# Effects of dark matter on the nuclear and neutron star matter

H. C. Das1,2 , Ankit Kumar1,2, Bharat Kumar3, S. K. Biswal4, Takashi
Nakatsukasa3, Ang Li4, S. K. Patra1,2,
1 Institute of Physics, Sachivalya Marg, Bhubaneswar-751005, India
2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400085, India
3 Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577,
Japan
4 Department of Astronomy, Xiamen University, Xiamen 361005, P. R. China
E-mail<EMAIL_ADDRESS>
###### Abstract

We study the dark matter effects on the nuclear matter parameters
characterising the equation of states of super dense neutron-rich nucleonic-
matter. The observables of the nuclear matter, i.e. incompressibility,
symmetry energy and its higher-order derivatives in the presence dark matter
for symmetric and asymmetric nuclear matter are analysed with the help of an
extended relativistic mean-field model. The calculations are also extended to
$\beta$-stable matter to explore the properties of the neutron star. We
analyse the dark matter effects on symmetric nuclear matter, pure neutron
matter and neutron star using NL3, G3 and IOPB-I forces. The binding energy
per particle and pressure are calculated with and without considering the dark
matter interaction with the nuclear matter systems. The influences of dark
matter are also analysed on the symmetry energy and its different
coefficients. The incompressibility and the skewness parameters are affected
considerably due to the presence of dark matter in the nuclear matter medium.
We extend the calculations to the neutron star and find its mass, radius and
the moment of inertia for static and rotating neutron star with and without
dark matter contribution. The mass of the rotating neutron star is
considerably changing due to rapid rotation with the frequency in the mass-
shedding limit. The effects of dark matter are found to be important for some
of the nuclear matter parameters, which are crucial for the properties of
astrophysical objects.

###### keywords:

dark matter– equation of state– stars: neutron

##  1 Introduction

It is well known that our Universe has only $\sim$ 6% visible matter and the
remaining $\sim$ 94% is considered to be dark matter (DM) and dark energy.
Zwicky estimated the total mass of the Universe (Zwicky, 2009; Kouvaris &
Tinyakov, 2010) and found that something is missing, which are termed as DM
($\sim$ 26%) and dark energy ($\sim$ 68%). Many theoretical, experimental and
observational efforts have been put to know the mystery of DM and dark energy.
Several DM candidates are hypothesized, like weakly interacting massive
particles (WIMPs) (Kouvaris & Tinyakov, 2011; Quddus et al., 2019), feebly
interacting massive particles (FIMPs) (Bernal et al., 2017; Hall et al.,
2010), Neutralino (Hooper & Wang, 2004; Han et al., 2014; Das et al., 2019)
and axions (Duffy & van Bibber, 2009) etc. The WIMPs are expected to produce
in the early hot Universe and annihilate in pairs, and these are the thermal
relics of the Universe (Ruppin et al., 2014). The WIMPs might have decayed in
the dense region of the Universe to yield standard model (SM) particles, gamma
rays, leptons and neutrinos. Many experiments have already been performed to
find out the direct and indirect consequences of DM. The direct experimental
searches like DAMA (Bernabei et al., 2008, 2010), Xenon (Angle et al., 2008)
and CDMS (CDMS, 2010) are set up to find the cross-section between WIMPs and
nucleons. The indirect detection experiments like Fermi large area telescopes
and imaging air Cherenkov telescopes have also established (Conrad, 2014). In
addition to that, the effects of DM on compact stars such as neutron star (NS)
and white dwarfs have been studied with different DM models (Kouvaris, 2008;
Bertone & Fairbairn, 2008). For example, the self-annihilating DM inside the
NS heat the stars, and it would affect the cooling properties of compact stars
(Kouvaris, 2008; Bhat & Paul, 2019). On the other hand, non-self-annihilating
DM is accumulated inside the stars and affects the stellar structure (De
Lavallaz & Fairbairn, 2010; Ciarcelluti & Sandin, 2011). In this paper, we
consider fermionic DM interacting with nucleonic matter via the Higgs portal
mechanism and constraints the nuclear matter (NM) and NS properties through DM
parameters.

