# Constraining Bosonic Dark Matter-Baryon Interactions from Neutron Star
Collapse

Chih-Ting Lu<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China  Arvind
Kumar Mishra ${\href https://orcid.org/0000-0001-8158-6602}$
<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical
Physics, Nanjing Normal University, Nanjing, 230021, China  Lei Wu
<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, 230021, China

(August 28, 2024)

###### Abstract

Dark matter (DM) may be captured around a neutron star (NS) through DM-nucleon
interactions. We observe that the enhancement of such capturing is
particularly significant when the DM velocity and/or momentum transfer depend
on the DM-nucleon scattering cross-section. This could potentially lead to the
formation of a black hole within the typical lifetime of the NS. As the black
hole expands through the accretion of matter from the NS, it ultimately
results in the collapse of the host. Utilizing the existing pulsar data
J0437-4715 and J2124-3858, we derive the stringent constraints on the DM-
nucleon scattering cross-section across a broad range of DM masses.

##  I Introduction

The presence of dark matter (DM) is firmly established through observational
evidence, indicating its gravitational interaction with baryonic matter (see
recent review, e.g. Bertone:2004pz ). However, despite substantial progress,
the nature of the interaction between DM and baryonic matter remains elusive.
Ongoing DM searches have yet to reveal any conclusive evidence for its
particle nature instead offering constraints on DM-baryon interactions
Klasen:2015uma ; Liu:2017drf ; PerezdelosHeros:2020qyt . DM detection
experiments encounter limitations, whether in size or technical constraints
Cebrian:2022brv ; MarrodanUndagoitia:2015veg . Therefore, it becomes
imperative to explore extreme cosmic locales that demonstrate sensitivity to
the interaction between DM and ordinary matter Bramante:2023djs . This
exploration promises to shed light on the unexplored realm of DM microphysics.

Neutron stars (NS), characterized by strong gravity and high density,
represent exotic environments ideal for investigating the interaction between
DM and ordinary baryonic matter Zhitnitsky:2023fhs . Neutron stars situated in
DM-dominated regions have the capability to capture surrounding DM particles,
consequently modifying their internal structure. The captured DM particles
interact with the matter in the NS, undergo thermalization, and can even lead
to black hole formation Goldman:1989nd ; 1990PhLB..238..337G ; Kouvaris:2010vv
; deLavallaz:2010wp ; Bramante:2013nma ; Bramante:2013hn ; Bell:2013xk ;
Guver:2012ba ; Garani:2018kkd ; Kouvaris:2018wnh ; Lin:2020zmm ;
Garani:2021gvc ; Singh:2022wvw . Therefore, the absence of neutron stars in
DM-dominated regions serves as a constraint on DM microphysics
Bramante:2014zca . For a detailed exploration of the impact of DM particles on
neutron stars and other compact objects, please refer to Refs. Bertone:2007ae
; Kouvaris:2010jy ; 2011PhRvD..84j7301L ; Gresham:2018rqo ;
Deliyergiyev:2019vti ; Dasgupta:2020mqg ; Dasgupta:2020dik ; Kain:2021hpk ;
Bramante:2022pmn ; Bramante:2023djs ; Ray:2023auh ; Bhattacharya:2023stq ;
Niu:2024nws  and the references therein.

In the literature, DM capturing was explored in the presence of constant DM-
nucleon scattering cross section Zentner:2009is ; McDermott:2011jp ;
Bramante:2014zca . However, for a more complete treatment, one needs to
include the DM velocity and/or momentum transfer depend on the DM-nucleon
scattering cross-section. These effects have been explored in the context of
the sun Guo:2013ypa ; Vincent:2013lua ; Vincent:2014jia ; Vincent:2015gqa ;
Vincent:2016dcp ; Busoni:2017mhe , white dwarfs Steigerwald:2022pjo  and
neutron stars Bell:2018pkk ; Bell:2020lmm ; Garani:2020wge ; Fujiwara:2022uiq
. However, previous studies are mostly focused on DM capturing and its
influence on the thermal conductivity and heating of the star.

In this study, we explore the DM velocity and momentum transfer dependent
parametrization for DM-nucleon scattering cross-sections and aim to delve into
DM microphysics through the survival analysis of neutron stars. Employing a
power-law parametrization for the DM-nucleon scattering cross section in the
DM velocity and momentum transfer dependent case, we estimate the DM capture
rate and investigate its dependency on DM parameters. Our findings reveal
that, in certain scenarios, both DM velocity and momentum transfer dependence
lead to an enhanced capturing rate compared to the constant scattering cross-
section case.

Following the capture, DM particles interact with the NS materials and
gravitate towards the core. We observe that the augmentation of the DM-nucleon
scattering cross-section results in a rapid thermalization of the DM particles
within the NS. This thermalization suggests the formation of gravitational
bound objects, potentially leading to black hole (BH) formation and the
destruction of the NS after accreting surrounding material.

Furthermore, by leveraging observational data from neutron stars J0437-4715
and J2124-3858 Kargaltsev:2003eb ; Manchester:2004bp , we constrain the DM
parameters. Our analysis indicates that positive DM velocity dependence
enhances DM accretion, imposing stringent constraints on DM parameters
compared to the constant scattering cross-section case and momentum-dependent
cross-section case. Notably, for some DM mass ranges, the constraints are even
stronger than the existing astrophysical and cosmological constraints
Vincent:2015gqa ; Vincent:2016dcp ; Busoni:2017mhe ; Boddy:2018wzy ;
Ooba:2019erm ; Maamari:2020aqz ; Buen-Abad:2021mvc .

The arrangement of our work is as follows: In section II, we discuss the
capture rate of the DM particles and our parametrization of the DM-Nucleon
scattering cross-section. Further, we estimate the number of captured
particles and present analytical and numerical results. In Section III, we
calculate the thermalization time scale for our models. Then, in Section IV,
we explore the black hole formation and discuss the fate of the NS in the
light of formed BH accretion and Hawking evaporation. In Section V, we present
and discuss our main results. Finally, in Section VI, we conclude our work.

##  II  Accretion of DM on neutron star

We assume that the neutron star is surrounded by dark matter particles. Here,
we focus on bosonic asymmetric dark matter (ADM) so that the annihilation of
the DM particles in the current universe can be neglected Petraki:2013wwa . We
further assume that DM primarily interacts with the nucleon and neglects any
self-interaction within the DM itself. In our analysis, we also neglect the
evaporation effect, which is important for
$m_{\chi}<2T_{\mathrm{NS}}/v^{2}_{\mathrm{esc}}$ where $T_{\text{NS}}$ and
$v_{\mathrm{esc}}$ are the typical temperature and escape velocity of the NS
McDermott:2011jp . So, we focus our study on comparably higher DM masses,
i.e., $m_{\chi}\geq{\cal{O}}(1)$ keV, where evaporation does not occur.
Therefore, using the above assumptions, the number of bosonic ADM particles
accumulating in the NS due to elastic DM-neutron interaction is given by

$\frac{dN_{\chi}}{dt}=C_{\chi\mathrm{N}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (1)

where $C_{\chi\mathrm{N}}$ is DM capturing rate due to the DM-neutron
interaction. The solution provides us the collected number of the DM particles
as $N_{\chi}=C_{\chi\mathrm{N}}\leavevmode\nobreak\ t$, where $t$ is the
accretion time which is typically the lifetime of the NS.

To calculate $N_{\chi}$, we need to provide $C_{\chi\mathrm{N}}$. The
calculation of $C_{\chi\mathrm{N}}$ has been extensively discussed in Refs.
1985ApJ…296..679P ; 1987ApJ…321..571G ; Bell:2020obw ; Robles:2022llu ;
Bell:2023ysh . In this work, we follow the approach given in Refs.
1985ApJ…296..679P ; 1987ApJ…321..571G , in which expression of
$C_{\chi\mathrm{N}}$ is given by 111For a more improved estimation of
$C_{\chi\mathrm{N}}$, please see Refs. Bell:2020obw ; Robles:2022llu ;
Bell:2023ysh  and references therein.

