# Influence of Dark Matter on the Magnetized Neutron Star

Vishal Parmar1<EMAIL_ADDRESS>H. C. Das2,3<EMAIL_ADDRESS>M.
K. Sharma1 S. K. Patra2,3 1 School of Physics and Materials Science, Thapar
Institute of Engineering and Technology, Patiala 147004, India 2Institute of
Physics, Sachivalaya Marg, Bhubaneswar 751005, India 3Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

Over the past two decades, significant strides have been made in the study of
Dark Matter (DM) admixed neutron stars and their associated properties.
However, an intriguing facet regarding the effect of DM on magnetized neutron
stars still remains unexplored. This study is carried out to analyse the
properties of DM admixed magnetized neutron stars. The equation of state for
the DM admixed neutron star is calculated using the relativistic mean-field
model with the inclusion of a density-dependent magnetic field. Several
macroscopic properties such as mass, radius, particle fractions, tidal
deformability, and the $f$-mode frequency are calculated with different
magnetic field strengths and DM configurations. The equation of state is
softer with the presence of DM as well as for the parallel components of the
magnetic field, and vice-versa for the perpendicular one. Other macroscopic
properties, such as mass, radius, tidal deformability, etc., are also affected
by both DM and the magnetic fields. The change in the magnitude of different
neutron star observables is proportional to the amount of DM percentage and
the strength of the magnetic field. We observe that the change is seen mainly
in the core part of the star without affecting the crustal properties.

##  I Introduction

In recent years, the investigation of magnetars and pulsars, characterized as
highly magnetized neutron stars, has emerged as a fascinating research field
at the juncture of nuclear physics and astrophysics. These enigmatic celestial
objects exhibit magnetic fields of remarkable strength ($B\sim\
10^{17}-10^{18}$ G), surpassing those typically observed in neutron stars by
several orders of magnitude Makishima  _et al._ (2014). Such immensely
powerful field conditions are presently beyond the reach of terrestrial
laboratories. Consequently, pulsars and magnetars serve as extraterrestrial
laboratories for examining and advancing physical theories. These objects
present a plethora of exhilarating phenomena, including manifestations of
exotic QED mechanisms like photon splitting and magnetic pair creation Baring
and Harding (2001), outburst and quiescent emissions Ellison  _et al._ (2000),
seismic activity Lander (2023), dissipative processes in the magnetospheres Hu
_et al._ (2022), axion-like particles Fortin  _et al._ (2021), and dense
matter physics Ferreira and Fraga (2023), among others. The exploration of
these physical phenomena establishes the study of magnetized neutron stars as
a pivotal research area in astrophysics, offering valuable insights into the
behaviour of matter and radiation in extreme environments Kaspi and
Beloborodov (2017).

It is a well-known fact with compelling evidence that most of the matter in
the Universe is dark matter (DM) Bertone  _et al._ (2005). Since neutron stars
are highly compact and dense, the collision between the DM particles and
constituents of the neutron star results in the loss of energy for the DM to
become bound to the gravitational pull of the neutron star. Therefore, neutron
stars have long been used as a tool in the quest to uncover the particle
nature of DM and their scattering cross sections Bertone and Fairbairn (2008);
Bell  _et al._ (2021); Kain (2021). The DM admixture neutron star results in
significant deviation in the neutron star observables such as mass-radius
profile, tidal deformation, luminosity Kouvaris (2008), accretion de Lavallaz
and Fairbairn (2010), etc., and hence, can act as a probe to measure the DM
properties indirectly.

In the last two decades, there have been numerous attempts to study the DM
admixed neutron star and associated properties Sandin and Ciarcelluti (2009);
Ciarcelluti and Sandin (2011); Ellis  _et al._ (2018); Hooper and Wang (2004);
Bertone and Fairbairn (2008); Kouvaris (2008); Das  _et al._ (2019, 2021a,
2022, 2021b, 2021c); Routaray  _et al._ (2023); Lopes and Das (2023). However,
despite significant progress, the impact of DM on magnetized neutron stars
remains a scientific question that necessitates immediate exploration. Since
most of the observed neutron stars are either pulsars or magnetars Manchester
_et al._ (2005), it becomes essential to study the influence of DM on these
compact stars. Recently strange star admixed with fermionic DM in a strong
magnetic field (MF) was analyzed using the MIT bag model Ferreira and Fraga
(2023). It was shown that tidal deformability gets intensely affected by such
stars.

In this paper, we aim to present an analysis of the unexplored influence of DM
on pure hadronic magnetized neutron stars over a MF range
($10^{17}-10^{19}$G). We use the magnetized neutron star formalism as
described in Broderick  _et al._ (2000); Strickland  _et al._ (2012); Parmar
_et al._ (2023). We use the effective relativistic mean field (E-RMF) model
for the nuclear interaction. The E-RMF theory has been successfully applied to
a wide range of nuclear physics problems ranging from finite nuclei to the
neutron star Patra  _et al._ (2002); Kumar  _et al._ (2018, 2020a); Das  _et
al._ (2021b); Parmar  _et al._ (2022). This theory has recently been used to
address the DM admixed neutron star Das  _et al._ (2021d, 2020). In the
present work, we use two E-RMF models, namely, BigApple Fattoyev  _et al._
(2020) and IOPB Kumar  _et al._ (2018). The BigApple force is designed to
account for the $2.6M_{\odot}$ and original constraint on the tidal
deformability of a $1.4M_{\odot}$ neutron star in accordance with the
secondary component of GW190814 Abbott  _et al._ (2020). At the same time, the
IOPB parameter set reproduces the maximum mass from massive pulsars such as
PSR J0740+6620, which estimate that the neutron star mass should be greater
than $2~{}M_{\odot}$ ($M=2.14_{-0.09}^{+0.10}\ M_{\odot}$) Cromartie  _et al._
(2020)). These parameter sets also reproduce the nuclear matter and finite
nuclei observables in agreement with empirical and observational constraints
Kumar  _et al._ (2018); Das  _et al._ (2021e).

