# Investigating Dark Matter-Admixed Neutron Stars with NITR Equation of State
in Light of PSR J0952-0607

Pinku Routaray  Sailesh Ranjan Mohanty  H.C. Das  Sayantan Ghosh  P.J. Kalita
V. Parmar  Bharat Kumar 11footnotetext: Corresponding author.

###### Abstract

The heaviest pulsar, PSR J0952-0607, with a mass of $M=2.35\pm 0.17\
M_{\odot}$, has recently been discovered in the disk of the Milky Way Galaxy.
In response to this discovery, a new RMF model, "NITR" has been developed. The
NITR model’s naturalness has been confirmed by assessing its validity for
various finite nuclei and nuclear matter (NM) properties, including
incompressibility, symmetry energy, and slope parameter values of 225.11,
31.69, and 43.86 MeV, respectively. These values satisfy the
empirical/experimental limits currently available. The maximum mass and
canonical radius of a neutron star (NS) calculated using the NITR model
parameters are 2.35 $M_{\odot}$ and 12.73 km, respectively, which fall within
the range of PSR J0952-0607 and the latest NICER limit. This study aims to
test the NITR model consistency by applying it to different systems and,
consequently, calibrate its validity extensively. Subsequently, the NITR model
equation of state (EOS) is employed to obtain the properties of a dark matter
admixed neutron star using two approaches: non-gravitational (single fluid)
and two-fluid. In the two-fluid model, the dark matter (DM) particles only
interact with each other via gravity rather than the nucleons. In both cases,
the equation of state becomes softer due to DM interactions, which reduces
various macroscopic properties such as maximum mass, radius, tidal
deformability, etc. Additionally, through the use of various observational
data, efforts are made to constraint the quantity of DM within the NS.

##  1 Introduction

The neutron star is an extremely compact and complex object in the Universe,
where extreme gravity meets extreme matter [1]. Astrophysicists are especially
interested in determining its structure and dynamics, particularly after the
detection of gravitational waves (GWs) from binary NS mergers [2, 3].
Multifaceted data on NSs [4] allows for the testing of various proposed EOSs,
which are not feasible in terrestrial laboratories [5]. The nuclear EOSs,
which are critical in NS simulations, enable the understanding of several
complex phenomena such as quark deconfinement [6], phase transitions [7, 1],
very high magnetic fields [8, 9, 10], appearances of hyperons [11, 12, 13,
14], accretion of dark matter [15, 16, 17, 18, 19], pasta structures inside
the crust [20], etc. with regards to the NS. Among various properties of the
NS, its mass and radius are crucial in various astrophysical phenomena and
their associated effects. Precise knowledge of these macroscopic properties of
NSs enables the probing of nuclear interactions in unconventional and extreme
environments. Furthermore, since NSs are the primary source of GWs, their
mass, and radii sensitively affect the merger dynamics, timescale, neutrino
bursts, etc. [21, 22].

Recently, a Galactic NS named PSR J0952-0607 has been detected in the disk of
the Milky Way, which is the fastest and heaviest of its kind with a mass
$M=2.35\pm 0.17\ M_{\odot}$ [23]. This discovery is in continuation of the
pulsar PSR J0740+6620 ($M=2.08\pm 0.07\ M_{\odot}$ [24, 25]). Among several
hundred available EOSs that are based on various forms of nuclear interactions
such as relativistic mean field theory (RMF), Skyrme density functional (SHF),
Gogny interaction, etc., only a few of them can reproduce the $M=2.35\pm 0.17\
M_{\odot}$ mass limit while satisfying nuclear matter constraints
simultaneously [26, 27]. Therefore, it has become essential to revisit the
optimization of the parameters of nuclear models to satisfy new constraints on
the NS mass.

In this paper, we propose a new model based on the extended RMF (E-RMF) which
is consistent with the underlying symmetry arising in Quantum Chromodynamics
(QCD) and solves the renormalization problem in conventional RMF theory [28,
29]. The E-RMF formalism has been successfully applied to various nuclear
physics problems, including nuclear structure, reaction, and NS structure. To
maintain consistency between the crust and the core, we calculate the unified
EOS by employing the compressible liquid drop model [20, 30] for the inner
crust calculation and obtain the outer crust EOS using the most recent
experimental masses and the Hartree-Fock-Bogoliubov-26 mass model [31, 32].
Our proposed EOS reproduces the mass of the NS as $2.35\ M_{\odot}$ and
satisfies the nuclear matter (NM) constraints, making it a better alternative
to the existing NS EOS. In this study, we use the NITR EOS to explore
different properties of the NS with dark matter as an additional component
inside it.