$C_{\chi\mathrm{N}}=\int^{R_{\mathrm{NS}}}_{0}4\pi r^{2}\leavevmode\nobreak\
\xi\int^{\infty}_{0}\frac{f(u)}{u}\leavevmode\nobreak\
w\Omega^{-}(w)\leavevmode\nobreak\ dudr\leavevmode\nobreak\ ,$ (2)

where $R_{\mathrm{NS}}$ is the radius of NS, $u$ is the DM velocity at very
large distance, and $f(u)$ is the DM velocity distribution function. The
quantity, $w$ is the DM speed at distance $r$ from the center of the NS,
defined by $w^{2}=u^{2}+v^{2}_{\mathrm{esc}}(r)$, where
$v_{\mathrm{esc}}(r)=\sqrt{GM(r)/r}$ is the escape velocity and $M(r)$ is the
NS mass enclosed inside the radius $r$. The term $\Omega^{-}(w)$ represents a
function related to the scattering kinematics and quantity, $w\Omega^{-}(w)$
is proportional to the probability per unit time of a collision that leads to
reduce the speed of DM particles from $w$ to less than $v_{\mathrm{esc}}$. In
the above equation, $\xi$ represents the nucleon degeneracy effects which
depend on the momentum transfer, and given by McDermott:2011jp

$\xi=\mathrm{min}\left[\frac{q}{p_{\mathrm{F}}},1\right]\leavevmode\nobreak\
,$ (3)

where
$q\simeq\sqrt{2}\left(\frac{m_{\chi}m_{\mathrm{N}}}{m_{\chi}+m_{\mathrm{N}}}\right)v_{\mathrm{esc}}$
is momentum transfer, $m_{\mathrm{N}}$ is the nucleon mass and
$p_{\mathrm{F}}=(3\pi^{2}n_{\mathrm{B}})^{\frac{2}{3}}$ is the Fermi energy in
which $n_{\mathrm{B}}$ represents the baryon number density. As examples,
$v_{\mathrm{esc}}\sim 1.8\times 10^{5}$km$/$sec, $q\geq p_{\mathrm{F}}$ for
$m_{\chi}>2$ GeV and $q<p_{\mathrm{F}}$ for $m_{\chi}<2$ GeV. Therefore, for
larger DM masses, we use $\xi=1$.

Further, we assume that the DM particle velocity distribution function is
Maxwellian, given as (in the rest frame of the DM halo) 1985ApJ…296..679P

$f_{0}(u)=\left(\frac{3}{2}\right)^{\frac{3}{2}}\frac{4}{\sqrt{\pi}}\frac{\rho_{\chi}}{m_{\chi}}\frac{u^{2}}{u^{2}_{0}}e^{-\frac{3}{2}\frac{u^{2}}{u^{2}_{0}}},$
(4)

where $\rho_{\chi}$, $m_{\chi}$ represent energy density and mass of the DM.
Here $u_{0}=\sqrt{3T_{\chi}/m_{\chi}}$ is the average velocity and $T_{\chi}$
is the temperature of DM. In the frame of the neutron stars, the DM
distribution function becomes

$f(u)=f_{0}(u)e^{-\frac{3}{2}\frac{u^{2}_{\mathrm{s}}}{u^{2}_{0}}}\left[\frac{\sinh(3uu_{\mathrm{s}}/u^{2}_{0})}{(3uu_{\mathrm{s}}/u^{2}_{0})}\right],$

where $u_{\mathrm{s}}$ is the velocity of NS with respect to the rest frame of
DM halo. In the dimensionless quantity defined as
$x=\left(\frac{3}{2}\frac{u^{2}}{u^{2}_{0}}\right)^{\frac{1}{2}},\leavevmode\nobreak\
\leavevmode\nobreak\
\eta=\left(\frac{3}{2}\frac{u^{2}_{\mathrm{s}}}{u^{2}_{0}}\right)^{\frac{1}{2}}$,
and DM distribution function becomes

$f(x)=\frac{4n_{\chi}}{\sqrt{\pi}}x^{2}e^{-x^{2}-\eta^{2}}\left(\frac{\sinh
2\eta x}{2\eta x}\right),$ (5)

where $n_{\chi}$ represents the DM number density surrounding the NS.

Moreover, to capture DM particles inside of the NS, DM particles need to
transfer enough energy to the NS so that after the interaction, their velocity
reduces to typically lower than the escape velocity. This condition requires

$\frac{u^{2}}{w^{2}}\leq\frac{E_{\mathrm{R}}}{E}\leq\frac{\mu}{\mu^{2}_{+}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (6)

where $E_{\mathrm{R}}$ is the energy transferred during the DM and nucleon
collision, and $E$ is the energy associated with the incoming DM particles.
Here $\mu\equiv m_{\chi}/m_{\mathrm{N}}$ and $\mu_{\pm}\equiv(\mu\pm 1)/2$ in
which $m_{\mathrm{N}}$ is the mass of nucleon. On the other hand, we also
assume that a single scattering between the DM and nucleon is sufficient to
transfer the energy required for DM capture. This is generally the case for
$m_{\chi}<10^{6}$ GeV. However, for higher masses, a single scattering will
not be sufficient for enough energy transfer, and multiple scattering is
needed for capturing, see Refs. Steigerwald:2022pjo ; Bramante:2017xlb .

Further, using the above assumptions and considering the interaction cross-
section of the DM particles with the nucleons to be constant and isotropic,
$\Omega^{-}(w)$ is given by Refs. 1985ApJ…296..679P ; 1987ApJ…321..571G

$\Omega^{-}(w)=n_{\mathrm{N}}\sigma\leavevmode\nobreak\
w\frac{\mu^{2}_{+}}{\mu}\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right)\Theta\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right)\leavevmode\nobreak\
,$ (7)

where $\sigma$ represents the DM-Nucleon scattering cross-section. Now, we
consider a finite-size effect of the nucleon in the DM-Nucleon interaction.
This consideration is crucial because, at the NS surface, the DM velocity is
quite high, approaching the escape velocity $v_{\mathrm{esc}}$. Consequently,
the imparted energy to the nucleon will be large, and DM-Nucleon interaction
may no longer behave as a point-like particle. Therefore, DM will interact
with the nucleon substructure, i.e., quarks and gluons, and this consideration
is relevant for massive DM particles. Therefore, to incorporate this effect,
one needs a nucleon form factor in the DM-Nucleon cross-section, i.e.,
$\sigma=\sigma_{\mathrm{point}}F(q^{2})$, where $\sigma_{\mathrm{point}}$ and
$F(q^{2})$ represent the cross-section when the DM and nucleons are point-like
and form factor, respectively. In this work, we follow a conservative approach
and consider the dipole form factor for DM-nucleon interaction 222In general,
the form factor depends on the models of the DM-nucleon interactions
Cirelli:2013ufw Ema:2020ulo  (in context of the NS, see Refs.Bell:2020obw
Anzuini:2021lnv ) and references therein. In this work, we have parameterized
the DM-Nucleon interaction in the power of relative velocity and momentum
transfer, so different models require different form factors. Therefore, to
keep our discussion simple and explore the effect of form factor in the
capture rate, we adopt a conservative approach and assume a dipole form factor
for our every parametrization. (considered in the next subsection), given by
Ema:2020ulo

$F(q^{2})=\frac{\Lambda^{4}}{(q^{2}+\Lambda^{2})^{2}},$ (8)

where $\Lambda$ is related to the scale below which the nucleon may no longer
behave like a point-like object. Considering the typical size of the nucleon
is $0.8$ fm ParticleDataGroup:2022pth , we assume $\Lambda\simeq 0.25$ GeV
throughout our analysis. Our typical estimation suggests that the higher
momentum transfer (DM masses) reduces the effective cross-section by two
orders of magnitude.

###  II.1 Parametrization of the DM-baryon elastic scattering

Now, we discuss our parametrization of the elastic DM-Nucleon interaction. In
literature, various types of elastic DM-nucleon scattering have been explored
extensively in the context of particle physics Chang:2009yt ; Fan:2010gt ;
Kumar:2013iva  and cosmology Vincent:2015gqa ; Vincent:2016dcp ; Ooba:2019erm
; Maamari:2020aqz ; Buen-Abad:2021mvc ; Boddy:2022tyt . We mainly focus on the
DM-nucleon scattering cross-section, which depends on their relative velocity
and momentum transfer during the collision. Here, a model-independent approach
is applied, and we parameterize the relative velocity-dependent cross-section
(VDCS) as Vincent:2015gqa

$\sigma^{\mathrm{vd}}_{\chi\mathrm{N}}(v_{\mathrm{rel}})=\sigma_{\chi\mathrm{N}}\left(\frac{v_{\mathrm{rel}}}{u_{0}}\right)^{2\alpha}\simeq\sigma_{\chi\mathrm{N}}\left(\frac{w}{u_{0}}\right)^{2\alpha},$
(9)

where $v_{\mathrm{rel}}$ is the relative velocity of DM-nucleon scattering and
$u_{0}$ is an average DM velocity. Here, $\alpha$ depends on the DM
parametrization; however, to keep our analysis general, we consider their
representative value $\alpha=(-1,0,1,2)$ and will not discuss their possible
particle physics models. For our calculation purposes, we consider typical
value, $u_{0}=220$ km$/$sec. As $w>u_{0}$, the scattering cross-section will
increase for positive $\alpha$ and decrease for negative $\alpha$.

