within the star can be expressed as
$L=2R_{\text{star}}|\cos{\theta}|$, where $R_{\text{star}}$ denotes the radius
of the star. Additionally, $n_{\rm H}(r)$ denotes the hydrogen number density
at a distance $r$ from the center of the star, while $\sigma_{\chi\rm H}$
represents the cross-section for DM-hydrogen interaction. Therefore, with a
fixed angle $\theta$, the maximum initial velocity for a DM particle to be
captured is

$v_{\max}(\theta)=\frac{v_{e}}{\left(1-z\beta_{\rm H}\right)^{N(\theta)/2}}.$
(2)

Here $v_{e}$ is the escape velocity of the star,
$v_{e}=\sqrt{{2GM_{\text{star}}}/{R_{\text{star}}}}$, with $M_{\text{star}}$
and $R_{\text{star}}$ represent the mass and radius of the star, respectively.

For the DM velocity distribution, we adopt a Maxwell-Boltzmann distribution in
the galactic frame, given by

$f_{\text{gf}}(v)=\frac{4}{\sqrt{\pi}}\frac{v^{2}}{v_{0}^{3}}\exp{\left(-\frac{v^{2}}{v_{0}^{2}}\right)},$
(3)

where $v_{0}$ is taken to be the circular velocity $v_{\text{cir}}$ in the
halo, which should also be the virial velocity $v_{\text{vir}}$ according to
Virial Theorem. When considering a star moving at velocity $\vec{v}_{\rm
star}$ relative to the galactic center, the DM velocity in the star’s frame
requires adjustment through a Galilean transformation. Moreover, as DM
particles approach the surface of the star, they experience acceleration due
to the gravitational potential. For a DM particle with velocity $\vec{u}$ at
infinity in the star’s frame, its velocity upon entering the stellar region
becomes $v^{2}=|\vec{u}|^{2}+v_{e}^{2}$. With these adjustments made, we
derive the velocity distribution of DM particles before they enter the stellar
region, and the fraction of DM particles that can be captured $F_{\rm cap}$
upon encountering the star can be subsequently computed, with detailed
calculation provided in the Supplementary Materials. Since we are consider DM
in the relatively strong interacting region, $F_{\rm cap}$ is very close to 1
for the parameter space we are interested.

At last, the rate of dark matter accretion in a star is given by Gould (1987);
Ray (2023)

$\frac{\mathrm{d}M_{\text{acc}}}{\mathrm{d}t}=F_{\text{cap}}\ \rho_{\chi}\pi
R_{\text{star}}^{2}\langle
v_{\text{gf}}\rangle\sqrt{\frac{8}{3\pi}}\left(1+\frac{3v_{e}^{2}}{2\langle
v_{\text{gf}}\rangle^{2}}\right)$ (4)

Here $\langle v_{\text{gf}}\rangle\equiv\int\mathrm{d}vvf_{\text{gf}}(v)$,
which is the average velocity over the DM velocity distribution $f_{\rm
gf}(v)$ in the galatic frame. $\rho_{\chi}$ is the dark matter density in the
adjacent region of the star. This will be properly calculated in the later
section.

If a star is located in a binary system, the DM capture rate can be enhanced
by the gravitational slingshot effect from the companion star. However, this
enhancement becomes significant only when the stars are in close proximity
within the binary. For example, a binary of two $1.3M_{\odot}$ stars with a
short orbital period of approximately 32 hours may increase the capture rate
by a factor of 1.5 Brayeur and Tinyakov (2012). Consequently, we omit this
effect from our study, as the DM capture process typically occurs long before
the binary enters the close inspiral stage.

## Dark Matter Collapse into a black hole

In the parameter space we are interested in, the mass of the DM particles is
very large, i.e. $m_{\chi}\gg m_{\rm H}\approx\mathrm{GeV}$, and consequently
exceeds the star temperature $T_{\text{star}}$. As a result, DM particles,
upon capture while traversing through the star, will continue to interact with
stellar matter, leading to further energy loss. Ultimately, they will
thermalize with the stellar environment.

For an MS star with mass as $4M_{\odot}$, the thermalization timescale is
related to the DM-Hydrogen scattering cross-section as Acevedo  _et al._
(2021)

$t_{\text{th}}\approx 2\times
10^{-6}\mathrm{year}\left(\frac{m_{\chi}}{10^{8}\,{\rm
GeV}}\right)\left(\frac{10^{-26}\mathrm{cm}^{2}}{\sigma_{\chi\mathrm{H}}}\right).$
(5)

For the cross-section we consider here, the thermalization happens within a
very short timescale, much shorter than the lifetime of the star.

After DM particles have thermalized, they become concentrated in the stellar
core with a characteristic radius $r_{\text{th}}$. Within this region, these
particles can form thermal bound states under the influence of the star’s
gravitational potential $V(r)$.22endnote: 2The gravitational potential of
accreted DM may be neglected, as the energy density of DM can only reach
$\rho_{\chi}\sim\pi^{2}/12\rho_{\text{star}}$ before the gravitational
collapse of DM starts. The gravitational potential of the star can be
approximately written as $\langle
V(r)\rangle=\frac{2}{3}\pi\rho_{\text{star}}Gm_{\chi}r^{2}$. Through the
Virial Theorem, this potential is related to the DM kinetic energy as $\langle
E_{k}\rangle=\frac{3}{2}T_{\text{star}}$, thereby offering an estimation for
the thermal radius

$r_{\text{th}}\approx\sqrt{\frac{9T_{\text{star}}}{4\pi
G\rho_{\text{star}}m_{\chi}}}.$ (6)

As time progresses, the number density of DM within the sphere continues to
increase until it reaches the threshold of gravitational instability,
triggering collapse. Criteria for particles collapsing into a BH have been
extensively examined in the literature. These criteria typically fall into
three categories: Jeans instability, self-gravitating instability, and the
Chandrasekhar limit. Following the analyses in Acevedo  _et al._ (2021); Ray
(2023), we outline these three critical conditions that must collectively be
satisfied for collapse to occur:

  * •

Jeans instability: the free-fall time for particles should be shorter than the
time required for the system to rebound, which is characterized by the sound
crossing time with a sound speed
$c_{s}\approx\sqrt{T_{\text{star}}/m_{\chi}}$.