There is a plethora of evidence supporting the hypothesis that dark matter
(DM) exists in the Universe, including rotation curves of galaxies, velocity
dispersions, galaxy clusters, gravitational lensing, and cosmic microwave
background. Moreover, recent cosmological findings suggest that DM is not
baryonic matter and must be a new type of matter that interacts only weakly
with particles in the standard model. Despite substantial research into DM
models from a particle physics perspective, the precise characteristics of DM
particles are still unknown, although the weakly interacting massive particles
(WIMPs) scenario has gained popularity. WIMPs can easily account for the known
relic abundance of DM at the weak interaction scale and can be captured by NSs
through elastic scattering with nucleons. The high density inside NSs means
that the collisional energy loss of DM particles can be substantial, allowing
them to be effectively captured inside such compact objects [16, 33, 34, 35,
36, 37]. The interaction of DM inside NSs has been studied using various
models. In some models, DM interacts non-gravitationally with normal matter
(NM) and is considered a single fluid system [38, 17, 39, 40, 41]. Other
models ignore non-gravitational interaction between DM and NM, considering DM
to interact solely with itself through gravity, which results in a two-fluid
system [42, 43, 44, 15, 45].

In the present work, we explore the properties of DM admixed neutron star
(DMANS) by utilizing our newly developed nuclear EOS, "NITR". We investigate
both single-fluid and two-fluid approaches for DMANS. In the single-fluid
case, DM particles interact with ordinary matter by exchanging standard model
Higgs, and we calculate the properties of DMANS by varying the DM Fermi
momentum to observe its effect on the EOS, mass, radius, tidal deformability,
and other parameters. On the other hand, in the two-fluid approach, we vary
the mass fractions (ratio of DM mass to NS mass) as well as their coupling to
obtain its effects. Our aim is to constrain the amount of DM inside the NS by
comparing our findings with various observational data.

##  2 RMF Formalism

The E-RMF formalism is based on the effective field theory-driven RMF model,
which has resolved the renormalization issue in RMF theory while adhering to
the fundamental QCD symmetries. This framework has been widely used in the
past few years to study various nuclear physics issues, as demonstrated in
Refs. [46, 47, 48, 49, 28, 29, 50, 18]. The E-RMF Lagrangian, which includes
the fourth-order interaction between different mesons such as $\sigma$,
$\omega$, $\rho$, and $\delta$ is given by [51, 48, 52, 28, 29].

$\displaystyle{\cal L}_{\rm nucl.}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\bar{\psi}_{\alpha}\Bigg{\\{}\gamma_{\mu}\bigg{(}i\partial^{\mu}-g_{\omega}\omega^{\mu}-\frac{1}{2}g_{\rho}\vec{\tau}_{\alpha}\\!\cdot\\!\vec{\rho}^{\,\mu}\bigg{)}-\bigg{(}M_{\rm
nucl.}-g_{\sigma}\sigma\bigg{)}\Bigg{\\}}\psi_{\alpha}$ (2.1)
$\displaystyle+\frac{1}{2}\partial^{\mu}\sigma\,\partial_{\mu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{\zeta_{0}}{4!}g_{\omega}^{2}(\omega^{\mu}\omega_{\mu})^{2}-\frac{\kappa_{3}}{3!}\frac{g_{\sigma}m_{\sigma}^{2}\sigma^{3}}{M_{nucl.}}-\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}m_{\sigma}^{2}\sigma^{4}}{M_{nucl.}^{2}}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}-\frac{1}{4}W^{\mu\nu}W_{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\bigg{(}\vec{\rho}^{\mu}\\!\cdot\\!\vec{\rho}_{\mu}\bigg{)}-\frac{1}{4}\vec{R}^{\mu\nu}\\!\cdot\\!\vec{R}_{\mu\nu}$
$\displaystyle-\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}\big{(}\omega^{\mu}\omega_{\mu}\big{)}\big{(}\vec{\rho}^{\,\mu}\\!\cdot\\!\vec{\rho}_{\mu}\big{)}\,.$

Here, the relevant coupling constants are $g_{\sigma}$, $g_{\omega}$ and
$g_{\rho}$, while the corresponding masses are
$m_{\sigma},m_{\omega},m_{\rho}$ for $\sigma$, $\omega$, and $\rho$ mesons,
respectively. The spatial component of the energy-momentum tensor provides
pressure, and its zeroth component gives the energy density of the system. The
details can be found in Ref. [28, 29, 53, 54, 55]. Using the energy density
and pressure for the systems, one can estimate the different properties of the
finite nuclei, nuclear matter, and NS, as discussed in Sec. 5.