To model the DM interaction with the neutron star, we consider the Neutralino
as a DM candidate, which belongs to WIMP as taken in Ref. Panotopoulos and
Lopes (2017). Further, DM is treated analogous to a neutron, considering the
DM as a charge-less fermion. Other types of DM candidates are also
hypothesized, such as Bosonic, asymmetric, etc., having different properties
compared to the Fermionic one. Several works have been done to explore such
types of scenarios that explore the effects of DM on the different NS
properties Panotopoulos and Lopes (2017); Das  _et al._ (2019, 2021a, 2022,
2021c, 2021b); Routaray  _et al._ (2023). However, there are still windows to
explore the DM effects on other properties of the compact star, its particle
nature, etc.

Henceforth, we aim to investigate the possible changes in the DM admixed
magnetized neutron star equation of state (EOS), its composition, and neutron
star observables, which include mass-radius relations, tidal deformability,
and $f$-mode oscillation. It is seen that with an increasing DM mass, the
maximum mass of the neutron star decreases Das  _et al._ (2019). On the other
hand, it was shown that the maximum mass of the neutron star is an
increasing/decreasing function of the MF depending on the
perpendicular/parallel pressure Patra  _et al._ (2020); Rather  _et al._
(2021); Bordbar and Karami (2022). Therefore, it is interesting to examine the
combined effect of DM admixed neutron star. Although the neutron star should
be deformed due to the anisotropic pressure in the presence of the MF, we
consider the neutron star to be spherically symmetric, which allows us to use
the Tolman-Oppenheimer-Volkoff (TOV) solution. This assumption holds good for
the range of MF $10^{15}-10^{18}$ as the deformation from spherical symmetry
turns out to be less than 1% Patra  _et al._ (2020); Bordbar and Karami
(2022); Chu  _et al._ (2015). Moreover, in the present work, we present
separate results for the perpendicular and parallel pressure in line with
Refs. Chu  _et al._ (2015); Huang  _et al._ (2010); Patra  _et al._ (2020).

The paper’s organization is as follows: In Sec II, we describe the effect of
MF on the EOS employing the E-RMF framework and DM Model. The results
indicating the EOS, composition, mass-radius relations, tidal deformability,
and $f$-mode oscillation are discussed in Sec. III. Finally, we summarize our
results in Sec. IV.

##  II Formalism

###  II.1 RMF model

The effective Lagrangian in the E-RMF, which include $\sigma$, $\omega$,
$\rho$, $\delta$, and photon in association with the baryons can be written as
Patra  _et al._ (2002); Müller and Serot (1996); Wang (2000); Kumar  _et al._
(2020b); Das  _et al._ (2021a),

$\displaystyle\mathcal{E}(r)$ $\displaystyle=$
$\displaystyle\psi^{\dagger}(r)\quantity{i\alpha\cdot\gradient+\beta[M-\Phi(r)-\tau_{3}D(r)]+W(r)+\frac{1}{2}\tau_{3}R(r)+\frac{1+\tau_{3}}{2}A(r)-\frac{i\beta\alpha}{2M}\quantity(f_{\omega}\gradient
W(r)+\frac{1}{2}f_{\rho}\tau_{3}\gradient R(r))}\psi(r)$ (1) $\displaystyle+$
$\displaystyle\quantity(\frac{1}{2}+\frac{k_{3}\Phi(r)}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi(r)^{2}+\frac{1}{2g^{2}_{s}}\quantity\Big(1+\alpha_{1}\frac{\Phi(r)}{M})(\gradient\Phi(r))^{2}-\frac{1}{2g^{2}_{\omega}}\quantity\Big(1+\alpha_{2}\frac{\Phi(r)}{M})(\gradient
W(r))^{2}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi(r)}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}(r)-\frac{1}{2e^{2}}(\gradient
A^{2}(r))^{2}-\frac{1}{2g^{2}_{\rho}}(\gradient
R(r))^{2}-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi(r)}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}(r)$
$\displaystyle-$
$\displaystyle\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W(r)^{4}-\Lambda_{\omega}(R^{2}(r)W^{2}(r))+\frac{1}{2g^{2}_{\delta}}(\gradient
D(r))^{2}+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D(r))^{2}.$

Here $\Phi(r)$, $W(r),R(r),D(r)$ and $A(r)$ are the fields corresponding to
$\sigma$, $\omega$, $\rho$ and $\delta$ mesons and photon, respectively. The
$g_{s}$, $g_{\omega}$, $g_{\rho}$, $g_{\delta}$ and $\frac{e^{2}}{4\pi}$ are
the corresponding coupling constants and $m_{s}$, $m_{\omega}$, $m_{\rho}$ and
$m_{\delta}$ are the corresponding masses. The zeroth component $T_{00}=H$ and
the third component $T_{ii}$ of energy-momentum tensor

$T_{\mu\nu}=\partial^{\nu}\phi(x)\frac{\partial\mathcal{E}}{\partial(\partial_{\mu}\phi(x))}-\eta^{\nu\mu}\mathcal{E},$
(2)

yields the energy and pressure density, respectively.