On the other hand, the DM particles can interact with the baryons with the
momentum transfer dependent cross-section (MDCS). This kind of scattering
cross-section originates in some special types of interaction Fan:2010gt ;
Kumar:2013iva  and can lead to important effects Chang:2009yt . Here, we
consider model-independent approach of MDCS parametrized by Vincent:2015gqa

$\sigma^{\mathrm{md}}_{\chi\mathrm{N}}(q)=\sigma_{\chi\mathrm{N}}\left(\frac{q}{q_{0}}\right)^{2\beta},$
(10)

where $q$ is the momentum transfer from DM to the nucleon. To keep our
analysis general, we assume some representative values of $\beta$ without
discussing their possible particle physics models. In the above, $q_{0}$ is a
normalized momentum, and for our purpose, we consider its typical value,
$q_{0}=40$ MeV (relevant for the DM direct detection experiment searches
Vincent:2015gqa ).

Finally, we note that the scattering cross-sections can also depend on the
combination of the relative velocity and momentum transfer. However, for
simplicity, we will ignore this possibility in this study.

###  II.2 Estimation of the captured DM particles

Now, we calculate the number of collected DM particles, $N_{\chi}$ in the
typical lifetime of the NS, $t$, using the velocity and momentum-transfer
dependent cross-sections. The analytic forms of $N_{\chi}$ in certain cases
are obtained, and for the detail derivations, we refer readers to see
appendices A (for VDCS case) and B (for MDCS case). All the analytic
expressions are summarized below:

(1) Calculation of $N_{\chi}$ in constant cross-section case: For constant
cross-section case, the expression for $N_{\chi}$ is obtained as

$\displaystyle
N^{\mathrm{const}}_{\chi}(t)=C^{\mathrm{const}}_{\chi\mathrm{N}}\leavevmode\nobreak\
t=\frac{B}{16\eta}P_{0}(A,\mu,\eta)\leavevmode\nobreak\ t$
$\displaystyle=2.7\times
10^{38}\left(\frac{\rho_{\chi}}{10^{2}\mathrm{GeV\leavevmode\nobreak\
cm^{-3}}}\right)\left(\frac{\mathrm{GeV}}{m_{\chi}}\right)\left(\frac{\mu^{2}_{-}}{\mu}\right)$
$\displaystyle\times\left(\frac{\sigma_{\chi\mathrm{N}}}{2\times
10^{-45}\mathrm{cm^{2}}}\right)F(q^{2})P_{0}(A,\mu,\eta)\left(\frac{t}{10^{10}\leavevmode\nobreak\
\mathrm{years}}\right)$ (11)

where $C^{\mathrm{const}}_{\chi\mathrm{N}}$, $B$ and $P_{0}(A,\mu,\eta)$ are
given in Eqs. (33), (34) and (35), respectively. In the above equation,
$\rho_{\chi}$ is the DM density surrounding the NS, which depends on the
position of the star and DM profile adapted.

(2) Calculation of $N_{\chi}$ in VDCS case:

  * •

For $\alpha=1$ (i.e., $\sigma\propto v^{2}_{\mathrm{rel}}$) case,

$\displaystyle N^{\mathrm{vd}}_{\chi}(\alpha=1,t)=$ $\displaystyle
C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=1)\leavevmode\nobreak\
t=\frac{B}{48\eta\mu}P_{1}(A,\mu,\eta)\leavevmode\nobreak\ t,$
$\displaystyle=\frac{1}{3\mu}\frac{P_{1}(A,\mu,\eta)}{P_{0}(A,\mu,\eta)}N^{\mathrm{const}}_{\chi}(t)$
(12)

where $C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=1)$ and $P_{1}(A,\mu,\eta)$ are
given in Eqs. (36) and (37), respectively.

  * •

For $\alpha=2$ (i.e., $\sigma\propto v^{4}_{\mathrm{rel}}$) case,

$\displaystyle N^{\mathrm{vd}}_{\chi}(\alpha=2,t)=$ $\displaystyle
C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=2)\leavevmode\nobreak\
t=\frac{B}{144\eta\mu^{2}}P_{2}(A,\mu,\eta)\leavevmode\nobreak\ t,$
$\displaystyle=\frac{1}{9\mu^{2}}\left(\frac{P_{2}(A,\mu,\eta)}{P_{0}(A,\mu,\eta)}\right)N^{\mathrm{const}}_{\chi}(t)$
(13)

where $C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=2)$ and $P_{2}(A,\mu,\eta)$ are
given in Eqs.(38) and (39), respectively.

(3) Calculation of $N_{\chi}$ in MDCS case: The number of the collected DM
particles in the momentum-transfer dependent case is given by

$\displaystyle
N^{\mathrm{md}}_{\chi}(\beta,t)=C^{\mathrm{md}}_{\chi\mathrm{N}}(\beta)\leavevmode\nobreak\
t=\left(\frac{q}{q_{0}}\right)^{2\beta}\frac{B}{16\eta}P_{0}(A,\mu,\eta)\leavevmode\nobreak\
t$
$\displaystyle=\left(\frac{q}{q_{0}}\right)^{2\beta}N^{\mathrm{const}}_{\chi}(t)\leavevmode\nobreak\
.$ (14)

Although the captured DM particles increase with DM-Nucleon scattering cross-
section, the cross-section will be maximum at geometric limit characterized by
$\sigma_{\mathrm{max}}=\pi R^{2}_{\mathrm{NS}}/N_{\mathrm{B}}$, where
$R_{\mathrm{NS}}$ and $N_{B}$ represent the NS radius and the total number of
the nucleons in the NS, respectively. Assuming $R_{\mathrm{NS}}=10.6$ km and
$N_{B}=1.7\times 10^{57}$, we obtain $\sigma_{\mathrm{max}}\sim 2\times
10^{-45}$ cm2. In the situation when DM-Nucleon scattering cross-section is
higher than the geometric limit, the effective value of cross-section
saturates, viz
$\sigma^{\mathrm{max}}_{\chi\mathrm{N}}=\mathrm{min}\left[\sigma_{\chi\mathrm{N}},\sigma_{\mathrm{max}}\right]$.
Therefore, the maximum capture rate is dictated by the saturation limit and
estimated as

$C^{\mathrm{sat}}_{\chi\mathrm{N}}=\pi
R^{2}_{\mathrm{NS}}\int^{\infty}_{0}\frac{f(u)}{u}\leavevmode\nobreak\
w^{2}\leavevmode\nobreak\ du.$ (15)

After integration, we get

$\displaystyle C^{\mathrm{sat}}_{\chi\mathrm{N}}=$
$\displaystyle\frac{\pi}{3}\frac{\rho_{\chi}}{m_{\chi}}R^{2}_{\mathrm{NS}}\bigg{[}\sqrt{\frac{6}{\pi}}u_{0}e^{-\frac{3}{2}\frac{u^{2}_{\mathrm{s}}}{u^{2}_{0}}}$
$\displaystyle+\frac{6G_{\mathrm{N}}M_{\mathrm{NS}}+R_{\mathrm{NS}}(u^{2}_{0}+3u^{2}_{\mathrm{s}})}{R_{\mathrm{NS}}u_{\mathrm{s}}}\mathrm{erf}\left(\sqrt{\frac{3}{2}}\frac{u_{\mathrm{s}}}{u_{0}}\right)\bigg{]}$
(16)

where $G_{\mathrm{N}}$ is the Newton’s gravitational constant, and
$M_{\mathrm{NS}}$ is the mass of NS.