The Lagrangian density parameters given by Eq. (2.1) have been calibrated
using the simulated annealing method to optimize the Lagrangian density for a
specified parameter space. The parameterization process has been elaborately
explained in Refs. [56, 57, 28, 29]. We used experimental data for binding
energies and charge radii of several nuclei, including 16O, 40Ca, 48Ca, 68Ni,
90Zr, 100,132Sn, and 208Pb, to fit the coupling constants or parameters and
constrain the values of the nuclear matter incompressibility (K) and symmetry
energy coefficient ($J$) within 210$-$245 MeV and 28$-$35 MeV, respectively.
Additionally, to ensure that the maximum NS mass is around 2.35 $M_{\odot}$,
we allowed the parameter $\zeta_{0}$ corresponding to the self-coupling of
$\omega$ mesons to vary between 1.0 and 1.5. The obtained parameter set is
given in Table 1.

However, recent experiments, such as CREX and PREX-II, have reported higher
values for the skin thickness of 48Ca and 208Pb, with a measured value of
$0.121\pm 0.026$ fm [58] and $0.283\pm 0.071$ fm[59] respectively. This
discrepancy can be attributed to the need for a higher value of symmetry
energy and its slope parameter, which current models and theories are unable
to reproduce [60, 61]. Therefore, there is still room for improvement in
developing a new parameter set that can solve these issues. In particular, a
higher value of the symmetry energy coefficient ($J$) is needed to better
match the experimental results. Further research and refinement of the model
are necessary to accurately predict the properties of nuclei and neutron
stars.

Table 1: Masses of the $\sigma$, $\omega$, and $\rho$ mesons, the coupling constants for nucleons with mesons are tabulated for the NITR model. The mass of the nucleons ($M_{\rm nucl.}$) is taken as 939 MeV. Some of the nuclear matter properties are also given in the lower panel. Parameter | $m_{\sigma}$ | $m_{\omega}$ | $m_{\rho}$ | $g_{\sigma}$ | $g_{\omega}$ | $g_{\rho}$ | $\kappa_{3}$ | $\kappa_{4}$ | $\zeta_{0}$ | $\Lambda_{\omega}$
---|---|---|---|---|---|---|---|---|---|---
NITR | 492.988 | 782.5 | 763.0 | 9.869 | 12.682 | 14.192 | 1.724 | -5.764 | 1.189 | 0.045
$\rho_{0}$ (fm-3) |  |  $\mathcal{E}_{0}$ (MeV) |  |  $K$ (MeV) |  |  $J$ (MeV) |  |  $L$ (MeV) |  |
0.155 |  | -16.34 |  | 225.11 |  | 31.69 |  | 43.86 |  |

##  3 Unified EOS

We construct the core and crust EOS for NS using the effective field theory-
driven RMF model and the NITR EOS. The core EOS is obtained by imposing both
the $\beta$-equilibrium and charge neutrality conditions on top of the E-RMF
Lagrangian density (in Eq. 2.1). For the crust EOS, we adopt the methodology
given in Ref. [32, 20]. The unified EOS is necessary for consistency and to
better ascertain the NS properties [62, 20]. The outer crust, which is
stratified into different layers of nuclei, is determined using the most
recent experimental masses [32]. For the inner crust, we assume the Wigner-
Seitz approximation, considering nuclear clusters in a body-centered cubic
lattice structure employing the compressible liquid drop model approach, and
calculate the energy of the cluster and surrounding neutron gas using the NITR
EOS. The transition from the heterogeneous inner crust to a homogeneous core
occurs at a density of $0.10115$ fm-3 and a pressure of $0.41340$ MeV fm-3 for
the NITR EOS, which is consistent with the available constraints on NS crust
[4, 63].