The energy spectrum of the proton, which gets modified due to the Landau
level, is written as Broderick  _et al._ (2000); Strickland  _et al._ (2012)

$E_{p}=\sqrt{k_{z}^{2}+\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{p^{2}}}+W-R/2,$ (3)

and for charged leptons (electron and muon) as

$E_{e,\mu}=\sqrt{k_{z}^{2}+\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{e,\mu}^{2}}},$ (4)

where

$\displaystyle\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{(p)^{2}}=M_{(p)}^{*^{2}}+2\Big{(}n+\frac{1}{2}-\frac{1}{2}\frac{q}{|q|}\sigma_{z}\Big{)}|q|B.$
(5) $\displaystyle\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{(e,\mu)^{2}}=M_{(e,\mu)}^{2}+2\Big{(}n+\frac{1}{2}-\frac{1}{2}\frac{q}{|q|}\sigma_{z}\Big{)}|q|B.$
(6)

Here, $\sigma_{z}$ is the spin along the axis of the MF $B$, $n$ is the
principal quantum number, and $k_{z}$ is the momentum along the direction of
the MF. $M^{*}$ is the effective mass for the proton. The neutron spectrum is
similar to the Dirac particle and takes form.

$E_{n}=\sqrt{k^{2}+M_{n}^{*^{2}}}+W+R/2.$ (7)

The number and energy density at zero temperature and in the presence of a MF
is given by Broderick  _et al._ (2000)

$\rho_{i=e,\mu,p}=\frac{|q|B}{2\pi^{2}}\sum_{\sigma_{z}}\sum_{n=0}^{n_{max}}k_{f,n,\sigma_{z}}^{i},$
(8) $\displaystyle E_{i=e,\mu,p}$ $\displaystyle=$
$\displaystyle\frac{|q|B}{4\pi^{2}}\sum_{\sigma_{z}}\sum_{n=0}^{n_{max}}$ (9)
$\displaystyle\crossproduct$
$\displaystyle\Big{[}E_{f}^{i}k_{f,n,\sigma_{z}}^{i}+\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}\ln\Big(\Big{|}\frac{E_{f}^{i}+k_{f,n,\sigma_{z}}^{i}}{\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}}\Big{|}\Big{missing})\Big{]}.$

respectively. In above equations, $k_{f,n,\sigma_{z}}^{i}$ is defined by

$k_{f,n,\sigma_{z}}^{i^{2}}=E_{f}^{i^{2}}-\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}},$ (10)

where the Fermi energies are fixed by the respective chemical potentials given
by

$\displaystyle E_{f}^{l=e,\mu}=\mu_{\mu,e},$ (11) $\displaystyle
E_{f}^{b=p,n}=\mu_{b}-W\pm R/2.$ (12)

In Eqs. (8) and (9), the $n_{\rm max}$ is the integer for which the Fermi
momentum remains positive in Eq. (10) and is written as

$\displaystyle n_{\rm
max}=\Bigg{[}\frac{E_{f}^{i^{2}}-M^{*^{2}}}{2|q|B}\Bigg{]},\;{\rm proton}$
(13) $\displaystyle n_{\rm
max}=\Bigg{[}\frac{E_{f}^{i^{2}}-M^{2}}{2|q|B}\Bigg{]},\;\;{\rm
electron}\;\&\;{\rm muon}\,.$

Here $[x]$ represents the greatest integer less than or equal to $x$. The
scalar density for the protons is further determined as

$\rho_{p}^{s}=\frac{|q|BM^{*}}{2\pi^{2}}\sum_{\sigma_{z}}\sum_{n=0}^{n_{max}}\ln\Big(\Big{|}\frac{E_{f}^{i}+k_{f,n,\sigma_{z}}^{i}}{\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}}\Big{|}\Big{missing}).$ (14)

The number, scalar, and energy density for the neutrons are similar to the
field-free case and can be found in Kumar  _et al._ (2018); Patra  _et al._
(2002) and reference therein. The total energy density is the sum of matter-
energy density and the contribution from the electromagnetic field,
$\frac{B^{2}}{8\pi}$.

Now the parallel ($P_{\parallel}$) and perpendicular ($P_{\perp}$) components
of the pressure can be written as Chakrabarty (1996)

$\displaystyle P_{\parallel}=\frac{|q|B}{4\pi^{2}}\sum_{\sigma_{z}=\pm
1}\sum_{n=0}^{n=n_{max}}\Bigg{[}E_{f}^{i}k_{f,n,\sigma_{z}}^{i}-\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}$ (15)
$\displaystyle\times\ln\Big(\Big{|}\frac{E_{f}^{i}+k_{f,n,\sigma_{z}}^{i}}{\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}}\Big{|}\Big{missing})\Bigg{]}.$

It also follows from Eq. (9) and (15) that

$P_{\parallel}=\sum_{i=n,p}\mu_{i}\rho_{i}-E\,.$ (16)

The transverse pressure is written as

$\displaystyle P_{\perp}=\frac{|q|^{2}B^{2}}{2\pi^{2}}\sum_{\sigma_{z}=\pm
1}\sum_{n=0}^{n=n_{max}}n\ln\Big(\Big{|}\frac{E_{f}^{i}+k_{f,n,\sigma_{z}}^{i}}{\mskip
1.19998mu\overline{\mskip-1.19998mu{M}\mskip-1.19998mu}\mskip
1.19998mu_{n,\sigma_{z}}^{{i^{2}}}}\Big{|}\Big{missing}).$ (17)

The pure electromagnetic contribution $\propto B^{2}$ to the energy and
pressure density is taken as $\frac{B^{2}}{8\pi}$ Broderick  _et al._ (2000).