To understand equation of state (EoS) of NS matter, it is imperative to
analyze the NM parameters at different proton-neutron compositions
($\alpha=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}})$, where $\rho_{n}$ and
$\rho_{p}$ are the neutron and proton densities respectively. The NM
parameters, such as binding energy per particle (BE/A), incompressibility
($K$), symmetry energy ($S$) and its derivatives ($L$-slope parameter,
$K_{sym}$-isovector incompressibility and $Q_{sym}$-skewness parameter) are
the key quantities for the study of an EoS. The NSs are the extreme object
with a high degree of density and isospin asymmetry. Hence it is interesting
to study the NM parameters at different conditions, from low to high density
and at different asymmetric factor $\alpha=$ 0 to 1 in the presence of DM. In
Ref. (Alam et al., 2016), it was shown that the linear combinations of the
isoscalar and isovector NM parameters are strongly correlated with NS radii
over a wide range of NS mass. These correlations are particularly important
for the canonical mass 1.4$M_{\odot}$ of the NS. With the help of GW170817
observation, a similar better correlation exists between the tidal
deformability $\Lambda$ and the Love number $k_{2}$ with the linear
combination of the $M_{0}$ and the curvature of the symmetry energy
$K_{sym,0}$ at saturation density (Malik et al., 2018; Carson et al., 2019).
Also, recently it is reported by various authors (Sandin & Ciarcelluti, 2009;
Kouvaris & Tinyakov, 2010; De Lavallaz & Fairbairn, 2010; Ciarcelluti &
Sandin, 2011; Leung et al., 2011; Li et al., 2012; Panotopoulos & Lopes, 2017;
Ellis et al., 2018; Bhat & Paul, 2019; Das et al., 2019; Ivanytskyi et al.,
2019; Quddus et al., 2019) that the NS core contains an admixture of DM
including many exotic baryonic species.

The internal structure of a NS is not well known till now, so we believe many
phenomena like quark deconfinement (Collins & Perry, 1975; Orsaria et al.,
2014; Mellinger et al., 2017), kaon condensation (Kaplan & Nelson, 1988;
Glendenning & Schaffner-Bielich, 1998; Glendenning & Schaffner-Bielich, 1999;
Pal et al., 2000; Gupta & Arumugam, 2012), phase transition (Glendenning,
1992; Sharma et al., 2007) and hyperons production (Ambartsumyan & Saakyan,
1960; Glendenning, 1985; Schaffner & Mishustin, 1996; Schulze et al., 2006;
Ðapo et al., 2010; Bhowmick et al., 2014; Biswal et al., 2016; Fortin et al.,
2017; Bhuyan et al., 2017; Biswal, 2019; Biswal et al., 2019) are occur inside
the star. The EoS plays a vital role to predict all the star parameters such
as mass ($M$), radius ($R$), tidal deformability ($\Lambda$) and moment of
inertial ($I$) of the NS. Many theoretical and observational studies have been
devoted to constraint these parameters. The binary NS merger event GW170817
(Abbott et al., 2017, 2018) provides a strong constraint on the EoS. The
recently reported massive NS (PSR J0740+6620) (Cromartie et al., 2019) with
the mass of $2.14^{+0.20}_{-0.18}$ $M_{\odot}$ within the $95.4\%$ confidence
limits, also puts a strong constraint on the nature of EoS.

Recently the simultaneous measurements of the $M$ and $R$ for NS are done by
the NASA Neutron star Interior Composition ExploreR (NICER) (Miller et al.,
2019; Riley et al., 2019; Bogdanov et al., 2019; Bilous et al., 2019;
Raaijmakers et al., 2019; Guillot et al., 2019), which constraint the EoS. For
that, we extend our calculations for rotating NS to measure the $M$, $R$ and
$I$ in the presence of DM. The theoretical observations allow the Keplerian
frequency of the rotating NS is more than 2000 $Hz$, but two fastest pulsar
detected have frequencies 716 $Hz$ (Hessels et al., 2006) and 1122 $Hz$
(Kaaret et al., 2007). Many calculations related to the Keplerian frequencies
(Stergioulas, 2003; Dhiman et al., 2007; Jha et al., 2008; Krastev et al.,
2008; Haensel et al., 2009; Sharma & Jha, 2009; Koliogiannis & Moustakidis,
2020) are devoted to fix the frequency range within this limit. In this work,
we want to study the effect of mass-shedding frequency on the mass of the DM
admixture NS, which can be used to constraint the EoS.