  * •

Self-gravitating instability: when the total mass of DM within the thermal
sphere is too much, the stable virialized state cannot exist. This leads to a
gravitational collapse, unless it is supported by thermal pressure or quantum
pressure.

  * •

Chandrasekhar limit: the total DM mass within the thermal sphere must lead to
a gravitational force strong enough to overcome the quantum pressure exerted
by the DM particles.

A detailed numerical analysis of these three criteria is provided in the
Appendix. We note that the critical DM mass required for Jeans Instability
(JI) consistently shares the same parametric dependence as that for self-
gravitating (SG) instability, but slightly stronger,

$M^{\text{JI}}_{\text{crit}}(m_{\chi})=\frac{\sqrt{3}\pi^{2}}{8}M^{\text{SG}}_{\text{crit}}=\sqrt{\frac{9\pi^{3}T_{\text{star}}^{3}}{64\rho_{\text{star}}G^{3}m_{\chi}^{3}}}.$
(7)

In the parameter space we investigated, these values are both significantly
higher than the Chandrasekhar limit, $M^{\text{Ch}}_{\text{crit}}$, for both
fermions and bosons. Consequently, the Jeans Instability sets the threshold
for the DM mass needed for the BH formation at the center of the star,

$\displaystyle M_{\text{crit}}=$ $\displaystyle 1.4\times
10^{-12}M_{\odot}\left(\frac{T_{\text{star}}}{3.0\times
10^{7}\mathrm{K}}\right)^{3/2}\left(\frac{m_{\chi}}{10^{8}\mathrm{GeV}}\right)^{-3/2}$
(8) $\displaystyle\left(\frac{\rho_{\text{star}}}{2.2\times
10^{-2}\mathrm{kg}/\mathrm{cm}^{3}}\right)^{-1/2}.$ $\displaystyle=$
$\displaystyle 1.4\times
10^{-12}M_{\odot}\left(\frac{M_{\text{star}}}{4M_{\odot}}\right)^{1.4125}\left(\frac{m_{\chi}}{10^{8}\mathrm{GeV}}\right)^{-3/2}$

The second equation is obtained through scaling relations between MS stars and
the Sun. Specifically, the radius and the temperature are given by
$R_{\text{star}}=(M_{\rm star}/M_{\odot})^{0.8}R_{\odot}$, and
$T_{\text{star}}=(M_{\rm star}/M_{\odot})^{0.475}T_{\odot}$, in accordance
with the Hertzsprung-Russell diagram. The star’s lifetime scales as
$\tau_{\text{star}}=(M_{\rm star}/M_{\odot})^{-2.5}\tau_{\odot}$ according to
Refs. Lecchini (2007); Hansen  _et al._ (2012). Additionally, we assume the
core density of the star follows the scaling of the average density, thus the
stellar core density can be written as $\rho_{\text{star}}\propto
M_{\text{star}}/R_{\text{star}}^{3}\propto(M_{\rm
star}/M_{\odot})^{-1.4}\rho_{\odot}$.

After the formation of the mini BH, its mass can grow through the accretion of
both stellar material and newly captured DM particles. The former is
characterized by Bondi accretion Bondi and Hoyle (1944) and the latter depends
on the DM capture rate. Additionally, the mini BH may evaporate via Hawking
radiation. Incorporating these processes, the mass evolution of the mini BH at
the core of the star can be expressed as

$\frac{\mathrm{d}M_{\mathrm{BH}}}{\mathrm{d}t}=\frac{4\pi\rho_{\text{star}}\left(GM_{\mathrm{BH}}\right)^{2}}{c_{\text{star}}^{3}}+\frac{\mathrm{d}M_{\text{acc}}}{\mathrm{d}t}-\frac{f\left(M_{\mathrm{BH}}\right)}{\left(GM_{\mathrm{BH}}\right)^{2}}.$
(9)

Here $c_{\text{star}}\approx\sqrt{T_{\text{star}}/m_{\mathrm{H}}}$ is the
speed of sound in the stellar matter. $f\left(M_{\mathrm{BH}}\right)$ is the
Page factor, and it characterizes the strength of Hawking radiation. The
temperature of the BH determines the number of SM particle species that it can
radiate. We take it to be $1/(74\pi)$ as the most aggressive choice, assuming
the emission of all species of SM particles with gray-body corrections Arbey
and Auffinger (2021); Acevedo  _et al._ (2021). A more common parameter in the
literature is $(1/15360\pi)$ assuming only photon emissions and neglecting
corrections from spin for simplicity Arbey and Auffinger (2019); Acevedo  _et
al._ (2021); Arbey and Auffinger (2021); Ray (2023), leading to a weaker
Hawking radiation.

To gain a qualitative understanding, we take the largest DM mass in our
parameter space, as $m_{\chi}=10^{9}\text{GeV}$, which leads to a mini BH with
smallest mass according to Eq. (8). The smaller the mass of a mini black hole,
the lower its accretion rate and the higher its Hawking radiation rate. With
this conservative choice, we find both stellar and DM accretion rates are
always much larger than the Hawking radiation rate. For example, for a
progenitor star with 4$M_{\odot}$, the first two terms in Eq. (9) are
$4.5\times 10^{28}\,{\rm GeV}/\mathrm{s}$ and $4.2\times 10^{31}\,{\rm
GeV}/\mathrm{s}$ respectively. 33endnote: 3We note that as the accretion
progresses, the BH grows larger, and the accretion of stellar matter becomes
dominant over the accretion of DM. Additionally, the effect of Hawking
radiation becomes increasingly negligible. On the other hand, the Hawking
radiation rate is only $6.2\times 10^{10}\,{\rm GeV}/\mathrm{s}$. For a
smaller DM particle mass, the Hawking radiation is even less important
compared to the accretion contributions. Thus for the parameter space we are
interested in, the mini BH induced by DM capture always persists, potentially
altering the fate of the host MS star.