Figure 1: The particle fractions (upper panel) of different species with the
presence of MF and DM having Fermi momenta 0.00, 0.02, and 0.04 GeV,
respectively for BigApple (left panel), and IOPB-I (right panel) parameter
sets and their EoSs (lower panel).

###  II.2 Density-dependent MF

In this work, the MF ($B$) is parametrized from the surface to the center of
the star as Rabhi  _et al._ (2009); Bandyopadhyay  _et al._ (1997); Mallick
and Schramm (2014); Bordbar and Karami (2022)

$B\Bigg{(}\frac{\rho}{\rho_{0}}\Bigg{)}=B_{\rm
surf}+B_{c}\Bigg{(}1-\exp{-\beta\Bigg{(}\frac{\rho}{\rho_{0}}\Bigg{)}^{\gamma}}\Bigg{)}.$
(18)

Here, $\rho_{0}$ is the saturation density, $B_{\rm surf}$ is the surface MF
taken to be $10^{15}$ G and $B_{c}$ is the MF at the center of the star. The
parameters $\beta=0.02$ and $\gamma=3.00$ are chosen to reproduce the
observational MF Casali  _et al._ (2014).

###  II.3 DM Model

In this section, we provide the formalism for the DM admixed neutron star. The
DM interacts with nucleons through the Higgs portal. In this study, we choose
Neutralino as a DM candidate, which belongs to WIMPS. The interacting
Lagrangian is in the following Panotopoulos and Lopes (2017); Das  _et al._
(2019, 2020, 2021a, 2021c):

$\displaystyle{\cal{L}}_{\rm DM}$ $\displaystyle=$
$\displaystyle\bar{\chi}\Big{[}i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\Big{]}\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(19) $\displaystyle-$ $\displaystyle\frac{1}{2}M_{h}^{2}h^{2}+\sum
f\frac{m}{v}\bar{\psi}h\psi,$

$\psi_{{}_{B}}$ and $\chi$ are the baryons and DM wave functions respectively.
Here, we choose values of the DM-Higgs coupling ($y$), proton-Higgs form
factor ($f$), and vacuum value ($v$) of Higgs are 0.07, 0.35, and 246 GeV,
respectively, as considered in Refs. Das  _et al._ (2020, 2021a). The free
parameters are constrained with the help of DM detection data available to
date Das  _et al._ (2021a). With this preliminary information, we can
calculate the DM scalar density ($\rho_{s}^{\rm DM}$), energy density, and
pressure using the mean-field approximation as done in Refs. Das  _et al._
(2019); Panotopoulos and Lopes (2017)

$\rho_{s}^{\rm
DM}=\langle\bar{\chi}\chi\rangle=\frac{\gamma}{2\pi^{2}}\int_{0}^{k_{f}^{\rm
DM}}dk\ \frac{M_{\chi}^{\star}}{\sqrt{M_{\chi}^{\star}{{}^{2}}+k^{2}}},$ (20)

where $k_{f}^{\rm DM}$ is the Fermi momentum for DM. $\gamma$ is the spin
degeneracy factor with a value of 2 for neutron and proton.

The energy density (${\cal{E}}_{\rm DM}$) and pressure ($P_{\rm DM}$) for
neutron star with DM can be obtained by solving the Eq. (19)

$\displaystyle{\cal{E}}_{\rm DM}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int_{0}^{k_{f}^{\rm DM}}k^{2}\
dk\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2},$ (21)

and

$\displaystyle P_{\rm DM}$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}\int_{0}^{k_{f}^{\rm DM}}\frac{k^{4}\
dk}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2},$ (22)

$M_{h}$ is the Higgs mass equal to 125 GeV, and $h_{0}$ is the Higgs field.

The total EoS for the DM admixed neutron star in the presence of MF is given
by

$\mathcal{E}=\mathcal{E}_{\rm nucl.}+\mathcal{E}_{\rm DM},P=P_{\rm
nucl.}+P_{\rm DM}.$ (23)

The MF does not affect the DM since it is charged less, similar to a neutron.
However, with the Higgs contribution, the effective mass of the system becomes

$M^{*}_{n,p}=M-\Phi-\sum_{n,p}\frac{fM_{n,p}}{v}h.$ (24)

The effective mass then dictates the property of the system through $n_{max}$
in Eq. (13).

##  III Results and Discussion

Figure 2: Mass-Radius relations for the DM admixed magnetized neutron star
with varying the MF strength for BigApple (left) and IOPB-I (right). The
overlaid bands are the observational data given by different observations (see
text for details).

###  III.1  Particle Fractions and EoS

The particle fraction (PF) of the species, such as neutrons, protons,
electrons, muons, and DM, can be calculated using the formula

$X_{i}=\rho_{i}/\rho_{b}\,,$ (25)

where $\rho_{i}$ is the density of each species, and $\rho_{b}$ is the baryon
density.

In Fig. 1, we calculate the value of $X_{i}$ for DM admixed magnetized neutron
star for three different DM momenta $k_{f}^{\rm DM}=0.00,0.02$, and $0.04$ GeV
with the variation of the core MF. The PF for neutron and DM are not taken as
they are charge-less particles. Hence, the MF has no interactions with them.
In this study, we have not included the anomalous magnetic moment (AMM) of the
neutrons and protons for simplicity of the results.