The EoS, which is the main input to the Tolman-Oppenheimer-Volkoff (TOV)
equations (Tolman, 1939; Oppenheimer & Volkoff, 1939) determine the stable
configurations of a static NS, are constructed in several ways. The non-
relativistic formalism with various Skyrme parametrizations (Skyrme, 1956,
1958; Vautherin & Brink, 1972; Chabanat et al., 1998; Alex Brown, 1998; Stone
& Reinhard, 2007; Dutra et al., 2012; Dechargé & Gogny, 1980) and three-body
potential of Akmal-Pandheripande (Akmal et al., 1998) are very successful in
describing the nuclear EoS, including the NS. The relativistic mean-field
(RMF) model, which gives a good description not only explains well the finite
nuclei in the $\beta$-stability line but also reproduce the experimental data
for exotic and superheavy nuclei (Rashdan, 2001; Bhuyan & Patra, 2012; Bhuyan,
2018; Kumar et al., 2018). In the present paper, we use the extended RMF
(E-RMF) models for the study of effects of DM on NM and NS properties with the
well known NL3 (Lalazissis et al., 1997), G3 (Kumar et al., 2017) and IOPB-I
(Kumar et al., 2018) parameter sets.

The effects of the DM on the NS have already discussed in some recent works,
for example, Panotopoulos et al. (Panotopoulos & Lopes, 2017), Das et al. (Das
et al., 2019) and Quddus et al. (Quddus et al., 2019). But our analysis gives
a better and wider platform to discuss the DM effects on the bulk properties
of the NM and the NS. Here, we briefly describe their works. Panotopouls et
al. (Panotopoulos & Lopes, 2017) have calculated the NS EoS with simple
$\sigma$-$\omega$ model (Walecka, 1974), and they have added the DM with
static NS to find its $M$ and $R$. They assumed that the DM particles interact
with nucleons via SM Higgs. They considered Neutralino as a DM candidate,
which has mass 200 GeV and Higgs mass is 125 GeV. Das et al. (Das et al.,
2019) have taken the same RMF model but with NL3 parameter set (Lalazissis et
al., 1997) and calculate the EoS, $M$, $R$ and $\Lambda$ of the static NS.
Quddus et al. (Quddus et al., 2019) have taken the E-RMF formalism, which is
suitable for constraints on the properties of the NS in the presence of WIMP
dark matter. The main difference between earlier two work and Quddus et al. is
that they have taken light Higgs mass 40 GeV and DM mass up to 8 GeV. In the
present calculations, we take three non-linear parameter sets, whose nuclear
matter properties cover a wide range. Though simple $\sigma$-$\omega$ model
gives a qualitative picture of the RMF model, still many important information
are missing in the linear $\sigma$-$\omega$ model. But E-RMF calculation
covers a full set of non-linear model contain various important interaction
terms as in Refs. (Kumar et al., 2017, 2018). We add the DM with the NM and
calculate NM parameters like BE/A, $K$, $S$, $L$, $K_{sym}$ and $Q_{sym}$ for
the whole density range. Finally, we apply the $\beta$-equilibrium condition
to the NM EoS. When we add the DM to the NM, we follow the same formalism by
the Panotopouls et al. Panotopoulos & Lopes (2017) and Das et al. (Quddus et
al., 2019). Then we calculate the $M$, $R$ and $I$ both for static and
rotating NS and we compare both the results.

The paper is organised as follows: the formalism used in this work is
presented in Sec. 2. In Sub-Sec. 2.1, we explain the basic formalism of RMF
model using NL3 and the recently developed G3 and IOPB-I forces for the
calculations of nucleonic EoS. In Sub-Sec. 2.2, we take the interaction of DM
with NM and calculate the EoS of nucleons with DM. In Sub-Sec. 2.3, we
calculate different parameters of NM. The Sub-Sec. 2.4 the calculation of EoS
of NS using $\beta$-equilibrium and charge-neutrality conditions. In Sub-Sec.
2.5, the observables of the NS are calculated like $M$, $R$, $I$ etc. for
static NS (SNS) and rotating NS (RNS). The results and discussions are
detailed in Sec. 3. Finally, summary and our concluding remarks are outlined
in Sec. 4.