##  III Thermalisation

We have seen earlier that the NS accretes the DM particles. The accreted DM
particles start interacting with NS materials, get thermalized, and form a
self-gravitating system. The energy loss by DM particles via its interaction
with the nucleon is McDermott:2011jp

$\frac{dE}{dt}=-\xi n_{\mathrm{N}}\sigma v\delta E\leavevmode\nobreak\ ,$ (17)

where $\xi$, $n_{\mathrm{N}}$, and $v$ represent the neutron degeneracy,
nucleon number density, and the relative velocity between the DM and nucleon,
respectively. Here $\xi$ is defined by Eq. (3) and $\delta
E=q^{2}/2m_{\mathrm{N}}$ is the energy lost by the DM particles to the
nucleon. Therefore, using the above approximation, the thermalization time
scale, $t_{\mathrm{th}}$, is obtained as

$t_{\mathrm{th}}=\frac{2m_{\mathrm{N}}E_{\mathrm{th}}}{n_{\mathrm{N}}\xi\sigma
vq^{2}}$ (18)

where $E_{\mathrm{th}}$ is the thermal energy. We then replace relative
velocity with the thermal velocity obtained as
$\frac{1}{2}m_{\chi}v^{2}_{\mathrm{th}}=\frac{3}{2}T=E_{\mathrm{th}},$ so
$v_{\mathrm{th}}=\sqrt{\frac{3T}{m_{\chi}}}$, where $T$ is the NS temperature.
Furthermore, we point out that if the thermalization time scale is smaller
than the lifetime of the NS ($t_{\mathrm{th}}<t_{\mathrm{NS}}$), DM particles
will be thermalized inside the NS in its lifetime.

In case the captured DM thermalizes inside the NS, it forms an isothermal
distribution, which is characterized by the thermal radius as

$r_{\mathrm{th}}=\left(\frac{9T}{8\pi
G_{\mathrm{N}}\rho_{\mathrm{B}}m_{\chi}}\right)^{\frac{1}{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (19)

where $\rho_{\mathrm{B}}$ is the energy density of the baryons.

##  IV Black hole formation from DM accretion

Now we explore the evolution of DM particles after they are captured inside
the NS. If capturing is sufficiently large, a system of DM particles can
collapse and form a black hole. Following Ref. McDermott:2011jp , we will
briefly discuss the mechanisms for the BH formation and explore the possible
fate of neutron stars in the light of formed BH accretion and Hawking
evaporation.

###  IV.1 Black hole formation

In a system of DM particles, black hole formation occurs when the number of
particles becomes larger than the number defined by the Chandrashekhar limit.
In this work, we assumed the asymmetric scalar DM Petraki:2013wwa . For non-
interacting bosonic DM particles, the Chandrashekhar limit is defined as

$N^{\mathrm{Boson}}_{\mathrm{Chandra}}\simeq 1.5\times
10^{34}\left(\frac{100\mathrm{GeV}}{m_{\chi}}\right)^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (20)

Here, we will discuss two mechanisms for BH formation:

(1) BH formation via Bose-Einstein Condensation (BEC): We have assumed that DM
particles are bosonic; therefore, in the extremely dense region of the NS, DM
particles may form a BEC. It happens when the central temperature becomes less
than the critical temperature, $T_{c}$. Here $T_{c}$ is defined as

$T_{c}=\frac{2\pi}{m_{\chi}}\left(\frac{n_{\chi}}{\zeta(3/2)}\right)^{\frac{2}{3}}\leavevmode\nobreak\
.$ (21)

where $\zeta$ is Riemann-Zeta function. The above equation shows that low mass
(high DM number density, $n_{\chi}$) leads to a large critical temperature. In
such a situation, BEC formation takes place smoothly. One can define a
critical number inside the thermal radius $N^{\mathrm{crit}}_{\chi}=1.06\times
10^{36}\left(T/10^{5}\mathrm{K}\right)^{3}$ by using Eq. (21). Therefore, it
is clear that whenever the number of captured DM inside the NS is greater than
the typical number, $N^{\mathrm{crit}}_{\chi}$, some DM particles will move to
the ground state and lead to BEC. The number of particles in the condensed
ground state (when $T<T_{c}$) is given by McDermott:2011jp

$N^{0}_{\chi}=N_{\chi}\left[1-\left(\frac{T}{T_{c}}\right)^{\frac{3}{2}}\right]\simeq
N_{\chi}-1.06\times 10^{36}\left(T/10^{5}\mathrm{K}\right)^{3}.$ (22)

In the case of BEC, the BH formation will take place when the number of
captured DM particles in the condensed ground state is larger than the bosonic
Chandrashekhar limit. i.e.,
$N^{0}_{\chi}>N^{\mathrm{Boson}}_{\mathrm{Chandra}}$. From the above equation,
we define a lower limit on $N_{\chi}$, given by

$N_{\mathrm{BEC}}=N^{\mathrm{Boson}}_{\mathrm{Chandra}}+N^{\mathrm{crit}}_{\chi}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (23)

In this case, the typical mass of the formed BH is given by
$M_{\mathrm{BH}}\sim m_{\chi}N^{\mathrm{Boson}}_{\mathrm{Chandra}}$.

(2) BH formation without a BEC (via self-gravitation): We have seen earlier
that DM particles may be thermalized inside the NS after the capturing.
Further, the system of DM particles starts self-gravitating whenever
$\rho_{\chi}$ becomes larger than $\rho_{\mathrm{B}}$ inside the thermal
radius. Therefore, the condition for the self-gravitation suggests

$\rho_{\chi}>\rho_{\mathrm{B}}\quad\mathrm{for}\quad r\leq r_{\mathrm{th}}.$
(24)

Using the above condition, one can define the number of DM particles,
$N_{\mathrm{self}}$, required for the self-gravitation inside the NS as

$N_{\mathrm{self}}\simeq 4.8\times
10^{41}\left(\frac{100\mathrm{GeV}}{m_{\chi}}\right)^{\frac{5}{2}}\left(\frac{T}{10^{5}\mathrm{K}}\right)^{\frac{3}{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (25)

Whenever the accreted DM particles in the NS is greater than this value, i.e.,
$N_{\chi}\geq N_{\mathrm{Self}}$, DM particles start self-gravitating. In this
condition, the BH formation requires
$N_{\mathrm{Self}}>N^{\mathrm{Boson}}_{\mathrm{Chandra}}$. According to Eqs.
(25) and (20), this happens for $m_{\chi}\leq
10^{17}\mathrm{GeV}\left(T/10^{5}\mathrm{K}\right)^{3}$. In this case, the
typical mass of the formed BH is given by $M_{\mathrm{BH}}\sim
m_{\chi}N_{\mathrm{Self}}$.

Figure 1: Effect of the DM velocity dependent (left panel) and DM momentum
transfer dependent (right panel) DM-Nucleon scattering cross-section on the DM
capture. The red dashed line is the saturation limit of captured DM particles.

Figure 2: Effect of the velocity dependent (left panel) and momentum transfer
dependent (right panel) DM-Nucleon scattering cross-section on the
thermalization scale.

###  IV.2 Stability of the formed black hole

As soon as the BH forms, two opposite processes start happening
simultaneously: first, accretion of the NS material and ambient DM particles
surrounding it, which grows BH mass, and second, Hawking evaporation, which
reduces BH mass. As the formed BH mass is small (temperature is high), Hawking
evaporation plays a dominant role, and if evaporation is large, it can destroy
the BH. Therefore, for stability, BH should consume enough material. In light
of the above mechanisms, the evolution of the BH with mass $M_{\mathrm{BH}}$
is obtained as

$\displaystyle\frac{dM_{\mathrm{BH}}}{dt}\simeq$ $\displaystyle
4\pi\lambda_{s}\left(\frac{G_{\mathrm{N}}M_{\mathrm{BH}}}{v^{2}_{s}}\right)^{2}\rho_{\mathrm{B}}v_{s}+\left(\frac{dM_{\mathrm{BH}}}{dt}\right)_{\mathrm{DM}}$
$\displaystyle-\frac{1}{1560\pi
G^{2}_{\mathrm{N}}M^{2}_{\mathrm{BH}}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (26)

Here, the first term on the right-hand side represents the Bondi-Hoyle
accretion rate in which $\lambda_{s}$, $v_{s}$ correspond to the accretion
eigenvalue and sound speed, respectively McDermott:2011jp ; Bell:2013xk .
Considering the contribution of the ambient DM accretion is small, from Eq.
(26), we find that the newly formed BH with the mass $M_{\mathrm{BH}}$ will
not be evaporated completely from the Hawking radiation whenever