From the upper panel of Fig. 1, we notice that protons and electrons appear
almost at the same density. However, the muon appears $\approx 0.1$ fm-3 for
both BigApple and IOPB-I cases. It is observed that a lower magnitude of MF
(for example, $10^{17}$ G) doesn’t change the PF significantly. However, with
an increase in the MF strength at the core, the population density changes and
shows oscillating behavior due to the subsequent filling of Landau levels.
This is because, with an increase in MF strength, the mass of the charged
particle becomes heavier, and it oscillates prominently, especially in the
core of the neutron star, as mentioned in Ref. Patra  _et al._ (2020). We have
also calculated the PF for the parallel ($\parallel$) and perpendicular
($\perp$) components of the MF and found that it is almost similar for both
components. In addition to MF, it is also noticed that the neutron star with a
finite DM fraction has does not affect the PF.

The EOSs for magnetized neutron stars with DM admixture are calculated and
presented in the lower panel of Figure 1, illustrating the dependence on the
MF strength for various DM fractions. The EOSs exhibit increased stiffness (or
softness) for the $\perp$ ($\parallel$) components of the MF. Additionally,
the presence of DM introduces further softening effect on the EOSs, with the
degree of softness primarily determined by the DM content within the neutron
star. Comparing the models employed in this study, BigApple demonstrates a
stiffer EOS than IOPB-I. However, the influence of the MF on the softness or
stiffness of the BigApple model is relatively less prominent as compared to
the IOPB-I case. This model dependency of the MF effect on EoS stems from
their effective masses which control the number of Landau levels (see Eq. (24)
and (13)).

In a previous study Parmar  _et al._ (2023), we computed the equation of
states (EOSs) for magnetized crusts using the CLDM model. In the present
investigation, we employ the same model to determine the crust EOS. However,
for the core EOS, we consider robust mean-field (RMF) models, namely BigApple
and IOPB-I. Subsequently, we construct unified EOSs encompassing the BigApple
and IOPB-I cases, as depicted in Figure 1. It is worth noting that the
percentage of DM remains nearly constant throughout the neutron star,
predominantly concentrated within the core region. The MF, on the other hand,
exerts negligible influence on the crust EOS. Consequently, the lower-density
EOSs for both the BigApple and IOPB-I models exhibit minimal variation as a
function of MF strength and DM fractions.

###  III.2 Mass-Radius relations, tidal deformability

The mass-radius ($M-R$) relations are obtained with Tolmann-Oppenheimer-
Volkoff Tolman (1939); Oppenheimer and Volkoff (1939) equations for the range
of central densities, which is shown in Fig. 2. We calculate the $M-R$
profiles for both BigApple and IOPB-I EOSs with $\parallel$ and $\perp$
components of MF by varying DM momenta 0.00, 0.02, and 0.04 GeV. The magnitude
of the maximum mass and its corresponding radius decreases for parallel MF
components and vice-versa for the perpendicular fields. In addition to MF, the
DM also reduces the magnitude of $M$ and $R$ values, which depend on its
percentage inside the star. In the case of the BigApple (IOPB-I) case, the
maximum mass is $2.60\ (2.15)M_{\odot}$, and its corresponding radius is
$12.41\,(11.91)$ km without the inclusion of the DM and MF.

With the inclusion of MF/DM, the magnitude of the maximum mass and its
corresponding radius decreases $\sim 4-5\%$. For $k_{f}^{\rm DM}=0.00,0.02$
GeV with all MF components, the curves corresponding to the BigApple model
reasonably satisfy the overlaid observational data. However, with $k_{f}^{\rm
DM}=0.04$ GeV, all curves satisfy only the maximum mass constraint given by
PSR J0740+6620 Cromartie  _et al._ (2020). Moreover, the canonical radii
corresponding to 0.04 GeV for the BigApple case don’t pass through NICER and
revised NICER limits Miller  _et al._ (2019, 2021). In the case of the IOPB-I
parameter set (right panel of Fig. 2) with $k_{f}^{\rm DM}=0.00,0.02$ GeV,
only parallel components of MF satisfy all the constraints imposed by
Cromartie et al., pulsar, and NICER. However, for $k_{f}^{\rm DM}=0.04$ GeV,
none of the constraints is satisfied for IOPB-I set. Hence, from this study,
we observe that one can put constraints on the amount of DM percentage and the
strength of the MF by employing diverse observational data.

Figure 3: Maximum mass of magnetized neutron star admixed with the DM as a
function of DM momentum for different MF values.

Through the preceding analysis, it is evident that the EOS and mass-radius
profile of DM admixed magnetized neutron stars are governed by two competing
mechanisms. Firstly, the EOS experiences stiffening (softening) due to the
$\perp$ ($\parallel$) pressure, while secondly, it undergoes softening as a
consequence of increased DM content. To investigate the potential influence of
MF on the rate at which the neutron star mass decreases due to the presence of
DM, we construct a plot of the maximum mass as a function of DM Fermi momenta
for varying MF strengths, as illustrated in Fig. 3. Remarkably, for higher MF
strengths, the rate of maximum mass reduction caused by DM content exhibits a
slight decrease. In other words, the MF exerts an attenuating effect on the DM
within the neutron star. Nonetheless, this effect is primarily notable under
high MF strengths, indicating that the MF does not significantly impact the
DM’s influence on the properties of neutron stars.

Figure 4: Dimensionless tidal deformability for the DM admixed magnetized
neutron star with varying the MF strength for BigApple (left) and IOPB-I
(right). The value of $\Lambda_{1.4}=190_{-120}^{+390}$, and
$\Lambda_{1.4}=616_{-158}^{+273}$ taken from the GW170817, and GW190814 events
respectively.