##  2 Theoretical Framework

From the last four decades, the RMF approaches are extremely popular to
describe finite and infinite NM properties. The Lagrangian is constructed by
taking the interaction of few numbers of mesons with nucleons and their self
and cross-couplings. The parameters are constructed, taking into account the
experimental data of few finite nuclei, saturation properties of the infinite
NM and NS properties.

###  2.1 Construction of RMF approach to find EoS for nucleons

The RMF Lagrangian is built from the interaction of mesons-nucleons and their
self ($\Phi^{3}$,$\Phi^{4}$, $W^{4}$) and cross-couplings ($\Phi^{2}-W^{2}$,
$R^{2}-W^{2}$, $\Phi-W^{2}$ and $\Phi-R^{2}$) of the mesons fields $\Phi$,
$W$, $R$, $D$. Where $\Phi$, $W$, $R$ and $D$ are the redefined fields for
$\sigma$, $\omega$, $\rho$ and $\delta$ mesons as $\Phi=g_{s}\sigma^{0}$,
$W=g_{\omega}\omega^{0}$, $R$ = g${}_{\rho}\vec{\rho}$ 0 and
$D=g_{\delta}\delta^{0}$ respectively. The RMF Lagrangian is discussed in
these Refs. (Miller & Green, 1972; Serot & Walecka, 1986; Furnstahl et al.,
1987; Reinhard, 1988; Furnstahl et al., 1997; Kumar et al., 2017, 2018). The
energy density (${\cal{E}}_{nucl.}$) and pressure ( $P_{nucl.}$) for a
nucleon-meson interacting system are given as (Kumar et al., 2018; Quddus et
al., 2019)

$\displaystyle{\cal{E}}_{nucl.}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{(2\pi)^{3}}\sum_{i=p,n}\int_{0}^{k_{i}}d^{3}kE_{i}^{\star}(k_{i})+\rho_{b}W+\frac{1}{2}\rho_{3}R$
(1)
$\displaystyle+\frac{m_{s}^{2}\Phi^{2}}{g_{s}^{2}}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi}{M_{nucl.}}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}-\frac{1}{4!}\frac{\zeta_{0}W^{4}}{g_{\omega}^{2}}$
$\displaystyle-\frac{1}{2}m_{\omega}^{2}\frac{W^{2}}{g_{\omega}^{2}}\Bigg{(}1+\eta_{1}\frac{\Phi}{M_{nucl.}}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}$
$\displaystyle-\Lambda_{\omega}(R^{2}\times
W^{2})-\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}\Phi}{M_{nucl.}}\Bigg{)}\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}$
$\displaystyle+\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}D^{2},$
$\displaystyle P_{nucl.}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{3(2\pi)^{3}}\sum_{i=p,n}\int_{0}^{k_{i}}d^{3}k\frac{k^{2}}{E_{i}^{\star}(k_{i})}+\frac{1}{4!}\frac{\zeta_{0}W^{4}}{g_{\omega}^{2}}$
$\displaystyle-\frac{m_{s}^{2}\Phi^{2}}{g_{s}^{2}}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi}{M_{nucl.}}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\frac{W^{2}}{g_{\omega}^{2}}\Bigg{(}1+\eta_{1}\frac{\Phi}{M_{nucl.}}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}$
$\displaystyle+\Lambda_{\omega}(R^{2}\times
W^{2})+\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}\Phi}{M_{nucl.}}\Bigg{)}\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}$
$\displaystyle-\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}D^{2}.$

The energy density of the nucleon in the meson medium is
$E_{i}^{\star}(k_{i})$=$\sqrt{k_{i}^{2}+{M_{i}^{\star}}^{2}}$, where
$M_{i}^{\star}$ is the effective mass calculated in Eq. (6) and $k_{i}$ is the
momentum of the nucleon, where i = p, n. The $\rho_{b}$ and $\rho_{3}$ in Eq.
(1) are the baryonic and iso-vector density respectively. $\gamma$ is the spin
degeneracy factor which is equal to 2 for individual nucleons. $M_{nucl.}$ is
the mass of the nucleon which is 939 MeV.