In Fig. 1, the left panel shows the unified EOS for NITR and three other EOSs,
while the right panel depicts the corresponding speed of sound. It is
noteworthy that NITR displays stiffer behavior than G3 and IOPB-I. At lower
density regimes, all four EOSs exhibit nearly the same pressure, resulting in
similar canonical radii. However, the pressure at higher density regimes
becomes stiffer, which significantly affects the maximum mass and its
corresponding radius. Among the EOSs, NL3 attains the highest value for the
speed of sound, as its pressure significantly supports more mass than the
other EOSs. Nevertheless, none of the EOSs violate causality limits
($c_{s}^{2}=1$). Table 2 presents some of the NS properties corresponding to
NITR EOS.

Table 2: The radius ($R_{1.4}$) in km, dimensionless tidal deformability ($\Lambda_{1.4}$), the moment of inertia in ($I_{1.4}\times 10^{45}$ g cm2) for the canonical star. The maximum mass ($M_{\rm max}$) in $M_{\odot}$ and its corresponding radius ($R_{\rm max}$) in km, $\Lambda_{\rm max}$, and $I_{\rm max}$ are given. Model | $R_{1.4}$ | $\Lambda_{1.4}$ | $I_{1.4}$ | $M_{\rm max}$ | $R_{\rm max}$ | $\Lambda_{\rm max}$ | $I_{\rm max}$
---|---|---|---|---|---|---|---
NITR | 12.73 | 686.84 | 1.70 | 2.35 | 12.01 | 08.97 | 3.19

Figure 1: Left: Unified EOSs for the NS for four parameter sets such as G3,
IOPB-I, NL3, and NITR EOSs. The orange shaded regions for the chiral EFT
bounds [64]. Right: The sound speed for considered EOSs.

##  4 DM inside the NS

During the evolution period of compact objects, such as NS, there is a certain
probability that some amount of DM is captured inside it due to its high
baryon abundance and huge gravitational potential. The accreted DM particles
lose enough amount of energy due to their interactions with nucleons [17, 65,
34]. Non-annihilating WIMPs were selected as the most popular DM candidate
because WIMPs are the most generous type of DM particle and the thermodynamic
residue. Other processes, such as the conversion of neutrons to scalar DM and
the generation of scalar DM via bremsstrahlung, contribute to an increase in
the abundance of DM contained within the NS [17, 44, 66]. The amount of DM
mainly affects both the evolutionary history and the properties of the NS.
There are two types of scenarios for DMANS: the single-fluid model and the
two-fluid model, which we will discuss in the following subsections.

###  4.1 Single Fluid DM Model

In the single-fluid model, DM particles interact with baryons by exchanging
standard model Higgs. The structure of the interacting Lagrangian can be
deduced as [38, 17, 39, 40, 41, 67]:

$\displaystyle{\cal{L}}_{\rm SFDM}$ $\displaystyle=$
$\displaystyle\bar{\chi}\left[i\gamma^{\mu}\partial\mu-
M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}M_{h}^{2}h^{2}+f\frac{M_{\rm
nucl.}}{v}\bar{\varphi}h\varphi,$ (4.1)

where, $\varphi$ and $\chi$ represent the nucleon and DM wave functions,
respectively, while $h$ is the Higgs field. The mass of the DM (Neutralino) is
assumed to be 200 GeV, and the Higgs mass ($M_{h}$) is 125 GeV. Since
Neutralino is a supersymmetric particle, different coupling parameters are
found in the electroweak sector of the standard model in the Higgs portal
case, resulting in values of the DM - Higgs coupling ($y$) in the range of
0.001-0.1 [68]. However, a value of $y=0.07$ is chosen in this study. The
Yukawa coupling between Higgs and nucleon is $fM_{\rm nucl.}/v$, where $f$ is
the proton-Higgs form factor, and $v$ is the vacuum expectation value assumed
to be 0.35 [69] and 246 GeV [40, 41, 17], respectively. The main assumption in
this model is that the DM density ($\rho_{DM}$) is $10^{3}$ times less than
the nucleon baryon density [40, 38, 17]. One can calculate the Fermi momentum
using the relation $k_{f}^{\rm DM}=(3\pi^{2}\rho_{DM})^{1/3}$. The value of
$k_{f}^{\rm DM}$ varied is obtained to be $\sim 0.033$ GeV, but in the current
work, it is varied between $0.00-0.05$ GeV to calculate the DM effects on
various NS properties.