Next, we calculate the dimensionless tidal deformability of the DM admixed
magnetized neutron star. The dimensionless tidal deformability ($\Lambda$) of
the star is calculated using the relation Hinderer (2008)

$\Lambda=\frac{2}{3}k_{2}C^{5}\,,$ (26)

where $k_{2}$ is the Love number for the quadrupole case. The solution for
$k_{2}$ can be found in Refs. Hinderer (2008); Das  _et al._ (2021e). $C$ is
the compactness defined as $M/R$. We calculate the $\Lambda$ for different DM
momenta with the variation of MF, which is shown in Fig. 4 for BigApple and
IOPB-I E-RMF sets. With the addition of DM, the values of $\Lambda$ decrease.
The curves corresponding to all the magnetized EOSs, including different DM
momenta, are observed to be relatively diminished in the lower mass regimes.
However, substantial changes have been observed at the maximum mass limit.
Different error bars are taken from the GW170817 and GW190814 events to
constrain the value of $\Lambda$. Except for DM momenta 0.04 GeV, none of the
curves passes through the GW170817 data. However, all the curves well
reproduced the GW190814 limit except for $k_{f}^{DM}=0.04$ GeV for both
BigApple and IOPB-I cases. In comparison to DM, the change in the value of
$\Lambda$ for both $\parallel$ and $\perp$ components are bleak for the
canonical mass. However, the $\Lambda$ changes considerably for maximum
neutron star mass depending on the strength of MF. Hence, we observed a
significant effect due to the DM for both BigApple and IOPB-I cases.

###  III.3 Calculation of $f$-mode oscillation of the magnetized neutron star

In the present section, we use the formalism required to calculate the
$f$-mode frequency as done in our previous study with the relativistic Cowling
approximation Das  _et al._ (2021b). The $f$-mode frequency for the quadrupole
case is calculated with the variation of MF strength for different DM
fractions, as shown in Fig. 5. We find the marginal changes in the $f$-mode
frequency in the case of the BigApple. With the increase in $k_{f}^{\rm DM}$,
the EOS becomes softer, which gives the lower values of the maximum mass and
its corresponding radius. Also, a relatively lower massive star oscillates
with a higher frequency and vice-versa. Therefore, the magnitude of $f$-mode
frequency for $k_{f}^{\rm DM}=0.04$ GeV is higher than other DM momenta. We
observe that the parallel and perpendicular MF components have marginal
effects on the $f$-mode frequency of the star.

In the case of IOPB-I (right panel in Fig. 5), there are significant changes
for both MF as well as DM. This is because the MF considerably affects the
EOSs for the IOPB-I. The magnitude of $f$-mode frequency is higher for IOPB-I
because it has a softer EOS compared to BigApple. Therefore, it oscillates
with a higher magnitude and radiates more $f$-mode frequency.

Figure 5: $f$-mode oscillation frequency of the magnetized neutron star with
different fractions of DM for BigApple (left) and IOPB-I (right) cases.

###  III.4 Relative change in the magnitude of neutron star properties

Figure 6: Relative change in the magnitude with respect to zero magnetic cases
in the center ($B_{c}$) for $\Delta M$, $\Delta R$, $\Delta\Lambda$,
$\Delta\bar{I}$, and $\Delta f$-mode frequency compare with different DM
fractions for BigApple (upper), and IOPB-I (lower) cases.

Given the significant impact of both MF and DM on neutron star properties, it
becomes imperative to thoroughly investigate and understand their relative
influences on various aspects of neutron stars. To examine this, in Fig. 6, we
calculate the percentage change in the magnitude of the mass, radius, tidal
deformability, dimensionless moment of inertia, and $f$-mode oscillation
frequency in comparison to zero MF strength and DM content for three different
DM percentage. The pink and cyan bars represent the perpendicular and parallel
components of the pressure corresponding to that central MF ($B_{c}$).

The relative changes in the neutron star properties are contingent upon three
key components: (i) the strength of the MF, (ii) the percentage of DM, and
(iii) the specific EOS employed. Notably, for MF strengths below $1\times
10^{18}$ G, both the parallel and perpendicular pressure components exhibit
negligible variations, as the emergence of anisotropy becomes evident only at
higher MF strengths. In the presence of higher MF strengths, the parallel
pressure component facilitates a decrease in the maximum mass due to the
influence of DM, while the perpendicular pressure component mitigates the rate
at which this decrement occurs. In contrast to the maximum mass, the radius of
the star exhibits the opposite behavior. Specifically, for higher MF
strengths, the perpendicular component of the pressure contributes to a
decrease in the radius, while the parallel pressure component attenuates this
effect. The maximum mass and corresponding radius of the star can exhibit
changes as high as 10%, highlighting significant variations in these
properties due to the presence of MF and DM.

The dimensionless tidal deformability ($\Lambda$) and the normalized moment of
inertia ($\bar{I}$) are notably more influenced by the presence of MF compared
to DM. Additionally, the changes in these properties demonstrate sensitivity
to the chosen EOS. The combined effects of DM and MF offer a range of
possibilities that prove valuable in reproducing specific observational data
associated with $\Lambda$ and $\bar{I}$.

Furthermore, the $f$-mode oscillation frequency is found to be influenced by
the presence of MF and DM. The introduction of DM significantly increases the
$f$-mode frequency, reaching up to 2% for $k_{f}^{\rm DM}=0.02$ and 10% for
$k_{f}^{\rm DM}=0.04$. The presence of MF further impacts this frequency
increase. The influence of MF on top of DM depends on the strength of the MF
and the chosen EOS used in the analysis.