###  2.2 Interaction between nucleons and DM candidates in NM

It is a well known fact that the NS is rotating along with the galaxy. The DM
halo in the Universe is also rotating so that the DM particles accreted mostly
in the NS core due to its very high gravitational field and high baryon
density (Goldman & Nussinov, 1989; Kouvaris, 2008; Xiang et al., 2014; Das et
al., 2019). When DM particles interact with nucleons, it loses energy and
helps in the cooling of the NS (Bhat & Paul, 2019). The cooling of the NS are
detailed in these Refs. (Gnedin et al., 2001; Page et al., 2004; Yakovlev &
Pethick, 2004; Yakovlev et al., 2005, 2010). The amount of DM inside the NS
depends on the evolution time of the NS in the Universe. In this context, we
consider the Neutralino (Martin, 1998; Panotopoulos & Lopes, 2017) as a
fermionic DM candidate which interact with nucleon via SM Higgs.The
interaction Lagrangian of DM and nucleons is given as (Panotopoulos & Lopes,
2017; Quddus et al., 2019; Das et al., 2019)

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{nucl.}+\bar{\chi}\left[i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(3)
$\displaystyle-\frac{1}{2}M_{h}^{2}h^{2}+f\frac{M_{nucl.}}{v}\bar{\varphi}h\varphi,$

where ${\cal{L}}_{nucl.}$ is the nucleon-mesons Lagrangian and $\varphi$ and
$\chi$ are the nucleonic and DM wave functions respectively. We take the mass
of Neutralino ($M_{\chi}$) is 200 GeV, and the coupling constants between DM
and SM Higgs is $y$, which can be found in the large Higgs mixing angle limit.
Since the Neutralino is the super symmetric particle, it has the various gauge
coupling constants in the electroweak sector of the standard model (Martin,
1998). So depending on the different parameters, the values of $y$ is in the
range 0.001 – 0.1. Thus we take the value of $y$ = 0.07 in our calculations.
The Higgs field directly couples to the nucleons with Yukawa interaction
$f\frac{M_{nucl.}}{v}$, where $f$ is proton-Higgs form factor. The detailed
analytical expression for $f$ can be found in (Cline et al., 2013). In lattice
calculations (Alarcón et al., 2012; Young, 2012), we can consider the value of
$f=0.35$, which is agreement with (Cline et al., 2013). The Higgs mass is
$M_{h}=$ 125 GeV. The vacuum expectation value ($v$) of Higgs is 246 GeV. From
the Lagrangian in Eq. (3), we get the total energy density (${\cal{E}}$) and
pressure ($P$) for NM with DM given as (Panotopoulos & Lopes, 2017; Das et
al., 2019; Quddus et al., 2019)

$\displaystyle{\cal{E}}$ $\displaystyle=$
$\displaystyle{\cal{E}}_{nucl.}+\frac{2}{(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}d^{3}k\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}$
(4) $\displaystyle+\frac{1}{2}M_{h}^{2}h_{0}^{2},$ $\displaystyle P$
$\displaystyle=$ $\displaystyle
P_{nucl.}+\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}\frac{d^{3}k\hskip
2.84526ptk^{2}}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}$ (5)
$\displaystyle-\frac{1}{2}M_{h}^{2}h_{0}^{2},$

where $k_{f}^{DM}$ is the DM Fermi momentum. We consider the baryon density
inside NS is 1000 times larger than the DM density, this imply that
$M_{\chi}/M$=1/6 (Panotopoulos & Lopes, 2017; Das et al., 2019), where $M$ is
the mass of the NS. One can get $k_{f}^{DM}$ is $\sim$ 0.03 GeV (Das et al.,
2019). So that we vary $k_{f}^{DM}$ from 0 – 0.06 GeV. The effective masses of
nucleon and DM are given as

$\displaystyle M_{i}^{\star}$ $\displaystyle=$ $\displaystyle
M_{nucl.}+g_{\sigma}\sigma_{0}\mp
g_{\delta}\delta_{0}-\frac{fM_{nucl.}}{v}h_{0},$ $\displaystyle
M_{\chi}^{\star}$ $\displaystyle=$ $\displaystyle M_{\chi}-yh_{0},$ (6)

where the $\sigma_{0}$, $\delta_{0}$ and $h_{0}$ are the meson field equations
of $\sigma$, $\delta$ and Higgs respectively and these are obtained by
applying mean field approximations, which are given in Ref. (Das et al.,
2019). The DM density $\rho_{\chi}$ is