The possibility of a DM-induced black hole forming and persisting within an MS
star depends on the rate at which DM is accreted. For the parameter space we
consider, DM effectively interacts with hydrogen, resulting in a nearly unity
capture probability. Under this approximation, we obtain an estimate for the
critical DM density in the vicinity of a star required to accumulate
sufficient DM to form a black hole within its lifetime $\tau_{\text{star}}$,

$\displaystyle\rho_{\text{crit}}$
$\displaystyle=\frac{M_{\text{crit}}}{\tau_{\text{star}}F_{\text{cap}}\pi
R_{\text{star}}^{2}\langle
v_{\text{gf}}\rangle\sqrt{\frac{8}{3\pi}}\left(1+\frac{3v_{e}^{2}}{2\langle
v_{\text{gf}}\rangle^{2}}\right)}$ (10) $\displaystyle\approx 4.10\
\mathrm{GeV/cm}^{3}\left(\frac{10^{6}\mathrm{GeV}}{m_{\chi}}\right)^{3/2}\left(\frac{M_{\text{star}}}{4M_{\odot}}\right)^{0.9}\left(\frac{\langle
v_{\text{gf}}\rangle}{440\mathrm{km}/\mathrm{s}}\right).$

Here in the second line we assume $\frac{3v_{e}^{2}}{2\langle
v_{\text{gf}}\rangle^{2}}\gg 1$, which is a valid approximation for the
typical value of $\langle v_{\text{gf}}\rangle$.

## Lower Mass Gap and a possible dark matter solution

Let us discuss how a mini BH induced by DM capture might impact the ultimate
fate of a star. First we consider a star within the mass range of
$[0.5,8]M_{\odot}$. If no DM-induced black hole forms in the center, such a
star will evolve into a red giant after depleting its hydrogen fuel,
eventually transforming into a WD. If a mini BH is formed during the MS phase
of the star, and if its mass is too small, the accretion rate may not be
significant enough to fully consume the star during this phase. This is due to
the core maintaining a relatively constant sound speed and density throughout
the MS phase. Also, radiation produced during accretion may support the in-
falling matter and decrease the accretion rate from Bondi accretion, referred
as Eddington Accretion in Bellinger  _et al._ (2023). However as stated in
Caplan  _et al._ (2024), such an effect could be softened when the mass of the
BH gets big enough, in which case most of the photons are gravitationally
trapped. This recovers the Bondi accretion which is more efficient. Following
their conclusion, a mini BH with mass $\gtrsim 10^{-10}M_{\odot}$ can consume
an MS star heavier than $1M_{\odot}$ within $\mathcal{O}(\mathrm{Gyr})$. To
form such a mini BH ($\gtrsim 10^{-10}M_{\odot}$) inside a star with mass
$3(5)M_{\odot}$, the DM mass needs to be $\lesssim 10^{6.7}(10^{6.9})\,{\rm
GeV}$. The consuming time will be shorter if the DM is lighter which induces a
heavier mini BH in the center. On the other hand, when the star transitions
into a red giant, the core’s contraction could result in a notably increased
accretion rate. In this study, we assume such a accretion is efficient enough
during the red giant phase, and the progenitor star will become a LMBH with
comparable mass if the DM induces a mini BH at the center.

MS stars with masses beyond $8M_{\odot}$ possess the energy necessary to
trigger supernova explosions, leading to the violent ejection of mass and
potentially resulting in the formation of a NS or an ABH. These massive stars
have relatively short lifetimes. During the MS phase, it is unlikely for a
mini BH to be formed at the center and further completely consume the whole
star. Moreover, the supernova process is rapid and drastic, with the mini BH
likely having minimal impact on the star’s evolution during this phase. Hence,
it is reasonable to assume that the evolution of these heavy MS stars remains
largely unaffected even in the presence of a mini BH at the center.

Finally, it’s worth noting that the presence of the Sun imposes a constraint
on this model. Taking the local dark matter density as
$\rho_{\chi}=0.4\mathrm{GeV/cm}^{3}$ and the local average velocity as
$270\mathrm{km/s}$, we derive an upper limit on the DM mass of $m_{\chi}\leq
10^{6.1}\mathrm{GeV}$, beyond which the Sun would already have been destroyed.
This result aligns with the results reported in Acevedo  _et al._ (2021); Ray
(2023).

## Lower Mass Black hole distribution

The spatial distribution of LMBHs depends on both the DM density profile and
the spatial distributions of MS stars. Consequently, this leads to a unique
prediction for the probability distribution of these LMBHs.

For the spatial distribution of the DM density $\rho_{\chi}(\vec{r}_{g})$, we
take the NFW profile Navarro  _et al._ (1997), normalized to the galaxy halo
mass $M_{\rm h}$. The total stellar mass is related to $M_{\rm h}$ through a
relationship derived from the Bolshoi-Planck simulation Behroozi  _et al._
(2019); Cui  _et al._ (2022). At last, the spatial distribution of stars
$\rho_{s}(\vec{r}_{g})$ depends on the type of the galaxy, either disk or
elliptical. The detailed descriptions of these quantities are provided in the
Supplemental Materials.

With everything prepared, we can calculate the probability of a star with mass
between $[3,5]M_{\odot}$ within a galaxy of mass $M_{\text{h}}$ becoming a
LMBH during its lifetime,

$\mathcal{P}(M_{\text{h}})=\frac{N_{\text{BH}}^{\text{LM}}(M_{\text{h}})}{N_{\text{star}}^{\text{LM}}(M_{\text{h}})}.$
(11)

The denominator is the total number of stars with mass between 3-5
$M_{\odot}$,

$N_{\text{star}}^{\text{LM}}(M_{\text{h}})=\int_{3M_{\odot}}^{5M_{\odot}}\frac{\mathrm{d}M_{\text{star}}}{\langle
M_{\text{star}}\rangle}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\int\mathrm{d}V\rho_{s}(\vec{r}_{g}),$
(12)

Here the volume integration extends to the virial radius of the galaxy
$R_{\text{vir}}(M_{\text{h}})$. We take the star mass distribution
$\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}$ following the
initial mass function (IMF) Kroupa (2001); Cui  _et al._ (2022),

$\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\propto
M_{\text{star}}^{-2.3},\ \ 0.5M_{\odot}\leq M_{\text{star}}<100M_{\odot}.$
(13)

The IMF is normalized by the averaged star mass $\langle
M_{\text{star}}\rangle=\int\mathrm{d}M_{\text{star}}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}M_{\text{star}}$,
whose integration range runs for all star masses. This normalization factor
will be cancelled eventually in our calculation. In addition, the numerator of
Eq. (11) is the number of stars in the same mass range but will become LMBHs
through DM capture,

$\displaystyle N_{\text{BH}}^{\text{LM}}(M_{\text{h}})=$
$\displaystyle\int_{3M_{\odot}}^{5M_{\odot}}\frac{\mathrm{d}M_{\text{star}}}{\langle
M_{\text{star}}\rangle}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\,\times$
(14)
$\displaystyle\int\mathrm{d}V\rho_{s}(\vec{r}_{g})\Theta\left[\rho_{\chi}(\vec{r}_{g})-\rho_{\text{crit}}(\vec{r}_{g},M_{\text{star}})\right].$

In Fig.1, we take several benchmarks for DM mass and show the probability of a
star with mass 3-5 $M_{\odot}$ to become a LMBH as a function of the halo
mass. We see that almost all stars in this mass range may become LMBHs if the
DM mass is too large. This is consistent with the DM mass upper limit imposed
by the survival of the Sun.