##  IV Summary and Conclusion

This study explores the different properties of the DM admixed magnetized
neutron star. The magnetized EOSs are calculated with the relativistic mean-
field model with density-dependent MF. The well-known RMF models, namely
BigApple, and IOPB-I, are used to obtain the EOSs for the magnetized neutron
star. In the case of the DM, we choose the simple DM model, where the DM
particle interacts with nucleons by exchanging Higgs. The MF strength in the
core is varied by fixing the surface MF. Moreover, the DM fraction is almost
constant throughout the neutron star. To see its effect on the magnetized
neutron star, we vary its fraction from $0.00-0.04$ GeV. We have calculated
various properties such as mass, radius, tidal deformability, $f$-mode
frequency with different interaction strengths for MF, and percentage of DM.

The EOSs of the DM admixed magnetized neutron star are found to be softer for
the parallel component of the pressure and stiffer for the perpendicular one.
The softness also depends on the DM contained inside the neutron star. The
particle fraction for the magnetized neutron star exhibits an oscillating
nature (predominantly in the core region of the star), due to an increase in
the proton mass and filling of Landau levels for the high MF. The macroscopic
properties are significantly affected by the DM as well as the MF. A higher DM
percentage having high MF strength in the core has pronounced effects on the
neutron star.

It has been observed that the maximum mass and its corresponding radius
decreases $\sim 4-5\%$ for DM admixed neutron star. The resulting maximum mass
and radius of the magnetized neutron star admixed with DM then result from the
competitive behavior of MF strength and DM percentage. Furthermore, the rate
at which the maximum mass decreases with increasing DM percentage is
attenuated by the presence of MF. The combined inclusion of both DM and MF
proves instrumental in reproducing certain observational data that could not
be predicted in the absence of these interactions.

The tidal deformability ($\Lambda$) and $f$-mode oscillation frequency are
obtained for both BigApple and IOPB-I cases by varying the MF strength and DM
percentage. The $\parallel$ components have a higher magnitude of $\Lambda$ in
comparison to $\perp$ one. We found similar results for the $f$-mode frequency
of the star. It has been observed that a significant change arises in
($\Lambda$) and $f$-mode oscillation frequency due to both MF and DM.

In future studies, it is possible to explore additional macroscopic properties
of both static and rotating DM admixed magnetized neutron stars. In this work,
a spherically symmetric neutron star model is considered as a simplifying
assumption. However, to calculate the properties of neutron stars more
efficiently, advanced techniques such as Lorene and numerical relativity can
be employed for both the static and rotating cases. Such techniques may
provide a comprehensive insight into the properties of DM admixed magnetized
neutron stars.