$\rho_{\chi}=\frac{\gamma}{(2\pi)^{3}}\int_{0}^{k_{f}^{\rm
DM}}\frac{M_{\chi}^{\star}}{\sqrt{M_{\chi}^{\star}{{}^{2}}+k^{2}}}d^{3}k,$ (7)

###  2.3 NM Parameters

The calculations of NM properties need the energy density and pressure as a
function of baryonic density. The energy density ${\cal{E}}$ can be expanded
in a Taylor series in terms of $\alpha$ (Horowitz et al., 2014; Baldo &
Burgio, 2016; Kumar et al., 2018).

${\cal{E}}(\rho,\alpha)={\cal{E}}(\rho,\alpha=0)+S(\rho)\alpha^{2}+{\cal{O}}(\alpha^{4}),$
(8)

where ${\cal{E}}(\rho,\alpha=0)$ is the energy of symmetric NM , $\rho$ is the
baryonic density and $S(\rho)$ is the symmetry energy, which is defined as

$S(\rho)=\frac{1}{2}\Bigg{(}\frac{\partial^{2}{\cal{E}}}{\partial\alpha^{2}}\Bigg{)}_{\alpha=0}$
(9)

The symmetry energy is also written as the energy difference between pure
neutron matter (PNM) and symmetric NM (SNM) or vice-versa through parabolic
approximation, i.e.

$S(\rho)=\frac{{\cal{E}}(\rho,\alpha=1)}{\rho}-\frac{{\cal{E}}(\rho,\alpha=0)}{\rho}.$
(10)

Although, the value of symmetry energy is fairly known at the saturation
density ($\rho_{0}$), its density dependence nature is not well known. The
behavior of $S(\rho)$ in high density, both qualitatively and quantitatively
shows a great diversion depending on the model used (Li et al., 2019). Similar
to the BE/A, the $S(\rho)$ can also be expressed in a leptodermous expansion
near the NM saturation density. The analytical expression of density
dependence symmetry energy is written as (Matsui, 1981; Kubis & Kutschera,
1997; Del Estal et al., 2001; Chen & Piekarewicz, 2014; Kumar et al., 2018):

$S(\rho)=J+L\zeta+\frac{1}{2}K_{sym}\zeta^{2}+\frac{1}{6}Q_{sym}\zeta^{3}+{\cal{O}}(\zeta^{4}),$
(11)

where $\zeta$=$\frac{\rho-\rho_{0}}{3\rho_{0}}$, $J$ = $S(\rho_{0}$) and the
parameters like slope ($L$), curvature ($K_{sym}$) and skewness ($Q_{sym}$) of
$S(\rho$) are

$\displaystyle L=3\rho\frac{\partial S(\rho)}{\partial\rho},$ (12)
$\displaystyle K_{sym}=9\rho^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}},$
(13) $\displaystyle
Q_{sym}=27\rho^{3}\frac{\partial^{3}S(\rho)}{\partial\rho^{3}}.$ (14)

The NM incompressibility ($K$) is defined as (Chen & Piekarewicz, 2014)

$\displaystyle
K=9\rho^{2}\frac{\partial}{\partial\rho}\Bigg{(}\frac{P}{\rho^{2}}\Bigg{)}.$
(15)

To estimate both symmetry energy and its slope parameters at saturation
density $\rho_{0}$, we use Eq. (10). Since the parameters $J$ and $L$ play
important roles like, formation of clusters in finite nuclei and normal star,
dynamic of heavy-ion collisions and cooling process of newly born NS. These
parameters are also crucial for the study of phase transition (finite/infinite
nuclear systems). Different approaches are available for the calculation of
$J$ and $L$ including their correlations (Centelles et al., 2009; Xu et al.,
2010; Fattoyev et al., 2012; Steiner & Gandolfi, 2012; Newton et al., 2012;
Dutra et al., 2012; Singh et al., 2013).