One can get the energy density and pressure by solving the Lagrangian density
(in Eq. 4.1) in the following [40, 38, 41],

$\displaystyle{\cal{E}}_{\rm SFDM}=\frac{2}{(2\pi)^{3}}\int_{0}^{k_{f}^{\rm
DM}}d^{3}k\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2}\
,$ (4.2) $\displaystyle P_{\rm SFDM}=\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{f}^{\rm
DM}}\frac{d^{3}k\hskip
2.84526ptk^{2}}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2}\,$
(4.3)

where $M_{\chi}^{*}(=M_{\chi}-yh_{0})$ is called the effective mass of the DM.
Now, the total energy density and pressure for DMANS can be written as,

$\displaystyle{\cal{E}}={\cal{E}}_{\rm NS}+{\cal{E}}_{\rm SFDM}\,,$
$\displaystyle{\rm and}\hskip 28.45274ptP=P_{\rm NS}+P_{\rm SFDM}\,.$ (4.4)

###  4.2 Two Fluid DM Model

The DM particles in the two-fluid DM (TFDM) model mainly self-interact through
gravity, and the nature of the DM candidate can either be fermionic or
bosonic. Here, we consider DM as a fermion and represent it by the field
$\psi_{D}$. The interactions between the dark scalar meson ($\phi_{D}$) and
the dark vector meson ($V_{D}^{\mu}$) with fermionic DM are represented by
$g_{\rm sd}\bar{\psi}\psi_{D}\phi_{D}$ and $g_{\rm
vd}\bar{\psi}\gamma_{\mu}\psi_{D}V_{D}^{\mu}$, where $g_{\rm sd}$ and $g_{\rm
vd}$ are the coupling constants corresponding to the dark scalar and dark
vector mesons, respectively. Similar to the baryons, one can define the
effective potential for DM with exchanging bosons [70]

$\displaystyle V_{eff}(r)=\frac{g_{\rm vd}^{2}}{4\pi}\frac{e^{-m_{\rm
vd}r}}{r}-\frac{g_{\rm sd}^{2}}{4\pi}\frac{e^{-m_{\rm sd}r}}{r}.$ (4.5)

Now, one can write the Lagrangian density for the DM model, which is given by
[42, 43]

$\displaystyle{\cal L_{\rm TFDM}}=$
$\displaystyle\bar{\psi_{D}}[\gamma_{\mu}(i\partial^{\mu}-g_{\rm
vd}V^{\mu})-(M_{D}-g_{\rm sd}\phi_{D})]\psi_{D}$
$\displaystyle+\frac{1}{2}[\partial_{\mu}\phi_{D}\partial^{\mu}\phi_{D}-m_{\rm
sd}^{2}\phi_{D}^{2}]-\frac{1}{4}V_{\mu\nu,D}V_{D}^{\mu\nu}+\frac{1}{2}m_{\rm
vd}^{2}V_{\mu,D}V_{D}^{\mu}\,,$ (4.6)

where, the mass of the DM candidate is represented by $M_{D}$, while $m_{sd}$
and $m_{vd}$ are the masses corresponding to the dark scalar meson and dark
vector meson, respectively. Additionally, we define the term $V_{\mu\nu,D}$ as
$\partial_{\mu}V_{\nu,D}-\partial_{\nu}V_{\mu,D}$.

In Section 2, we introduced the E-RMF model, and in Subsection 4.1, we
discussed the DM admixed single-fluid model. Similar to these models, the
mean-field approximation can be applied to the TFDM Lagrangian density given
in Equation 4.6. This allows us to obtain the energy density and pressure of
the TFDM model, denoted by ${\cal E}_{\rm TFDM}$ and $P_{\rm TFDM}$,
respectively. The expressions for ${\cal E}_{\rm TFDM}$ and $P_{\rm TFDM}$ can
be found in the literature [43, 42]

$\displaystyle{\cal E}_{\rm TFDM}=\frac{1}{\pi^{2}}\int_{0}^{k_{D}}dk\
k^{2}(k^{2}+M_{D}^{*2})^{1/2}+\frac{g_{\rm vd}^{2}}{2m_{\rm
vd}^{2}}\rho_{D}^{2}+\frac{m_{\rm sd}^{2}}{2g_{\rm
sd}^{2}}(M_{D}-M_{D}^{*})^{2}\,,$ (4.7) $\displaystyle P_{\rm
TFDM}=\frac{1}{3}\frac{1}{\pi^{2}}\int_{0}^{k_{D}}dk\
\frac{k^{2}}{(k^{2}+M_{D}^{*2})^{1/2}}+\frac{g_{\rm vd}^{2}}{2m_{\rm
vd}^{2}}\rho_{D}^{2}-\frac{m_{\rm sd}^{2}}{2g_{\rm
sd}^{2}}(M_{D}-M_{D}^{*})^{2}\,.$ (4.8)