$M_{\mathrm{BH}}>M^{\mathrm{crit}}_{\mathrm{BH}}\simeq\bigg{(}\frac{v^{3}_{s}}{6240\pi^{2}G^{4}_{\mathrm{N}}\lambda_{s}\rho_{\mathrm{B}}}\bigg{)}^{\frac{1}{4}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (27)

Assuming the equation of state of neutrons as $P=K\rho^{\gamma}$, one gets
$K=3^{3/2}\pi^{4/3}/(5m^{8/3}_{N})$ and $\lambda_{s}=0.25$ for non-
relativistic degenerate neutrons gas case ($\gamma=5/3$) 1983bhwd.book…..S .
From Eq. (27), we find that initial BH with the mass greater than
$M^{\mathrm{crit}}_{\mathrm{BH}}=1.2\times 10^{37}$ GeV will grow through the
accreting the surrounding materials and destroy the NS. Therefore, when
$M_{\mathrm{BH}}\sim
m_{\chi}N_{\mathrm{Self}}<M^{\mathrm{crit}}_{\mathrm{BH}}$ (when BH form
without BEC) and $M_{\mathrm{BH}}\sim
m_{\chi}N^{\mathrm{Boson}}_{\mathrm{Chandra}}<M^{\mathrm{crit}}_{\mathrm{BH}}$
(when BH form with BEC), the BH will evaporate.

##  V Results and discussions

Having equipped ourselves with the basic ingredients, we will now present our
main results. Firstly, we will explore the effect of the velocity and momentum
transfer dependent cross-section on the capture rate and the thermalization
inside the NS. Later, we will constrain the DM parameters by the observations
of the neutron stars. For the calculation purposes, we consider
$M_{\mathrm{NS}}=1.44M_{\odot}$, $R_{\mathrm{NS}}=10.6$ km,
$N_{\mathrm{B}}=1.7\times 10^{57}$, $u_{0}=220$ km$/$sec, $u_{\mathrm{s}}=230$
km$/$sec, $v_{\mathrm{esc}}=1.8\times 10^{5}$ km$/$sec, and
$\rho_{\chi}=0.3\mathrm{GeV\leavevmode\nobreak\ cm^{-3}}$ otherwise specified.

###  V.1 Effect of the VDCS and MDCS on DM capturing

The number of captured DM particles inside the NS as a function of DM mass is
shown in Fig. 1. The left panel in Fig. 1 corresponds to the velocity-
dependent DM-Nucleon scattering cross-section. Here, magenta, black, blue, and
red lines correspond to $\alpha=-1,0,1,2$, respectively. The red dashed line
corresponds to the saturation limit of the captured DM particles. Here, we
find that the number of the collected DM particles, $N_{\chi}$, is constant on
the low masses but decreases on the higher masses. The captured DM particles
increase for the positive DM velocity-dependent power case but decrease for
the negative one. This feature is obvious as $N_{\chi}\propto\sigma$, so the
number of the captured DM particles depends on the behavior of the DM-Nucleon
scattering cross-section. Further, $N_{\chi}$ estimated for all VDCS cases is
always lower than one obtained from the saturation limit.

Figure 3: Constraints on the DM-Nucleon scattering cross-section for constant
cross-section case. The left and right panels are obtained using the pulsar
data PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, olive, brown, and
sky-blue regions are excluded from the BH formation with a BEC, without a BEC,
and non-thermalization of the captured DM particles. The red dashed line
corresponds to the saturation limit of the thermalization time scale. Further,
the black region corresponds to the constraint on the DM-Nucleon scattering
cross-section obtained from the LUX-ZEPLIN (LZ) experiment LZ:2022lsv  for the
comparison, and purple and orange regions represent the BH evaporation. The
red dashed line is the saturation limit of the DM-Nucleon scattering cross-
section.

The right panel in Fig. 1 shows the captured DM particles as a function of DM
mass in the MDCS case. Magenta, black, blue, and red lines correspond to
$\beta=-1,0,1,2$, respectively. As mentioned earlier, the red dashed line
corresponds to the saturation limit of the captured DM particles. Here, we
find that collected DM particles decrease for high DM masses for all powers.
However, on low DM masses, unlike the VDCS case (where $N_{\chi}$ is constant
for all powers), $N_{\chi}$ decreases sharply for positive powers,i.e.,
$\beta=1,2$ and increases for $\beta=-1$ case. The striking feature of the
MDCS case is that for low-mass DM particles, the captured rate is smallest for
$\beta=2$ and largest for $\beta=-1$.

###  V.2 DM-thermalization

Fig. 2 shows the thermalization time scale as a function of DM mass. The left
and right panels correspond to the velocity dependent and momentum transfer
dependent DM-Nucleon scattering cross-section, respectively. Here we see that
for the velocity dependent cases, $t_{\mathrm{th}}$ increases with the DM
mass. In comparison with the constant cross-section, the time scale is smaller
for low DM masses, whereas it is larger for high DM masses. On the contrary,
for the $\alpha=-1$ case, the thermalization time scale is largest and
smallest on the low and high DM mass among all VDCS cases, respectively. In
the momentum transfer dependent case, the thermalization scale is largest for
$\beta=2$ and smallest for the $\beta=-1$ case. This is true for all DM mass
ranges.

Figure 4: Contraints on the DM-Nucleon scattering cross-section in VDCS with
$\alpha=1$. The left and right panels are obtained from the pulsar data
PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored regions carry
the same meanings as Fig. 3.

###  V.3 Constraints on the DM-Nucleon interaction in the velocity dependent
cross-section

Now, we will constrain the DM parameters from the survival of the NS (i.e.,
the gravitational collapse of the DM particles will not destroy the host NS).
However, before that, we will briefly summarize the conditions that restrict
the BH formation inside the neutron stars following Ref. McDermott:2011jp .
When the BEC formation does not occur, the BH formation is preferable on the
large DM masses. The survival of the NS requires,
$N_{\chi}<N^{\mathrm{Bosons}}_{\mathrm{Chandra}}$. In terms of
$N_{\mathrm{Self}}$ this condition suggests

$N_{\chi}<N_{\mathrm{Self}},\quad\mathrm{valid\leavevmode\nobreak\ for}\quad
m_{\chi}\leq 10^{17}\left(T/10^{5}\mathrm{K}\right)^{3}$ (28)

as $N_{\mathrm{Self}}>N^{\mathrm{Bosons}}_{\mathrm{Chandra}}$ for
$m_{\chi}\leq 10^{17}\left(T/10^{5}\mathrm{K}\right)^{3}$. In cases where the
BEC formation occurs, the BH formation is preferable on the small DM masses.
Here the servival of the neutron stars requires,
$N^{0}_{\chi}<N^{\mathrm{Bosons}}_{\mathrm{Chandra}}$. Using this, in terms of
collected DM numbers, the condition for the survival of the NS becomes

$N_{\chi}<N_{\mathrm{BEC}}\leavevmode\nobreak\ .$ (29)

We then consider the pulsar data and constrain the DM parameters. Here, two
cold and old neutron stars with known parameters are involved. First,
PSRJ0437-4715 at distance $139\pm 3$ pc, surface temperature, $T_{e}=1.2\times
10^{5}$K Kargaltsev:2003eb , $t_{\mathrm{age}}\sim 6.69\times 10^{9}$ years
Manchester:2004bp . Second, PSRJ2124-3858 at distance $270$ pc, surface
temperature, $T_{e}<4.6\times 10^{5}$K Kargaltsev:2003eb ,
$t_{\mathrm{age}}\sim 7.81\times 10^{9}$ years Manchester:2004bp . The central
temperature can be obtained from the analytic formula given in Ref.
1982ApJ…259L..19G  using the surface data. This provides us $T_{c}=2.1\times
10^{6}$K and $T_{c}=2.5\times 10^{7}$K for the neutron stars PSRJ0437-4715 and
PSRJ2124-3858, respectively McDermott:2011jp . We will constrain the DM
parameters from the above two NS data.

Figure 5: Constraints on the DM-Nucleon scattering cross-section for VDCS with
$\alpha=2$ case. The left and right panels are obtained from the pulsar data
PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored regions carry
the same meanings as Fig. 3.