Next we calculate the relative probability distribution of LMBHs as a function
of logarithmic halo mass ,

$\mathcal{E}(M_{\text{h}})=\frac{\frac{\mathrm{d}n}{\mathrm{d}\log{[M_{\text{h}}]}}N_{\text{BH}}^{\text{LM}}(M_{\text{h}})}{\int\mathrm{d}\log{[M_{\text{h}}]}\frac{\mathrm{d}n}{\mathrm{d}\log{[M_{\text{h}}]}}N_{\text{BH}}^{\text{LM}}(M_{\text{h}})}.$
(15)

Here the halo mass distribution
$\frac{\mathrm{d}n}{\mathrm{d}\log{[M_{\text{h}}]}}$ is taken as the Sheth-
Tormen distribution at $z=0$ Sheth and Tormen (1999); Cui  _et al._ (2022). We
show the relative probability distribution as a function of the logarithmic
halo mass in Fig.2. We emphasize that the relative probability distribution,
$\mathcal{E}(M_{\text{h}})$ in Eq.(15), is a unique prediction in this
mechanism, and it serves as an excellent discriminator to distinguish this
mechanism from other LMBH formation mechanisms.

Figure 1: With various choices of DM masses and galaxy types, we show the
probability for progenitor stars within mass range $[3,5]M_{\odot}$ to become
LMBHs, as a function of the halo mass. The black line labeled as “All Star”
denotes the scenario where all MS stars within the same mass range are
converted into LMBHs ($\mathcal{P}=1$).  Figure 2: With various choices of DM
masses and the galaxy types, we show the relative probability distribution for
$[3,5]M_{\odot}$ LMBHs as a function of the logarithmic halo mass. The black
line labeled as “All Star” denotes the scenario where all MS stars within the
same mass range are converted into LMBHs.

## Binary Event Rate Estimation

One useful way to search for a LMBH in a distant galaxy is through GW
radiation emitted when it merges with a compact object. The waveform during
the merger offers valuable insights for identifying the properties of the
binary system, especially the masses involved. In this section, we take LMBH-
NS binaries as a benchmark and estimate their merger rate.

The formation and evolution of binaries involve several astrophysical
uncertainties, leading to large error bars in the predictions for binary
merger rates. For instance, the merger rate for an ABH and NS binary can vary
widely, from $0.1$ to $800\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ for isolated
binaries Belczynski  _et al._ (2001); Samsing and Hotokezaka (2021);
Broekgaarden  _et al._ (2021); Abbott  _et al._ (2021), and from $0.1$ to
$100\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ for binaries in dynamical environments
such as globular clusters Portegies Zwart and McMillan (2000); Antonini  _et
al._ (2014); Hoang  _et al._ (2020); Sedda (2020); Abbott  _et al._ (2021).

In order to reduce the uncertainty, we compare the merger rates of two
distinct binary systems: the LMBH-NS binary and the ABH-NS binary. In the
LMBH-NS binary, the LMBH originates from a progenitor star with a mass of
$3-5M_{\odot}$, which collapses to form an LMBH through dark matter accretion.
Conversely, in the ABH-NS binary, the ABH arises from a progenitor star with a
mass of $20-80M_{\odot}$. Such a star undergoes standard supernova processes,
resulting in an ABH with a mass greater than $5M_{\odot}$ and less than
approximately $20M_{\odot}$.

To compare the LMBH-NS rate with the ABH-NS rate, several assumptions need to
be made. Firstly, during the formation of the LMBH and the ABH, the mass of
the BH is related to that of the progenitor star in different ways. We assume
that the LMBH and its progenitor star share approximately the same mass. This
assumption is motivated by the expectation that no drastic explosion should
occur when such a low-mass MS star evolves to its final stage. 44endnote:
4Particularly, when the dark matter mass $m_{\chi}<10^{6}\,{\rm GeV}$, the
mini BH inside the star can swallow the progenitor star within 1 Gyr Bellinger
_et al._ (2023); Caplan  _et al._ (2024), within the lifetime of the star
before it turns into a red giant. Conversely, when a massive MS star collapses
to form an ABH, it loses a significant fraction of its mass through a
supernova explosion. For simplicity, we choose
$M_{\text{ABH}}=M_{\text{star}}/4$ in our estimation Vink  _et al._ (2001,
2011); Chen  _et al._ (2015); Cui  _et al._ (2022). In addition, the mass
distribution of progenitor stars follows the IMF described in Eq. (13). In our
calculations, we assume that this scaling is preserved in binary systems
containing a NS before the progenitor star evolves into a BH.

Moreover, during the formation of an ABH, the supernova explosion typically
leads a significant kick. Such effects could mildly increase the merger rate,
as the kick induces a non-trivial eccentricity that accelerates the rate of
energy loss via gravitational waves. However, the kick may also create too
much kinetic energy, causing the binary to become unbound, thereby decreasing
the merger rate Belczynski  _et al._ (2001); Tang  _et al._ (2020); Abbott
_et al._ (2021). Overall, this effect may alter the merger rate of ABH-NS
binaries by a factor ranging from approximately 0.1 to 1.2. For an order-of-
magnitude estimation, we consider this factor to be 1. Under these
assumptions, we find that the number density of LMBH-NS binaries is
approximately 6.84 times greater than that of ABH-NS binaries.