The DM number density is denoted by $\rho_{D}$, while $M_{D}^{*}=M_{D}-g_{\rm
sd}\phi_{D}$ is referred to as the effective mass of the DM candidate. The
attractive potential $C_{\rm sd}=g_{\rm sd}/m_{\rm sd}$ and repulsive
potential $C_{\rm vd}=g_{vd}/m_{vd}$ are both defined in units of GeV-1. These
potentials play a crucial role in the two-fluid DM (TFDM) model, where DM
particles mainly self-interact through gravity. Specifically, they determine
the nature of the interactions between DM and the dark scalar meson
($\phi_{D}$) and dark vector meson ($V_{D}^{\mu}$).

##  5 Results and Discussions

In this section, we present our numerical results for NS. Various properties
obtained with the newly developed parameter set NITR for DMANS in single fluid
as well as two fluid approaches are also discussed.

###  5.1 Single fluid DM Model

In the case of SFDM, the DM particles directly couple to nucleons by
exchanging Higgs, resulting in the system energy density and pressure being
the sum of both nucleons and DM. By varying $k_{f}^{\rm DM}$ for the NITR
model, we calculate the DMANS EOS, and the unified EOS for DMANS is presented
in the upper panel of Fig. 2 for different DM Fermi momenta. We also overlay
the lower-density chiral EFT data [64] on top of it. The addition of DM
results in a softer EOS, with the degree of softness mainly dependent on the
amount of DM present inside the NS.

The mass-radius relationship for our newly developed EOS "NITR" is shown in
the lower left panel of Fig. 2 for varying $k_{f}^{\rm DM}$. The heaviest
pulsar PSR J0952-0607 [23] has a mass of $M=2.35\pm 0.17\ M_{\odot}$, which
our EOS satisfies very well, also predicting a mass of $2.35\ M_{\odot}$. NITR
not only predicts the heaviest pulsar mass but also satisfies both NICER [71,
72] and revised NICER data [73], with a radius of 12.73 km. However, as we
increase $k_{f}^{\rm DM}$, the maximum mass and corresponding radius decrease.
This is due to the presence of DM softening the EOS, and it depends on the
amount of DM present inside the star. We can constrain the amount of DM inside
the NS using observational data for mass and radius. For $k_{f}^{\rm DM}$ up
to 0.03 GeV, the $M-R$ behavior lies within observational limits and passes
through the NICER data. However, an increase in $k_{f}^{\rm DM}$ further
softens the NS, shifting the $M-R$ behavior outside of the observational
constraints.

Table 3: The maximum mass ($M_{\rm max}$) in $M_{\odot}$ and its corresponding radius ($R_{\rm max}$) in km and dimensionless tidal deformability $\Lambda_{\rm max}$ are given and also for the canonical NS, its corresponding radius ($R_{1.4}$) in km and dimensionless tidal deformability ($\Lambda_{1.4}$) given by varying the $k_{f}^{\rm DM}$. $k_{f}^{\rm DM}$ (GeV) | $M_{\rm max}$ | $R_{\rm max}$ | $\Lambda_{\rm max}$ | $R_{1.4}$ | $\Lambda_{1.4}$
---|---|---|---|---|---
0.00 | 2.35 | 12.73 | 8.97 | 12.01 | 686.84
0.01 | 2.35 | 12.61 | 8.96 | 11.96 | 664.65
0.02 | 2.32 | 12.32 | 8.49 | 11.77 | 567.49
0.03 | 2.25 | 11.74 | 7.94 | 11.32 | 397.77
0.04 | 2.12 | 10.92 | 6.92 | 10.61 | 228.93
0.05 | 1.96 | 9.99 | 6.29 | 9.71 | 111.24

Figure 2: Upper: Unified EOSs for DMANS with and without DM by varying the
$k_{f}^{\rm DM}$. Lower left: Mass-Radius relation for single fluid approach
by varying $k_{f}^{\rm DM}$. Different color bands signify the masses of the
NS observed from the various pulsars, such as PSR J0740+6620 [25], heaviest
pulsars J0952-0607 [23], and GW190814 [74]. The simultaneous mass-radius
constraint provided by NICER [71, 72] and revised NICER [73] are also shown.
Lower right: The tidal deformability as the function of the mass of the NS
with varying $k_{f}^{\rm DM}$. Two observational constraints GW190814 [75] and
GW170817 [2] are also used to validate our model.