Fig. 3 shows the constraint on the bosonic DM-Nucleon scattering cross-section
for the constant cross-section case by using the pulsar data, J0437-4715 (left
panel) and J2124-3858 (right panel). The olive and brown regions are excluded
from the BH formation with a BEC and BH formation without a BEC. Further, the
purple and orange colors represent the regions where the BH evaporation
dominates over the accretion rate, and BH loses its mass with time. Therefore,
in these regions, the constraints are relaxed. In the sky-blue region,
captured DM particles do not thermalize with the neutrons and cannot
distribute themselves within the thermal radius. Hence, in this region, bounds
will not be applicable. In addition, the black region is the constraint
obtained from the LUX-ZEPLIN (LZ) experiment LZ:2022lsv  for comparison 333It
is important to emphasize that the LZ (2022) constraint LZ:2022lsv  is
specifically shown for the constant cross-section case. However, as there are
no explicit constraints (using the LZ(2022) data) available for velocity
dependent and momentum transfer dependent DM-Nucleon scattering cross-section
cases in the literature, we will not compare our constraints (obtained from
VDCS and MDCS scenarios) with the LZ(2022) data., and the red dashed line
corresponds to the saturation limit of the DM-Neucleon scattering cross-
section.

From Fig. 3, we find that pulsar data J0437-4715, constrains the low DM mass
range, $5\times 10^{-3}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq
10^{2}$ GeV from BH formation with a BEC. The constraint on higher DM masses
$m_{\chi}\geq 5$ GeV are relaxed from the BH evaporation (purple region), and
saturation limit on cross-section. Hence, J0437-4715 constrains, $5\times
10^{-3}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 5$ GeV from a BH
formation via BEC. Further, J0437-4715 constrains the higher DM masses,
$m_{\chi}\geq 3\times 10^{5}$ GeV from the BH formation without a BEC.
However, due to the BH evaporation, constraints are relaxed for $m_{\chi}\geq
1\times 10^{7}$ GeV (orange region). Therefore, using the evaporation and
thermalization constraints, J0437-4715 limit, $3\times
10^{5}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 10^{7}$ GeV from the
BH formation without a BEC. On the other hand, pulsar J2124-3858 data
indicates that the BH formation with a BEC requires a DM-nucleon scattering
cross-section larger than saturation limits; therefore, this data rules out
the possibility of the BH formation via BEC. Further, the pulsar data
constrains $m_{\chi}\geq 1.5\times 10^{8}$ GeV; however, these constraints are
relaxed from the non-thermalization and the BH evaporation. Therefore,
J2124-3858 data does not constrain the DM physics in this model. In light of
the above, we point out that the constraints on $\sigma_{\chi\mathrm{N}}$ are
weaker in comparison with the constant cross-section case where the form
factor has not been taken into account (see Fig. 2 in Ref. McDermott:2011jp ).

Figure 6: Constraints on the DM-Nucleon scattering cross-section for VDCS with
$\alpha=-1$ case. The left and right panels are obtained from the pulsar data
PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored regions carry
the same meanings as Fig. 3.

Figure 7: Constraints on the bosonic DM-Nucleon scattering cross-section for
MDCS with $\beta=1$ case. The left and right panels are obtained from the
pulsar data PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored
regions carry the same meanings as Fig. 3.

Fig. 4 shows the constraint on the bosonic DM-Nucleon scattering cross-section
for the VDCS with $\alpha=1$ by using the pulsar data, J0437-4715 (left panel)
and J2124-3858 (right panel). The red-dotted line and colored regions carry
the same meanings as Fig. 4. Here, we see that J0437-4715 constrains,
$10^{-5}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 10^{6}$, from the
BH formation with a BEC. However, due to the BH evaporation, constraints are
relaxed for $m_{\chi}\geq 17$ GeV (purple region), so J0437-4715 constrains
$10^{-5}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}<17$ GeV from BH
formation with a BEC. Further, from the BH formation without a BEC case,
J0437-4715 data constrains $m_{\chi}\geq 38$ GeV; nevertheless, due to the BH
evaporation and thermalization constraint, parameter space is limited,
$38\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 5.7\times 10^{3}$ GeV.
Further, J2124-3858 data constrains the DM parameters,
$10^{-5}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}<16$ GeV and $4.3\times
10^{2}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 3.5\times 10^{4}$ GeV
from BH formation with a BEC and without a BEC, respectively. In this case,
due to the positive DM velocity dependence ($\sigma\propto
v^{2}_{\mathrm{rel}}$), the cross-section and collected DM number density
increase sharply, and the possibility of the BH formation becomes high.
Therefore, in this scenario, the DM parameter constraints are stronger than
the constant cross-section case (see Fig. 2 in Ref. McDermott:2011jp ).

In Fig. 5, we show the constraint on the bosonic DM-Nucleon scattering cross-
section for VDCS with $\alpha=2$ from the pulsar data J0437-4715 (left panel)
and J2124-3858 (right panel). The red-dotted line and colored regions carry
the same meanings as earlier. Here we find that J0437-4715 constrains
$10^{-7}\leavevmode\nobreak\ \mathrm{GeV}\leq m_{\chi}\leq 12.7$ GeV from the
BH formation with a BEC and $5\times 10^{-3}\leavevmode\nobreak\
\mathrm{GeV}\leq m_{\chi}\leq 6.6\times 10^{-2}$ GeV from the BH formation
without a BEC. Further, pulsar data J2124-3858 contrains
$10^{-7}\leavevmode\nobreak\ \mathrm{GeV}\leq m_{\chi}\leq 12$ GeV from the BH
formation with a BEC and $5\times 10^{-2}\mathrm{GeV}\leq m_{\chi}\leq
7.5\times 10^{2}$ GeV from the BH formation without BEC. In this scenario,
cross-section again depends on the positive power of DM velocity but more
strongly than in the previous case; therefore, the collected DM density and BH
formation probability will be even higher. Therefore, the DM mass constraint
for this scenario is stronger than that of the constant cross-section and
quadratic DM velocity dependent one.

Figure 8: Constraints on the bosonic DM-Nucleon cross-section for MDCS with
$\beta=2$ case. The left and right panels are obtained using the pulsar data
PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored regions carry
the same meanings as Fig. 3.

Figure 9: Constraints on the bosonic DM-Nucleon cross-section for MDCS with
$\beta=-1$. The left and right panels are obtained from the pulsar data
PSRJ0437-4715 and PSRJ2124-3858, respectively. Here, the colored regions carry
the same meanings as Fig. 3.

Finally, Fig. 6 shows the constraint on the bosonic DM-Nucleon scattering
cross-section for VDCS with $\alpha=-1$ from the pulsar data discussed above.
Again, the color of the plotted regions carries the same meanings as earlier.
Interestingly, accreted DM particles are insufficient for both neutron stars
to form black holes. This is because of the negative DM velocity dependence,
the cross-section, and hence, the collected DM number decreases (compared to
the constant cross-section). It reduces the possibility of BH formation, which
requires a cross-section larger than the saturation limit. Therefore, this
scenario fails to constrain DM parameters from the BH formation.

Furthermore, we emphasize that for the VDCS case, the accreted DM particles
are increased for the positive power, which enhances the probability of BH
formation in the host NS. Our analysis shows that the pulsar data provides a
strong constraint on the DM parameters for low mass DM particles, i.e.,
$10^{-7}\leavevmode\nobreak\ \mathrm{GeV}\leq m_{\chi}\leq 12.7$ GeV for model
$\sigma\propto v^{4}_{\mathrm{rel}}$. In addition, the scenario $\sigma\propto
v^{2}_{\mathrm{rel}}$ constrains the intermediate DM masses, $4.3\times
10^{2}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 3.5\times 10^{4}$ GeV
and the constant cross-section constrains the high mass DM particles, $3\times
10^{5}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq 10^{7}$ GeV. We point
out that these constraints are more stringent than the existing astrophysical
and cosmological constraints Vincent:2015gqa ; Vincent:2016dcp ;
Busoni:2017mhe ; Boddy:2018wzy ; Ooba:2019erm ; Maamari:2020aqz ; Buen-
Abad:2021mvc .