At last, for a BH-NS binary with a fixed initial orbit separation, it merges
faster when the BH mass is bigger due to a higher GW emission rate. This
affects the measured merger rates for a GW detector with an observation period
at O(1) years. The GW radiation power depends on the masses of the binary. The
maximum value of the orbital period for a BH-NS binary to merge within $\Delta
t^{\rm ob}=1$ year under the Post-Newtonian approximation is

$P^{\rm ob}_{\rm max}(M_{\rm BH})\approx\left(\frac{256\pi^{8/3}\Delta t^{\rm
ob}G^{5/3}M_{\rm BH}M_{\rm NS}}{5(M_{\rm BH}+M_{\rm NS})^{1/3}}\right)^{8/3}.$
(16)

In Tokovinin and Moe (2020), the probability distribution of close binary
periods $\frac{\mathrm{d}n}{\mathrm{d}\log P}$ follows a Gaussian
distribution, centered at $\log_{10}(P/\text{day})=4.8$, with a dispersion of
approximately 2.3. This distribution is universal across all BH masses. Taking
$M_{NS}=1.5M_{\odot}$, the difference in binary number densities and the
difference in merger efficiencies together yield a fudge factor $A\simeq 5.79$
when we relate the observed merger rate of ABH-NS binaries and that of the
LMBH-NS binaries. More details can be found in Supplementary Materials.
Consequently, after the convolution with the halo mass distribution, one can
write the rate ratio of these two types of mergers as

$\frac{\mathcal{R}^{\text{LMBH}}}{\mathcal{R}^{\text{ABH}}}=A\frac{\int\mathrm{d}\log{[M_{\text{h}}]}\frac{\mathrm{d}n}{\mathrm{d}\log{[M_{\text{h}}]}}N_{\text{BH}}^{\text{LM}}(M_{\text{h}})}{\int\mathrm{d}\log{[M_{\text{h}}]}\frac{\mathrm{d}n}{\mathrm{d}\log{[M_{\text{h}}]}}N_{\text{star}}^{\text{HM}}(M_{\text{h}})}.$
(17)

Here
$N_{\text{star}}^{\text{HM}}=\int_{20M_{\odot}}^{80M_{\odot}}\frac{\mathrm{d}M_{\text{star}}}{\langle
M_{\rm
star}\rangle}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\int\mathrm{d}V\rho_{s}(\vec{r}_{g})$,
which is the total number of heavy stars given a galaxy halo mass. Taking the
central value of the ABH-NS merger rate measured by LVK, as
$\mathcal{R}^{\text{ABH}}=130\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ Abbott  _et
al._ (2021), we show our predicted LMBH-NS binary merger rate as a function of
DM mass in Fig. 3. It is worth mentioning that a DM with mass smaller than
$10^{6}$ GeV gives a LMBH-NS merger rate consistent with the recent LVK’s
reported rate Abbott  _et al._ (2024),
$55_{-47}^{+127}\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ shown as the purple band in
Fig. 3.

Figure 3: For different galaxy types, we show the predicted LMBH-NS merger
rate as a function of DM mass. In comparison, the purple line (band)
corresponds to the merge rate (uncerntainty) by LVK’s recent observations
Abbott  _et al._ (2024). The region to the right of the orange vertical line
is excluded, which is the survival constraint of the Sun.

## Discussion

The detection of the LMBH-NS merger by LVK indicates the existence of LMBHs.
This is a surprising result and may require a detailed study on their possible
formation mechanism. We find that the strongly interacting DM model offers a
potential explanation for the formation of these LMBHs. A significant portion
of the parameter space gives a LMBH-NS merger rate consistent with LVK’s
measurement.

Moreover, whether a mini BH can form within a star and eventually convert it
into a LMBH depends on the properties of the DM in the star’s vicinity. This
leads to a unique feature in the LMBH distribution as discussed in Eq. (15).
Currently, the GW network is still in its beginning stage, and the angular
resolution is not good enough to identify the host galaxy of the merger on an
event-by-event basis. However, with the expansion of the GW detector network
and enhancements in individual detector sensitivity, we expect more LMBH-NS
mergers can be measured. Both event statistics and angular resolution have the
potential for significant improvements. If the host galaxy can be identified,
the distribution of LMBH-NS mergers can be directly tested. Even without
event-by-event identification of the host galaxy, statistical analyses can be
conducted based on the distribution of potential host galaxies from surveys
like the Dark Energy Spectroscopic Instrument Abareshi  _et al._ (2022); Adame
_et al._ (2024). It is conceivable that such LMBH-NS distribution could be
utilized to verify or refute the LMBH formation mechanism studied here.

## Acknowledgements

We thank Anil Seth, Dan Wik, Gail Zasowski and Zheng Zheng for useful
discussions. This work is supported by the National Key Research and
Development Program of China under Grant No. 2020YFC2201501. S.G. is supported
by the National Natural Science Foundation of China under Grant No. 12247147.
J.S. is supported by Peking University under startup Grant No. 7101302974 and
the National Natural Science Foundation of China under Grants No. 12025507,
No.12150015; and is supported by the Key Research Program of Frontier Science
of the Chinese Academy of Sciences (CAS) under Grants No. ZDBS-LY-7003. Y.Z.
is supported by the U.S. Department of Energy under Award No. DESC0009959.

###### Notes

  1. 1 There are still debates regarding the validity of this statement, particularly concerning the fallback of outgoing matter from supernovae, as discussed in Mandel and Müller (2020); Antoniadis  _et al._ (2022); Sukhbold  _et al._ (2016); Ertl  _et al._ (2019).
  2. 2 The gravitational potential of accreted DM may be neglected, as the energy density of DM can only reach $\rho_{\chi}\sim\pi^{2}/12\rho_{\text{star}}$ before the gravitational collapse of DM starts.
  3. 3 We note that as the accretion progresses, the BH grows larger, and the accretion of stellar matter becomes dominant over the accretion of DM. Additionally, the effect of Hawking radiation becomes increasingly negligible.
  4. 4 Particularly, when the dark matter mass $m_{\chi}<10^{6}\,{\rm GeV}$, the mini BH inside the star can swallow the progenitor star within 1 Gyr Bellinger  _et al._ (2023); Caplan  _et al._ (2024), within the lifetime of the star before it turns into a red giant.