###  2.4 The EoS of NS

In this section, we describe the NS EoS within a medium of nucleons, electrons
and muons. In the NS, the neutron decays to proton, electron and anti-neutrino
(Glendenning, 1997; Quddus et al., 2019; Bhat & Paul, 2019). There is also
inverse $\beta$–decay to maintain the beta equilibrium and charge neutrality
condition. This can be expressed as (Glendenning, 1997)

$\displaystyle n\rightarrow p+e^{-}+\bar{\nu},$ $\displaystyle
p+e^{-}\rightarrow n+\nu.$ (16)

The stability of NSs is followed by $\beta$-equilibrium and charge-neutrality
conditions as follow as

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e},$
$\displaystyle\mu_{e}$ $\displaystyle=$ $\displaystyle\mu_{\mu},$ (17)

where, $\mu_{n}$, $\mu_{p}$, $\mu_{e}$, and $\mu_{\mu}$ are the chemical
potentials of neutrons, protons, electrons, and muons, respectively, and the
charge neutrality conditions is

$\displaystyle\rho_{p}=\rho_{e}+\rho_{\mu}.$ (18)

The chemical potentials $\mu_{n}$, $\mu_{p}$, $\mu_{e}$, and $\mu_{\mu}$ are
given by

$\mu_{n}=g_{\omega}\omega_{0}+g_{\rho}\rho_{0}+\sqrt{k_{n}^{2}+(M_{n}^{\star})^{2}},$
(19)
$\mu_{p}=g_{\omega}\omega_{0}-g_{\rho}\rho_{0}+\sqrt{k_{p}^{2}+(M_{p}^{\star})^{2}},$
(20) $\mu_{e}=\sqrt{k_{e}^{2}+m_{e}^{2}},$ (21)
$\mu_{\mu}=\sqrt{k_{\mu}^{2}+m_{\mu}^{2}},$ (22)

where $M_{n}^{\star}$ and $M_{p}^{\star}$ is the effective masses of neutron
and proton respectively calculated in Eq. (6). To find the particle fraction,
we solve Eq. (17) and (18) followed by Eqs. (19 – 22) in a self-consistent way
for a given baryon density. The total energy density and pressure of NS are
given by,

$\displaystyle{\cal{E}}_{NS}$ $\displaystyle=$
$\displaystyle{\cal{E}}+{\cal{E}}_{l},$ $\displaystyle and\hskip
28.45274ptP_{NS}$ $\displaystyle=$ $\displaystyle P+P_{l},$ (23)

where,

${\cal{E}}_{l}=\sum_{l=e,\mu}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{l}}d^{3}k\sqrt{k^{2}+m_{l}^{2}},$
(24)

and

$P_{l}=\sum_{l=e,\mu}\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{l}}\frac{d^{3}k\hskip
5.69046ptk^{2}}{\sqrt{k^{2}+m_{l}^{2}}}.$ (25)

Where ${\cal{E}}_{l}$, $P_{l}$ and $k_{l}$ are the energy density, pressure
and Fermi momentum for leptons respectively. The Eq. (23) gives the total
energy, pressure and number density of the NS.

###  2.5 Observables of the NS

In the Sec. 1, we have already mention that NS is very complex structure and
therefore the detailed study require both theories of GR and dense matter. i.e
NS forms a link between two fundamental theory in the modern physics and this
connection already given by Einstein’s field equation (Einstein, 1916)

$\displaystyle G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=8\pi T^{\mu\nu},$
(26)

where $G^{\mu\nu}$, $R^{\mu\nu}$, $g^{\mu\nu}$ and $R$ are the Einstein
tensor, Ricci tensor, metric tensor and Ricci scalar respectively. The
$T^{\mu\nu}$ is the energy-momentum tensor for perfect fluid is given as
(Glendenning, 1997; Krastev et al., 2008)

$\displaystyle T^{\mu\nu}=({\cal E}_{NS}+P_{NS})u^{\mu}u^{\nu}+Pg^{\mu\nu},$
(27)

where ${\cal E}_{NS}$ and $P_{NS}$ are the energy density and pressure of the
NS. $u^{\mu}$ is the 4-velocity satisfying, $u^{\mu}u_{\mu}=-1$. The
$T^{\mu\nu}$ directly depends on the EoS of the stellar matter in the form of
$P_{NS}({\cal E}_{NS})$. To solve the Einstein’s Eq. 26, First, we have to
solve the short-range nuclear forces of many-body nuclear physics in a local
inertial frame in which space-time is flat. Second, the long range force of
the gravitational field which describe the curvature of space-time created by
the massive objects (Krastev et al., 2008). We calculate the many-body theory
using RMF formalism in Sub-Sec. 2.1. But in this present Sub-Sec. we calculate
the different observables like $M$, $R$ and $I$ etc. for the NS. In our whole
calculations we use $G=c=1$.