## References

  * Makishima  _et al._ (2014) K. Makishima, T. Enoto, J. S. Hiraga, T. Nakano, K. Nakazawa, S. Sakurai, M. Sasano, and H. Murakami, Phys. Rev. Lett. 112, 171102 (2014).
  * Baring and Harding (2001) M. G. Baring and A. K. Harding, The Astrophysical Journal 547, 929 (2001).
  * Ellison  _et al._ (2000) D. C. Ellison, E. G. Berezhko, and M. G. Baring, The Astrophysical Journal 540, 292 (2000).
  * Lander (2023) S. K. Lander, The Astrophysical Journal Letters 947, L16 (2023).
  * Hu  _et al._ (2022) K. Hu, M. G. Baring, A. K. Harding, and Z. Wadiasingh, The Astrophysical Journal 940, 91 (2022).
  * Fortin  _et al._ (2021) J.-F. Fortin, H.-K. Guo, S. P. Harris, E. Sheridan, and K. Sinha, Journal of Cosmology and Astroparticle Physics 2021, 036 (2021).
  * Ferreira and Fraga (2023) O. Ferreira and E. S. Fraga, Journal of Cosmology and Astroparticle Physics 2023, 012 (2023).
  * Kaspi and Beloborodov (2017) V. M. Kaspi and A. M. Beloborodov, Annual Review of Astronomy and Astrophysics 55, 261 (2017).
  * Bertone  _et al._ (2005) G. Bertone, D. Hooper, and J. Silk, Physics Reports 405, 279 (2005).
  * Bertone and Fairbairn (2008) G. Bertone and M. Fairbairn, Phys. Rev. D 77, 043515 (2008).
  * Bell  _et al._ (2021) N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. Thomas, and M. Virgato, Phys. Rev. Lett. 127, 111803 (2021).
  * Kain (2021) B. Kain, Phys. Rev. D 103, 043009 (2021).
  * Kouvaris (2008) C. Kouvaris, Phys. Rev. D 77, 023006 (2008).
  * de Lavallaz and Fairbairn (2010) A. de Lavallaz and M. Fairbairn, Phys. Rev. D 81, 123521 (2010).
  * Sandin and Ciarcelluti (2009) F. Sandin and P. Ciarcelluti, Astroparticle Phys. 32, 278–284 (2009).
  * Ciarcelluti and Sandin (2011) P. Ciarcelluti and F. Sandin, Phys. Lett. B 695, 19–21 (2011).
  * Ellis  _et al._ (2018) J. Ellis, G. Hütsi, K. Kannike, _et al._ , Phys. Rev. D 97, 123007 (2018).
  * Hooper and Wang (2004) D. Hooper and L.-T. Wang, Phys. Rev. D 69, 035001 (2004).
  * Das  _et al._ (2019) A. Das, T. Malik, and A. C. Nayak, Phys. Rev. D 99, 043016 (2019).
  * Das  _et al._ (2021a) H. C. Das, A. Kumar, B. Kumar, _et al._ , JCAP 2021, 007 (2021a).
  * Das  _et al._ (2022) H. C. Das, A. Kumar, B. Kumar, _et al._ , Galaxies 10 (2022), 10.3390/galaxies10010014.
  * Das  _et al._ (2021b) H. C. Das, A. Kumar, S. K. Biswal, and S. K. Patra, Phys. Rev. D 104, 123006 (2021b).
  * Das  _et al._ (2021c) H. C. Das, A. Kumar, and S. K. Patra, MNRAS 507, 4053 (2021c).
  * Routaray  _et al._ (2023) P. Routaray, H. C. Das, S. Sen, B. Kumar, G. Panotopoulos, and T. Zhao, Phys. Rev. D 107, 103039 (2023).
  * Lopes and Das (2023) L. L. Lopes and H. C. Das, J. Cosmol. Astropart. Phys. 2023, 034 (2023).
  * Manchester  _et al._ (2005) R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, The Astronomical Journal 129, 1993 (2005).
  * Broderick  _et al._ (2000) A. Broderick, M. Prakash, and J. M. Lattimer, The Astrophysical Journal 537, 351 (2000).
  * Strickland  _et al._ (2012) M. Strickland, V. Dexheimer, and D. P. Menezes, Phys. Rev. D 86, 125032 (2012).
  * Parmar  _et al._ (2023) V. Parmar, H. C. Das, M. K. Sharma, _et al._ , Phys. Rev. D 107, 043022 (2023).
  * Patra  _et al._ (2002) S. K. Patra, M. Centelles, X. Viñas, _et al._ , Phys. Rev. C 65, 044304 (2002).
  * Kumar  _et al._ (2018) B. Kumar, S. K. Patra, and B. K. Agrawal, Phys. Rev. C 97, 045806 (2018).
  * Kumar  _et al._ (2020a) A. Kumar, H. Das, S. Biswal, _et al._ , The European Physical Journal C 80, 1 (2020a).
  * Parmar  _et al._ (2022) V. Parmar, H. C. Das, A. Kumar, _et al._ , Phys. Rev. D 105, 043017 (2022).
  * Das  _et al._ (2021d) H. C. Das, A. Kumar, and S. K. Patra, Phys. Rev. D 104, 063028 (2021d).
  * Das  _et al._ (2020) H. C. Das, A. Kumar, B. Kumar, _et al._ , Monthly Notices of the Royal Astronomical Society 495, 4893 (2020).
  * Fattoyev  _et al._ (2020) F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, _et al._ , Phys. Rev. C 102, 065805 (2020).
  * Abbott  _et al._ (2020) B. P. Abbott, R. Abbott, T. D. Abbott, _et al._ , The Astrophysical Journal 892, L3 (2020).
  * Cromartie  _et al._ (2020) H. T. Cromartie, E. Fonseca, S. M. Ransom, _et al._ , Nature Astronomy 4, 72–76 (2020).
  * Das  _et al._ (2021e) H. C. Das, A. Kumar, B. Kumar, _et al._ , Int. J. Mod. Phys. E 30, 2150088 (2021e), arXiv:2009.10690 [nucl-th] .
  * Panotopoulos and Lopes (2017) G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 083004 (2017).
  * Patra  _et al._ (2020) N. K. Patra, T. Malik, D. Sen, _et al._ , The Astrophysical Journal 900, 49 (2020).
  * Rather  _et al._ (2021) I. A. Rather, U. Rahaman, V. Dexheimer, _et al._ , The Astrophysical Journal 917, 46 (2021), arXiv:2102.04067 [nucl-th] .
  * Bordbar and Karami (2022) G. H. Bordbar and M. Karami, The European Physical Journal C 82, 74 (2022).
  * Chu  _et al._ (2015) P.-C. Chu, X. Wang, L.-W. Chen, and M. Huang, Phys. Rev. D 91, 023003 (2015).
  * Huang  _et al._ (2010) X.-G. Huang, M. Huang, D. H. Rischke, and A. Sedrakian, Phys. Rev. D 81, 045015 (2010).
  * Müller and Serot (1996) H. Müller and B. D. Serot, Nuclear Physics A 606, 508 (1996).
  * Wang (2000) P. Wang, Phys. Rev. C 61, 054904 (2000).
  * Kumar  _et al._ (2020b) A. Kumar, H. C. Das, S. K. Biswal, _et al._ , The European Physical Journal C 80 (2020b).
  * Chakrabarty (1996) S. Chakrabarty, Phys. Rev. D 54, 1306 (1996).
  * Rabhi  _et al._ (2009) A. Rabhi, H. Pais, P. K. Panda, and C. Providência, Journal of Physics G: Nuclear and Particle Physics 36, 115204 (2009).
  * Bandyopadhyay  _et al._ (1997) D. Bandyopadhyay, S. Chakrabarty, and S. Pal, Phys. Rev. Lett. 79, 2176 (1997).
  * Mallick and Schramm (2014) R. Mallick and S. Schramm, Phys. Rev. C 89, 045805 (2014).
  * Casali  _et al._ (2014) R. H. Casali, L. B. Castro, and D. P. Menezes, Phys. Rev. C 89, 015805 (2014).
  * Tolman (1939) R. C. Tolman, Phys. Rev. 55, 364 (1939).
  * Oppenheimer and Volkoff (1939) J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374 (1939).
  * Miller  _et al._ (2019) M. C. Miller, F. K. Lamb, A. J. Dittmann, _et al._ , APJ 887, L24 (2019).
  * Miller  _et al._ (2021) M. C. Miller, F. K. Lamb, A. J. Dittmann, _et al._ , APJ 918, L28 (2021), arXiv:2105.06979 [astro-ph.HE] .
  * Hinderer (2008) T. Hinderer, The Astrophysical Journal 677, 1216 (2008).