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Supplementary Materials: Dark Matter-Induced Low-Mass Gap Black Hole Echoing
LVK Observations

The supplemental materials detail the mechanisms of dark matter capture within
stellar bodies. They provide explicit formulas for the criteria of
gravitational collapse, as mentioned in the main text, elucidating the
conditions under which such collapses are expected to occur. Additionally, the
materials offer insights into the galaxy number density distribution with
respect to halo mass, as well as the distribution of stellar and dark matter
within each galaxy. Finally, the modifications to the relative binary
formation rate—comparing dark matter-induced low-mass black holes with neutron
stars, and astrophysical black holes with neutron stars—are meticulously
calculated.

##  I Dark matter capture rate in the star body

In this section, we follow the calculations presented in Gould (1987); Jungman
_et al._ (1996); Acevedo  _et al._ (2021); Ray (2023). For the dark matter
(DM) velocity distribution, we assume a Maxwell-Boltzmann distribution in the
galactic frame, as given in Eq.(3). We consider a star velocity $|\vec{v}_{\rm
star}|=v_{0}$ within the galactic frame. Consequently, the velocity
distribution of DM particles at infinitely far in the star frame can be
derived through a Galilean transformation, denoted as $f(u)$:

$f(u,\phi)=\frac{1}{N^{*}}\frac{uv_{\text{gf}}^{2}(u)}{v_{\text{gf}}(u)-v_{0}\cos\phi}e^{-v_{\text{gf}}^{2}(u)/v_{0}^{2}},$
(S1)

where $\phi$ is the isotropic angle between the dark matter velocity
$\vec{v}_{\text{gf}}$ in the galactic frame and the star’s velocity
${\vec{v}_{\rm star}}$, in the galactic frame. The term $v_{\text{gf}}$
satisfies the Galilean transformation:

$v_{\text{gf}}^{2}+v_{0}^{2}-2v_{\text{gf}}v_{0}\cos\phi=u^{2}.$ (S2)

Here, $N^{*}$ is the normalization factor ensuring
$\int\mathrm{d}u\int\mathrm{d}\cos{\phi}f(u,\phi)=1$.

As described in the main text, dark matter particles slower than the maximum
initial velocity $v_{\max}(\theta)$ will be captured, eventually falling into
the star’s core. Thus, the maximum velocity at an infinite distance in the
star frame can be derived from a gravitational acceleration:

$u_{\max}(\theta)=\sqrt{v_{\max}(\theta)^{2}-v_{\text{esc}}^{2}}$ (S3)

Then, we derive the average capture rate for dark matter particles,
$F_{\text{cap}}$, which is the fraction of DM particles captured by the star:

$F_{\text{cap}}=\int_{0}^{1}\mathrm{d}\cos\theta\int_{0}^{u_{\max}(\theta)}\mathrm{d}u\int_{-1}^{1}\mathrm{d}\cos\phi
f(u,\phi).$ (S4)

We consider DM in the relatively strong interacting region, so that $F_{\rm
cap}$ is very close to 1 for the parameter space of interest, as discussed in
Acevedo  _et al._ (2021); Ray (2023).

##  II criterion of gravitational collapse

As outlined in the main text, the dark matter accreted within the star can
collapse into a black hole only if it satisfies the following three criteria:
Jeans instability, self-gravitating instability, and the Chandrasekhar limit.
We discuss these criteria individually below.

In the stable thermal state, before the onset of any instability, dark matter
particles reach a virialized distribution with a temperature equal to that of
the host star, $T_{\text{star}}$. At this juncture, the dark matter density is
sufficiently low to be considered negligible, allowing the star’s
gravitational potential to be approximated as
$V(r)=\frac{2}{3}\pi\rho_{\text{star}}Gm_{\chi}r^{2}$. According to the Virial
Theorem, this potential $\langle V(r)\rangle$ correlates with the DM kinetic
energy $\langle E_{k}\rangle=\frac{3}{2}T_{\text{star}}$, providing an
estimate for the thermal radius $r_{\text{th}}$, within which dark matter can
remain gravitationally bound in a virialized thermal state:

$r_{\text{th}}\approx\sqrt{\frac{9T_{\text{star}}}{4\pi
G\rho_{\text{star}}m_{\chi}}}.$ (S5)

### Jeans Instability

Jeans instability occurs when the dark matter self-gravitating free-fall time
$t_{\rm ff}$—the time it takes for a dark matter particle to fall from the
thermal radius to the star’s center—equals the sound-crossing time $t_{\rm
sc}$, assuming a sound speed $c_{s}=\sqrt{\frac{T_{\rm star}}{m_{\chi}}}$. The
free-fall time, given the star’s harmonic gravitational potential, is:

$t_{\rm ff}=\sqrt{\frac{3\pi}{16\rho^{\rm star}_{\chi}G}},$ (S6)

where $\rho^{\rm star}_{\chi}$ represents the dark matter density within the
thermal sphere of the star. The sound-crossing time can be expressed as:

$t_{\rm sc}=\frac{r_{\rm th}}{c_{s}}=\frac{3}{\sqrt{4\pi G\rho_{\rm star}}}.$
(S7)

Equating $t_{\rm ff}$ and $t_{\rm sc}$ yields:

$\rho^{\rm star}_{\chi}=\frac{\pi^{2}}{12}\rho_{\rm star},$ (S8)

leading to a critical mass for the onset of Jeans Instability:

$M^{\text{JI}}_{\text{crit}}(m_{\chi})=\sqrt{\frac{9\pi^{3}T_{\text{star}}^{3}}{64\rho_{\text{star}}G^{3}m_{\chi}^{3}}}.$
(S9)

### Self-gravitating Instability

When the mass of accreted dark matter, denoted as $M_{\rm acc}$, becomes
significantly large, the dark matter cannot maintain a virialized state at the
star’s temperature. In the absence of sufficient pressure from other
interactions, such as the thermal pressure accounted for in the Jeans
Instability scenario or the quantum pressure in the Chandrasekhar limit, the
dark matter would collapse to form a black hole. The virial theorem for a dark
matter particle in a bound state considers both the gravitational potential
from the stellar and dark matter contributions as:

$3T_{\rm star}=\frac{4}{3}\pi r^{2}\rho_{\rm star}Gm_{\chi}+\frac{GM_{\rm
acc}m_{\chi}}{r}.$ (S10)

A solution for $r$ becomes infeasible when $M_{\rm acc}$ reaches a critical
threshold, indicating that no thermal bound state can be formed within radius
$r$ without considering additional interactions, signifying gravitational
instability. The $r.h.s.$ above has a minimum value at $r=\left(\frac{3M_{\rm
acc}}{8\pi\rho_{\rm star}}\right)^{1/3}$, which sets the condition for self-
gravitating instability:

$3T_{\rm star}=\frac{3}{2}Gm_{\chi}\left(\frac{8}{3}\pi\rho_{\rm star}M_{\rm
acc}^{2}\right)^{1/3},$ (S11)

leading to the critical mass for instability:

$M^{\text{SG}}_{\text{crit}}(m_{\chi})=\sqrt{\frac{3T_{\text{star}}^{3}}{\pi\rho_{\text{star}}G^{3}m_{\chi}^{3}}}.$
(S12)

### Chandrasekhar Limit

The Chandrasekhar limit defines the mass threshold beyond which quantum
degenerate pressure can no longer counteract self-gravity. This criterion must
be met to prevent collapse under quantum pressure. For fermionic dark matter,
the critical mass is:

$M^{\text{Ch-F}}_{\text{crit}}(m_{\chi})\sim\frac{M_{\rm
pl}^{3}}{m_{\chi}^{2}},$ (S13)

and for bosonic dark matter:

$M^{\text{Ch-B}}_{\text{crit}}(m_{\chi})\sim\frac{M_{\rm pl}^{2}}{m_{\chi}},$
(S14)

where $M_{\rm pl}\sim G^{-1/2}$ represents the Planck mass.

The critical mass conditions for Jeans instability and self-gravitating
instability show similar parameter dependencies, albeit with slight
differences in coefficients. Therefore, to ascertain the critical mass for
dark matter collapse, one must compare the conditions for Jeans instability
and the Chandrasekhar limit. Considering a star of mass $3M_{\odot}$ as the
least massive star under consideration, the critical mass for Jeans
instability, $M^{\text{JI}}_{\text{crit}}(m_{\chi})\approx 10^{48}\,{\rm
GeV}\left(\frac{m_{\chi}}{10^{6}\,{\rm GeV}}\right)^{-3/2}$ for fermionic dark
matter, and $M^{\text{JI}}_{\text{crit}}(m_{\chi})\approx 1.8\times
10^{45}\,{\rm GeV}\left(\frac{m_{\chi}}{10^{6}\,{\rm GeV}}\right)^{-2}$. The
scaling relations for temperature and core density in the main sequence stars
indicate that an increase in star mass raises the critical mass for Jeans
instability but does not alter the Chandrasekhar limit. Additionally, bosonic
dark matter exhibits a lower Chandrasekhar limit. Consequently, the Jeans
instability criterion predominantly determines the critical mass for dark
matter to collapse into a black hole,
$M_{\text{crit}}=M^{\text{JI}}_{\text{crit}}$, across most of the parameter
space of interest.

##  III Halo Mass Distribution, Dark Matter Halo, and Stellar Matter
Distribution

We utilize the Sheth-Tormen distribution at $z=0$ for modeling the local
galaxy halo mass distribution Sheth and Tormen (1999); Cui  _et al._ (2022):

$M_{\text{h}}\frac{\mathrm{d}n}{\mathrm{d}M_{\text{h}}}=\Omega_{\mathrm{m},0}\rho_{\mathrm{cr},0}\frac{\mathrm{d}\sigma(M_{\text{h}})}{\sigma(M_{\text{h}})\mathrm{d}M_{\text{h}}}f(\sigma),$
(S15)

where
$\Omega_{\mathrm{m},0}\equiv\rho_{\mathrm{m},0}/\rho_{\mathrm{cr},0}\approx
0.3$ represents the current matter energy fraction, and
$\rho_{\mathrm{cr},0}\approx 10^{-26}\mathrm{kg/cm}^{3}$ denotes the current
critical energy density. The term $\sigma(M_{\text{h}})$ indicates the current
root-mean-square (rms) density fluctuation, which is modeled using numerical
simulations from the Bolshoi simulations, incorporating observational
parameters from WMAP5 and WMAP7 data Sheth and Tormen (2002); Klypin  _et al._
(2011); Prada  _et al._ (2012):

$\displaystyle\sigma(M_{\text{h}})$
$\displaystyle=\frac{16.9y^{0.41}}{1+1.102y^{0.20}+6.22y^{0.333}},$ (S16)
$\displaystyle y$
$\displaystyle\equiv\left[\frac{M_{\text{h}}}{10^{12}h^{-1}M_{\odot}}\right]^{-1}.$

The function $f(\sigma)$ represents the modified analytic fit of first-
crossing distribution function, which assumed to be an adaptation of the
original Press-Schechter function, refined by Klypin  _et al._ (2011):

$\displaystyle f(\sigma)=$ $\displaystyle
A_{f}\sqrt{\frac{2b_{f}}{\pi}}\left[1+\left(\frac{b_{f}}{\sigma^{2}}\right)^{-0.3}\right]\frac{1}{\sigma}\exp\left(-\frac{b_{f}}{2\sigma^{2}}\right),$
(S17) $\displaystyle A_{f}=0.322,\quad b_{f}=2.01.$

The structure of the dark matter halo is characterized by a Navarro-Frenk-
White (NFW) profile Navarro  _et al._ (1997):

$\rho_{\chi}(r_{g})=\frac{\rho_{0}}{r_{g}/R_{s}\left(1+r_{g}/R_{s}\right)^{2}},$
(S18)

where $r_{g}$ is the radial distance from the center of the halo, and $R_{s}$
is the scale radius linked to the virial radius $R_{\text{vir}}$ through the
concentration parameter $C(M_{\rm h})=R_{\text{vir}}/R_{s}$. The concentration
parameter, based on the Bolshoi and MultiDark simulation data for $z\approx
0$, is described as Prada  _et al._ (2012):

$\displaystyle C(M_{\rm h})=$ $\displaystyle
A_{C}\left[\left(\frac{\sigma(M_{\text{h}})}{b_{C}}\right)^{c_{C}}+1\right]\exp\left(\frac{d_{C}}{\sigma(M_{\text{h}})^{2}}\right),$
(S19) $\displaystyle A_{C}=2.881,\quad b_{C}=$ $\displaystyle 1.257,\quad
c_{C}=1.022,\quad d_{C}=0.06.$

The virial radius is defined as the radius within which the average density of
the halo is $\Delta$ times the critical energy density, with $\Delta$
typically set to 200 White (2001):

$R_{\text{vir}}=\sqrt[3]{\frac{3M_{\rm h}}{4\pi\Delta\rho_{\text{cr,0}}}}.$
(S20)

Consequently, $R_{s}$ is expressed as a function of the halo mass $M_{\rm h}$,
allowing for the determination of $\rho_{0}$ through the normalization of the
halo mass:

$M_{\rm h}=\int_{0}^{R_{\text{vir}}}\rho_{\chi}(r_{g})4\pi
r_{g}^{2}\mathrm{d}r_{g}.$ (S21)

However, it’s noted that the Milky Way’s halo does not perfectly align with
this model due to its complex structure and history McMillan (2016).