###  V.4 Constraints on the DM-Nucleon interaction in the momentum dependent
cross-section

In this subsection, we constrain the bosonic DM-Nucleon scattering cross-
section in the MDCS from the survival of the NS. Fig. 7 shows the constraint
on the DM-Nucleon scattering cross-section for MDCS with $\beta=1$ from the
pulsar data J0437-4715 (left panel) and J2124-3858 (right panel). The color of
the plotted regions carries the same meanings as Fig. 4. We see that
J0437-4715 constrains $0.01\mathrm{GeV}\leq m_{\chi}\leq 13$ GeV from the BH
formation via BEC but does constrain the DM parameters from the BH formation
without a BEC. Further, J2124-3858 constrains a small DM mass range, i.e.,
$0.1\mathrm{GeV}\leq m_{\chi}\leq 1.2$ GeV from BH formation with a BEC but it
fails to constrain the DM parameters from the BH formation without a BEC.

Fig. 8 shows the constraints on the DM-Nucleon scattering cross-section for
MDCS with $\beta=2$ from the pulsar data discussed earlier. We see that pulsar
data J0437-4715 and J2124-3858 constrain $0.06\mathrm{GeV}\leq m_{\chi}\leq
13$ GeV and $0.02\mathrm{GeV}\leq m_{\chi}\leq 12.6$ GeV from the BH formation
via BEC. However, for both pulsar data, DM parameters are unconstrained from
the BH formation without a BEC.

Finally, Fig. 9 shows the constraints on the DM-Nucleon scattering cross-
section for MDCS with $\beta=-1$ using the pulsar data. Here, we see that BH
formation occurs specifically at low DM masses, not at high DM masses. It is
because, for this scenario, the accreted DM particles increase for low DM
masses and decrease quickly for large DM mass (see right panel of Fig. 1). The
pulsar data J0437-4715 and J2124-3858 constrains $m_{\chi}\leq 0.6$ GeV and
$m_{\chi}\leq 0.03$ GeV from the BH formation via BEC. Further, similar to the
previous case, BH does not form without a BEC, so the DM parameters are
unconstrained.

Therefore, we emphasize that in MDCS case, pulsar data constrains the low mass
DM particles,$m_{\chi}\leq 0.6$ GeV for $\sigma\propto q^{-2}$ and slightly
higher mass DM particles, $0.02\mathrm{GeV}\leq m_{\chi}\leq 13$ GeV for the
positive power of momentum transfer dependence, i.e., $\sigma\propto q^{2}$
and $\sigma\propto q^{4}$.

##  VI Conclusion

Due to strong gravity and high density, neutron stars (NS) are a lucrative
place to explore dark matter (DM) microphysics. Neutron stars situated in the
DM-rich environment can capture the DM particles. The captured DM particles
can be thermalized, form a gravitationally bound core, and finally form a
black hole inside the NS. After accreting the surrounding material, the black
hole can destroy the NS. In this work, we constrain the DM microphysics using
the data from the survival of the NS.

Assuming the DM velocity and momentum transfer dependent parametrization of
the DM-nucleon scattering cross section, we estimated the capture rate of the
DM particles by the NS. In a few cases, both the DM velocity and momentum
transfer dependence on DM-nucleon scattering cross-section led to enhancement
in the capturing rate over the constant scattering cross-section case. We
report that the enhancement of DM-nucleon scattering cross-section leads to
quick thermalization of the DM particles inside the NS.

Further, using observational data of the neutron stars J0437-4715 and
PSRJ2124-3858, we have constrained the DM parameters. In our analysis, we
reported that the positive DM velocity dependence enhances the DM accretion
rate and, therefore, strongly constrains the DM parameter comparison to the
constant scattering cross-section case. We found that the velocity dependent
cross-section imposes the most stringent constraints on DM parameters.
Specifically, $\sigma\propto v^{4}_{\mathrm{rel}}$ constrains low mass DM
particles, ($10^{-7}\leavevmode\nobreak\ \mathrm{GeV}\leq m_{\chi}\leq 12.7$
GeV), $\sigma\propto v^{2}_{\mathrm{rel}}$ constrains the intermediate DM
masses, ($4.3\times 10^{2}\mathrm{GeV}\leavevmode\nobreak\ \leq m_{\chi}\leq
3.5\times 10^{4}$ GeV ), and the constant cross-section constrains the high
mass DM particles, ($3\times 10^{5}\mathrm{GeV}\leavevmode\nobreak\ \leq
m_{\chi}\leq 10^{7}$ GeV). Notably, these constraints are stronger than
existing astrophysical and cosmological constraints. Furthermore, we obtained
that the momentum transfer dependent cross-section places restrictions on low-
mass DM particles ($m_{\chi}\leq 0.6$ GeV) for $\sigma\propto q^{-2}$, and on
comparatively higher mass DM particles ($0.02$ GeV $\leq m_{\chi}\leq 13$ GeV)
for positive powers of momentum transfer dependence, i.e., $\sigma\propto
q^{2}$ and $\sigma\propto q^{4}$.

Furthermore, we also pointed out that in this work, we assumed certain
assumptions, such as the system is Newtonian, non-relativistic, baryon and DM
densities are constant inside the NS and escape velocity is constant.
Therefore, when all effects are taken into account, the constraint will be
modified. In the future, we would like to explore these aspects and apply our
refined formalism to investigate the properties of DM from other compact
objects.

##  VII ACKNOWLEDGEMENT

This work is supported by the National Natural Science Foundation of China
(NNSFC) under grants No. 12335005, No. 12275134, and No. 12147228. AKM would
like to thank Liangliang Su for the useful discussions.

##  Appendix A Derivation of accreted DM particles via VDCS

From Eq.(2), it is clear that the integration involves the velocity, $u$, and
$r$ integration. To simplify this, we assume that the DM and nucleon densities
are constant inside the NS, i.e., $n_{\mathrm{N}}(r)=n_{\mathrm{N}}$, and
$n_{\chi}(r)=n_{\chi}$. Further as the variation of escape velocity also
depends on NS density, so for simplicity we neglect the variation of the
escape velocity with $r$ and consider its value at the surface on the NS,
i.e., $v_{\mathrm{esc}}(r)\sim
v_{\mathrm{esc}}(R_{\mathrm{NS}})=v_{\mathrm{esc}}$. In this case the cross-
section depends on the DM velocity, and to simplify this, we rewrite the
$\sigma^{\mathrm{vd}}_{\chi\mathrm{N}}$ in terms of the dimensionless
variables, $x$ and
$A=\left(\frac{3}{2}\frac{u^{2}}{u^{2}_{0}}\frac{\mu}{\mu^{2}_{-}}\right)^{\frac{1}{2}}$.
After the simplification, we obtain
$w(x)=\left[\frac{2u^{2}_{0}}{3}\left(x^{2}+\frac{A^{2}\mu^{2}_{-}}{\mu}\right)\right]^{\frac{1}{2}}$,
and

$\sigma^{\mathrm{vd}}_{\chi\mathrm{N}}\simeq\sigma_{\chi\mathrm{N}}\left(\frac{w(x)}{u_{0}}\right)^{2\alpha}=\sigma_{\chi\mathrm{N}}\left[\frac{2}{3}\left(x^{2}+\frac{A^{2}\mu^{2}_{-}}{\mu}\right)\right]^{\alpha}.$
(30)

Further, in this case, $\Omega^{-}(w)$ is given by

$\Omega^{-}(w)=n_{\mathrm{N}}\sigma^{\mathrm{vd}}_{\chi\mathrm{N}}F(q^{2})\leavevmode\nobreak\
w\leavevmode\nobreak\
\frac{\mu^{2}_{+}}{\mu}\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right)\Theta\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right).$
(31)

In terms of the dimensionless variables, the above equation is simplified as

$\Omega^{-}(w)=n_{N}\leavevmode\nobreak\
\sigma^{\mathrm{vd}}_{\chi\mathrm{N}}F(q^{2})\leavevmode\nobreak\
\frac{1}{w}\leavevmode\nobreak\ \frac{v^{2}}{A^{2}}\leavevmode\nobreak\
\left(A^{2}-x^{2}\right)\Theta(A-x).$ (32)

Having been equipped with the basic equations, we can now apply Eq.(2),
Eq.(5), and Eq.(32) to obtain the capture rate of the DM particle. We
emphasize that even after this simplification, it is difficult to get the
general form of the $C_{\chi\mathrm{N}}$ for arbitrary velocity-dependent
cross-section, i.e., $C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha)$ but for some
special cases, viz. $\alpha=(0,1,2)$, we can get the analytic results.