The deformation of a neutron star in the presence of a companion star is
measured by the tidal deformability, denoted by $\lambda=(2/3)k_{2}/C^{5}$,
where $k_{2}$ and $C$ are the second Love number and compactness of the star
[76, 77]. We calculated the dimensionless tidal deformability
($\Lambda=\lambda/M^{5}$) for the NITR EOS and displayed it in the lower right
panel of Fig.2. The predicted $\Lambda_{1.4}$ satisfies the limit
($\Lambda_{1.4}\leq 800$) given in Ref. [2]; however, this limit is slightly
above the upper bound of $\Lambda_{1.4}=190_{-70}^{+120}$ [3]. The inclusion
of DM inside the NS results in a significant change in the tidal
deformability. When $k_{f}^{\rm DM}$ increases from 0.0 GeV to some finite
value, the tidal deformability starts to decrease due to its dependence on the
mass and radius of the star. We increased $k_{f}^{\rm DM}$ up to 0.05 GeV and
observed that the gravitational bound is satisfied for a canonical NS. The
properties of the neutron star for both the canonical and maximum mass cases
are summarized in Table 3.

###  5.2 Two fluid DM Model

Here, we calculate the properties of DMANS, including its density profiles,
$M-R$ relations, and tidal deformability, using the two-fluid model [42, 43].
The predicted properties are obtained by varying the mass and fractions of DM
inside the NS. We depict the TFDM EOSs in Fig. 3 for different potentials,
including the EOS without DM for the NITR model, for comparison. The values of
$C_{\rm sd}$ and $C_{\rm vd}$ are chosen such that they represent short-range
repulsive and long-range attractive interactions, respectively. The exact
values for these parameters are still unknown, but Ref. [42] predicts them
using the Bayesian method and finds a range of Csd =
3.90${}^{\rm+0.82}_{\rm-0.70}$ GeV-1 and Cvd = 11.88${}^{\rm+0.53}_{\rm-0.46}$
GeV-1, which are model-dependent. However, we roughly choose the values of Csd
= 0 and/or 4 GeV-1 and Cvd = 0 and/or 10 GeV-1 as done in [43]. It is
important to note that the smaller attractive potential slightly softens the
DM EOS, whereas stiffening depends on the larger repulsive potential. However,
the value of Csd should not be too large as it can make the pressure negative
and disrupt the hydrostatic equilibrium inside the star. Thus, we keep the
values of Csd and Cvd at 4 GeV-1 and 10 GeV-1, respectively, in our
calculations.

Figure 3: The TFDM EOSs for different choices of interaction strength or
potential accounted here. Also, we take our new EOS "NITR" for comparison. The
mass of the DM candidate is taken here as 1 GeV.

We have obtained number density profiles as a function of the radius for the
canonical star, which is depicted in Fig. 4, for different DM masses ranging
from 300 MeV to 1 GeV and varying DM fractions. Our observations show that for
lower DM masses, such as 300 MeV, an increase in the mass fraction results in
an expansion of the DM radius, which leads to the formation of a DM halo that
overpowers the NS radius. Conversely, for higher DM masses, such as 1 GeV, the
radius of the NM dominates the DM halo, which is primarily concentrated at the
center of the star. From this evidence, we can conclude that lighter DM
particles tend to form DM halos, whereas heavier DM particles mainly reside in
the core of DMANS [78, 79].

Figure 4: Density profile of the canonical NS for different DM masses by
varying DM fraction. The possibility of DM halo and DM core is also shown.
Here, the value of potential considered as $C_{\rm sd}=4$ GeV-1 and $C_{\rm
vd}=10$ GeV-1.

In Fig. 5, the upper panel shows the mass-radius relationship for different DM
masses with varying mass fractions, taking into consideration different
observational constraints to constrain the amount of dark matter present
inside the NS. The orange shaded line represents the PSR J0740+6620, which
suggests that any theoretical model’s maximum mass limit should be able to
estimate 2 $M_{\odot}$. In light of this, we vary the DM mass and mass
fraction to constrain the amount of DM inside the NS while considering the
impacts of DM on the maximum mass and radius of the NS. Interestingly, it is
observed that the impacts of less massive DM candidates on the radius are more
pronounced than those of heavier DM candidates. For example, when the DM mass
is 300 MeV, an increase in the mass fraction leads to a continuous increase in
the radius corresponding to the maximum mass, indicating the formation of a DM
halo, as seen in Fig. 4. Similarly, for DM mass 500 MeV, a halo is observed
for a higher mass fraction, while for a lower mass fraction, DM captures at
the core of the NS. To observe this continuum, the DM mass is increased up to
1 GeV with varying mass fractions, where the DM completely captures inside the
NS and resides at the core of the NS. Therefore, it can be concluded that less
massive DM candidates dominate the NS to form a DM halo, whereas heavier DM
candidates capture inside the NS and present as a DM core.