With the halo mass distribution established, we estimate the total stellar
mass $M_{s}$ in local galaxies (at $z\approx 0$) with a halo mass of
$M_{\text{h}}\equiv xM_{0}$, following the methodology in Behroozi  _et al._
(2019); Cui  _et al._ (2022):

$\log_{10}\left(\frac{M_{s}}{M_{0}}\right)=\epsilon_{0}-\log_{10}\left(10^{-\alpha_{0}x}+10^{-\beta_{0}x}\right)+\gamma_{0}\exp^{-\frac{1}{2}\left(\frac{x}{\delta_{0}}\right)^{2}},$
(S22)

where $M_{0}=10^{12.06}M_{\odot}$, $\epsilon_{0}=-1.459$, $\alpha_{0}=1.972$,
$\beta_{0}=0.488$, $\gamma_{0}=10^{-0.958}$, $\delta_{0}=0.391$.

In this paper, we adopt two typical structures for the stellar matter in
galaxies: elliptical and disk.

We employ the Hernquist Model Hernquist (1990) to describe the isotropic
distribution of elliptical galaxies:

$\rho_{s}^{e}(r_{g})=\frac{C^{e}}{2\pi}\frac{R^{e}}{r_{g}(r_{g}+R^{e})^{3}},$
(S23)

where $r_{g}$ represents the radial distance from the center of the halo, and
$R^{e}$, related to the half-mass radius, is defined as
$R^{e}=R_{1/2}/(1+\sqrt{2})$.

For disk galaxies, the stellar matter distribution is modeled using an
exponential disk profile Mo  _et al._ (2010); Cui  _et al._ (2022):

$\rho_{s}^{d}(R_{g},h_{g})=C^{d}\exp{\left(-\frac{R_{g}}{R^{d}}\right)}\exp{\left(-\frac{|h_{g}|}{h^{d}}\right)},$
(S24)

where $(R_{g},h_{g})$ denote the cylindrical coordinates in the galaxy, with
$R^{d}$ and $h^{d}$ being the characteristic scale lengths related to the
galaxy’s half-mass radius: $R^{d}\approx R_{1/2}/1.68$ and $h^{d}\approx
R_{1/2}/10$.

In both cases, the half-mass radius of the halo is proportionate to the virial
radius, expressed as $R_{1/2}\approx 0.015R_{\text{vir}}$ Kravtsov (2013). The
normalization of the total stellar mass for both profiles is used to determine
the normalization constants $C^{e}$ and $C^{d}$:

$M_{s}=\int\rho^{e(d)}_{s}(\vec{r})d^{3}\vec{r},$ (S25)

where the integration is performed over the entire volume of the galaxy.

##  IV modification to the relative binary formation rate

Here, we analyze the effects of black hole (BH) mass on the relative formation
rates of BH-Neutron Star (NS) binaries, highlighting two primary factors.

Firstly, BHs of varying mass ranges exhibit distinct number densities. This
variance stems from the progenitor stars’ adherence to an Initial Mass
Function (IMF) Kroupa (2001):

$\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\propto
M_{\text{star}}^{-2.3},\ \ 0.5M_{\odot}\leq M_{\text{star}}<100M_{\odot},$
(S26)

assuming $M_{\rm LMBH}=M_{\rm star}$ for lower mass black holes, which accrete
the majority of their progenitor star’s mass, and $M_{\rm
ABH}=\frac{1}{4}M_{\rm star}$ to account for mass loss during the evolution of
more massive stars.

Secondly, for a BH-NS binary with a given initial orbital separation, the
merger occurs more rapidly with increasing BH mass due to enhanced
gravitational wave (GW) emission. This impacts the observed merger rates for
GW detectors over observation periods of the order of 1 year. The orbital
period’s maximum value, allowing for a merger within $\Delta t^{\rm ob}=1$
year, is given by

$P^{\rm ob}_{\rm max}(M_{\rm BH})\approx\left(\frac{256\pi^{8/3}\Delta t^{\rm
ob}G^{5/3}M_{\rm BH}M_{\rm NS}}{5(M_{\rm BH}+M_{\rm NS})^{1/3}}\right)^{8/3},$
(S27)

with the period distribution for close binary systems modeled as a Gaussian
centered at $\log_{10}(P/\text{day})=4.8$ with a dispersion of 2.3 Tokovinin
and Moe (2020).

Integrating over the IMF and binary period distribution, we determine the
modification to the merge rate ratio $A$:

$A=\frac{\int_{3M_{\odot}}^{5M_{\odot}}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\mathrm{d}M_{\text{star}}\int^{{\rm
log}P^{\rm ob}_{\rm max}(M_{\text{star}})}_{{\rm log}P^{\rm ob}_{\rm
min}(M_{\text{star}})}\frac{\mathrm{d}n}{\mathrm{d}{\rm log}P}\mathrm{d}{\rm
log}P}{\int_{20M_{\odot}}^{80M_{\odot}}\frac{\mathrm{d}n_{\text{star}}}{\mathrm{d}M_{\text{star}}}\mathrm{d}M_{\text{star}}\int^{{\rm
log}P^{\rm ob}_{\rm min}(M_{\text{star}}/4)}_{{\rm log}P^{\rm ob}_{\rm
max}(M_{\text{star}})}\frac{\mathrm{d}n}{\mathrm{d}{\rm log}P}\mathrm{d}{\rm
log}P},$ (S28)

yielding $A=5.79$, under the assumption that $M_{\rm NS}=1.5M_{\odot}$ in all
cases. This ratio highlights the significant impact of BH mass on BH-NS binary
formation rates.


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