###  A.1 For constant cross-section case ($\sigma=\sigma_{\chi\mathrm{N}}$)

To obtain the $C_{\chi\mathrm{N}}$ for the constant DM-Nucleon scattering
cross-section, we consider $\alpha=0$. The analytic expression of
$C_{\chi\mathrm{N}}$ is given by

$C^{\mathrm{const}}_{\chi\mathrm{N}}=\frac{B}{16\eta}f_{0}(A,\mu,\eta),$ (33)

where

$\quad
B=\sqrt{\frac{32}{3}}\frac{\mu^{2}_{-}}{\mu}F(q^{2})N_{\mathrm{B}}n_{\chi}u_{0}\sigma_{\chi\mathrm{N}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (34) $\displaystyle\mathrm{and}\quad
P_{0}(A,\mu,\eta)=$
$\displaystyle\sqrt{\pi}\left(2A^{2}-2\eta^{2}-1\right)\text{erf}(A-\eta)$
$\displaystyle+e^{-(A+\eta)^{2}}\bigg{[}2A\left(e^{4A\eta}-1\right)+2\eta\left(e^{4A\eta}-2e^{A(A+2\eta)}+1\right)$
$\displaystyle+\sqrt{\pi}e^{(A+\eta)^{2}}\left(2A^{2}-2\eta^{2}-1\right)(\text{erfc}(A+\eta)-2\text{erfc}(\eta)+1)\bigg{]},$
(35)

where $\mathrm{erf(z)}$ and $\mathrm{erfc(z)}$ represent the error function,
and complimentary error function, defined via
$\mathrm{erf(z)}=\frac{2}{\sqrt{2}}\int^{z}_{0}e^{-y^{2}}dy$, and
$\mathrm{erfc(z)}=1-\mathrm{erf(z)}$, respectively. In the above equation,
$n_{\chi}$ is the number density of the DM surrounding the NS, and
$N_{\mathrm{B}}$ is the total number of baryons inside the NS. We checked that
in the rest frame of DM halo, i.e., $\eta\rightarrow 0$, and in an assumption
of the point-like structure of nucleon (neglecting the form factor), our
expression of $C^{\mathrm{const}}_{\chi\mathrm{N}}$ matches with the Ref.
1987ApJ…321..571G .

###  A.2 For $\alpha=1$ (i.e., $\sigma\propto v^{2}_{\mathrm{rel}}$) case

For $\alpha=1$, an analytic expression of $C_{\chi\mathrm{N}}$ is given by

$C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=1)=\frac{B}{48\eta\mu}P_{1}(A,\mu,\eta)$
(36)

where $P_{1}(A,\mu,\eta)$ is given by

$\displaystyle=\sqrt{\pi}\bigg{[}1-2\text{erfc}(\eta)+\text{erf}(A-\eta)+\text{erfc}(A+\eta)\bigg{]}$
$\displaystyle\times\bigg{[}\mu\left(4\left(A^{2}-3\right)\eta^{2}+2A^{2}-4\eta^{4}-3\right)$
$\displaystyle+2A^{2}\left(2A^{2}-2\eta^{2}-1\right)\mu_{-}^{2}\bigg{]}$
$\displaystyle+\eta\mu\left(2\eta^{2}+5\right)\left(e^{4A\eta}-2e^{A(A+2\eta)}+1\right)$
$\displaystyle+2A^{3}\left(e^{4A\eta}-1\right)\mu_{-}^{2}$
$\displaystyle+2A^{2}\eta\bigg{[}\left(e^{4A\eta}+1\right)\mu_{-}^{2}+2e^{A(A+2\eta)}\left(\mu-\mu_{-}^{2}\right)\bigg{]}$
(37)

###  A.3 For $\alpha=2$ (i.e., $\sigma\propto v^{4}_{\mathrm{rel}}$) case

For $\alpha=2$, an analytic expression of $C_{\chi\mathrm{N}}$ is given by

$C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=2)=\frac{B}{144\eta\mu^{2}}P_{2}(A,\mu,\eta)$
(38)

$\mathrm{where}\quad P_{2}(A,\mu,\eta)=$

$\displaystyle\sqrt{\pi}\bigg{[}(1-2\text{erfc}(\eta)+\text{erf}(A-\eta)+\text{erfc}(A+\eta)\bigg{]}$
$\displaystyle\times\bigg{[}\mu^{2}\left(A^{2}\left(8\left(\eta^{2}+3\right)\eta^{2}+6\right)-8\eta^{6}-60\eta^{4}-90\eta^{2}-15\right)$
$\displaystyle+4A^{2}\mu_{-}^{2}\bigg{\\{}\left(A^{2}\left(4\eta^{2}+2\right)-4\eta^{2}\left(\eta^{2}+3\right)-3\right)\bigg{\\}}\mu$
$\displaystyle+(-1+2A^{2}-2\eta^{2})\mu_{-}^{2}\bigg{]}$
$\displaystyle+2e^{-(A+\eta)^{2}}\bigg{[}A\mu^{2}\left(4\eta^{2}\left(\eta^{2}+6\right)+15\right)\left(e^{4A\eta}-1\right)$
$\displaystyle+\eta\mu^{2}\left(4\left(\eta^{2}+7\right)\eta^{2}+33\right)\left(e^{4A\eta}-2e^{A(A+2\eta)}+1\right)$
$\displaystyle+4A^{5}\left(e^{4A\eta}-1\right)\mu_{-}^{4}+4A^{3}\mu\left(e^{4A\eta}-1\right)\left(\mu+\left(2\eta^{2}+3\right)\mu_{-}^{2}\right)$
$\displaystyle+4A^{4}\eta\mu_{-}^{2}\left(\left(e^{4A\eta}+1\right)\mu_{-}^{2}-2e^{A(A+2\eta)}\left(\mu_{-}^{2}-2\mu\right)\right)$
$\displaystyle+4A^{2}\mu\eta\left(e^{4A\eta}+1\right)\left(2\mu+\left(2\eta^{2}+5\right)\mu_{-}^{2}\right)\left(2\eta^{2}+5\right)e^{A(A+2\eta)}$
$\displaystyle\left(\mu-2\mu_{-}^{2}\right)\bigg{]}$ (39)

###  A.4 For $\alpha=-1$ (i.e., $\sigma\propto v^{-2}_{\mathrm{rel}}$) case

In this case, no analytic solution has been obtained. Therefore, we adopt
numerical method to estimate the
$C^{\mathrm{vd}}_{\chi\mathrm{N}}(\alpha=-1)$.

##  Appendix B Derivation of the accreted DM particles via MDCS

In the MDCS case, the cross-section,
$\sigma^{\mathrm{md}}_{\chi\mathrm{N}}\propto q^{2\beta}$. Here we assume
$q\simeq\sqrt{2}\left(\frac{m_{\chi}m_{\mathrm{N}}}{m_{\chi}+m_{\mathrm{N}}}\right)v_{\mathrm{esc}}$
which suggests that the cross-section is independent of the initial DM
velocity. Further, $\Omega^{-}(w)$ in this case is given by

$\Omega^{-}(w)=n_{\mathrm{N}}\sigma^{\mathrm{md}}_{\chi\mathrm{N}}F(q^{2})\leavevmode\nobreak\
w\leavevmode\nobreak\
\frac{\mu^{2}_{+}}{\mu}\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right)\Theta\left(\frac{\mu}{\mu^{2}_{+}}-\frac{u^{2}}{w^{2}}\right)\leavevmode\nobreak\
.$ (40)

Now, rewriting the above equation in terms of the dimensionless variable, we
obtain

$\Omega^{-}(w)=n_{N}\sigma_{\chi\mathrm{N}}\left(\frac{q}{q_{0}}\right)^{2\beta}\leavevmode\nobreak\
F(q^{2})\frac{1}{w}\leavevmode\nobreak\
\frac{v^{2}}{A^{2}}\leavevmode\nobreak\ \left(A^{2}-x^{2}\right)\Theta(A-x).$
(41)

Further, using Eq. (2), Eq.(5), and Eq. (41), we calculate the captured rate
of the DM particles. For arbitrary power of $\beta$, a analytic expression of
$C^{\mathrm{md}}_{\chi\mathrm{N}}$ is given by

$C^{\mathrm{md}}_{\chi\mathrm{N}}(\beta)=\left(\frac{q}{q_{0}}\right)^{2\beta}\leavevmode\nobreak\
C^{\mathrm{const}}_{\chi\mathrm{N}}\leavevmode\nobreak\ .$ (42)

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