Figure 5: Upper: Mass-Radius relation for different DM particle mass ranging
from 300 MeV to 1 GeV plotted varying mass fraction. The shaded bands
represent different pulsars. Lower: The dimensionless tidal deformability is
shown for the same condition as the upper panel.

In the lower panel of Fig. 5, we present the predicted dimensionless tidal
deformability. It is observed that the tidal deformability is dependent upon
the radius of the star. When we increase the mass fraction of light-dark
matter particles, the tidal deformability rises beyond the observational
limits, as confirmed in Refs. [42]. For heavy DM particles such as 1 GeV, the
tidal deformability decreases steadily with increasing mass fraction, leading
to a canonical tidal deformability ($\Lambda_{1.4}$) that falls within the
accepted limits of observation. In summary, the tidal deformability depends on
the mass and mass fraction of the DM particles, and for heavier DM particles,
the tidal deformability stays within the observed limits of the canonical
tidal deformability.

##  6 Conclusions

We have developed a new parameter set called "NITR" using the E-RMF formalism
to support experimental evidence for finite nuclei, infinite nuclear matter,
and neutron stars in light of the recently observed heaviest pulsar, PSR
J0952-0607 ($M=2.35\pm 0.17\ M_{\odot}$). We optimized the new parameter set
using empirical evidence for eight spherical nuclei, and the observables such
as binding energy and radii for finite nuclei can now be predicted reasonably
accurately. Our calculations match the existing experimental data well for the
infinite nuclear matter at saturation densities ($\rho_{0}=0.155$ fm-3), and
we obtained incompressibility, symmetry energy, and slope parameter values for
the NITR set of 225.11, 31.69, and 43.86 MeV, respectively. The maximum mass
yielded by our parameter set is consistent with PSR J0952-0607, and we
obtained a maximum mass of $2.35M_{\odot}$ and a canonical radius appropriate
with theoretical results [80, 81] with the value of 12.73 km. To investigate
the NS macroscopic properties, we considered DMANS for our study, where we
considered DM interacting with nucleons inside the NS both non-gravitationally
(single fluid approach) and gravitationally (two fluid approach). In the
single fluid approach, we considered neutralino as the DM candidate that
interacts with nucleons by exchanging Higgs boson. By varying the $k_{f}^{\rm
DM}$, we observed that the maximum mass and its corresponding radius, as well
as canonical radius, decreases with an increase in $k_{f}^{\rm DM}$, and a
similar effect was observed for tidal properties as they scale with mass and
radius. We varied the $k_{f}^{\rm DM}$ from $0.00-0.05$ GeV and found that the
NS properties satisfy observational constraints up to a certain value of
$k_{f}^{\rm DM}$ ($\approx 0.03$ GeV), allowing us to constrain the amount of
DM present inside the NS. In the two-fluid approach, we investigated the
impacts of fermionic DM on the characteristics of the NS under the assumption
that DM and NM interact only via gravity. First, we investigate the two-fluid
EOS with different interaction strengths and finally choose their value in our
computation in such a way that they should be short-range repulsive (Csd= 4
GeV-1) and long-range attractive (Cvd= 10 GeV-1) interactions. We defined
multiple sets of mass and mass fraction and take them as free parameters in
our calculations and observed that the presence of DM leads to a broadening as
well as contracting of mass-radius correlations and tidal properties of the
NS. We established that the region of DM candidates that significantly affects
the dynamics of nuclear structure is located around 300 MeV. Interestingly,
when the mass of the DM is less, the DM exceeds the NM to form the DM halo,
which can be clearly understood from the density profile of the star. An
explicit DM halo may be likely to develop if there is a repulsive force
between DM particles. Hence, in this study, we try to constrain the amount of
DM inside the NS using the latest observational data. In the future, more
observational data can put stringent constraints on the DM particle nature and
their amount inside compact objects.

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