In case of static, spherically symmetric stars the metric is in the form of

$\displaystyle
ds^{2}=-e^{2\nu(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2},$ (28)

where $r$, $\theta$ and $\phi$ are the spherical co-ordinates. $\nu(r)$,
$\lambda(r)$ are the metric potential are given as (Krastev et al., 2008)

$\displaystyle e^{2\lambda(r)}=[1-\gamma(r)]^{-1},$ (29) $\displaystyle
e^{2\nu(r)}$ $\displaystyle=$ $\displaystyle
e^{-2\lambda(r)}=[1-\gamma(r)],\qquad r>R_{star}$ (30)

with

$\gamma(r)=\left\\{\begin{array}[]{l l}\frac{2m(r)}{r},&\quad\mbox{if
$r<R_{star}$}\\\ \\\ \frac{2M}{r},&\quad\mbox{if
$r>R_{star}$}\end{array}\right.$ (31)

For static star, the Einstein’s Eq. 26 reducess to

$\displaystyle\frac{dP_{NS}(r)}{dr}=-\frac{[P_{NS}(r)+{\cal{E}}_{NS}(r)][m(r)+4\pi
r^{3}P_{NS}(r)]}{r[r-2m(r)]},$ (32)

and

$\displaystyle\frac{dm(r)}{dr}=4\pi r^{2}{\cal{E}}_{NS}(r),$ (33)

where ${\cal{E}}_{NS}(r)$ and $P_{NS}(r)$ are the total energy density and
pressure appearing in Eq. (23) under the $\beta$-equilibrium condition. $m(r)$
is the gravitational mass, and $r$ is the radial parameter. These two coupled
equations are solved to get the $M$ and $R$ of the SNS at certain central
density.

The metric of RNS in equilibrium is described by a stationary and axisymmetric
metric of the form (Stergioulas, 2003)

$\displaystyle ds^{2}=-e^{2\nu}dt^{2}+e^{2\psi}(d\phi-\omega
dt^{2})+e^{2\alpha}(r^{2}d\theta^{2}+d\phi^{2}),$ (34)

where $\nu$, $\psi$, $\omega$ and $\alpha$ are metric functions which depend
on $r$ and $\theta$ only. The energy-momentum tensor same as in Eq. 27, but
the 4-velocity is changed as follow as

$u^{\mu}=\frac{e^{\mu}}{\sqrt{1-v^{2}}}(t^{\mu}+\Omega\phi^{\mu}),$ (35)

and the spatial velocity, $v=(\Omega-\omega)e^{\psi-\nu}$ , where $\Omega$ is
the angular velocity of the star. $t^{\mu}$ and $\phi^{\mu}$ are two killing
vectors in the space-time associated with time and transnational symmetries.

Many people have calculated the moment of inertia of the NS (Stergioulas,
2003; Jha et al., 2008; Sharma & Jha, 2009; Friedman & Stergioulas, 2013;
Paschalidis & Stergioulas, 2017; Quddus et al., 2019; Koliogiannis &
Moustakidis, 2020). The simple expression is $I=J/\Omega$, where $J$ is
angular momentum and $\Omega$ is the angular velocity of the NS. The
expression of $I$ of uniformly rotating NS with angular frequency $\omega$ is
given as (Lattimer & Prakash, 2000; Worley et al., 2008)

$I\approx\frac{8\pi}{3}\int_{0}^{R}({\cal E}_{NS}+P_{NS})\
e^{-\phi(r)}\Big{[}1-\frac{2m(r)}{r}\Big{]}^{-1}\frac{\bar{\omega}}{\Omega}r^{4}dr,$
(36)

where the $\bar{\omega}$ is the dragging angular velocity for a uniformly
rotating star. The $\bar{\omega}$ satisfying the boundary conditions are

$\bar{\omega}(r=R)=1-\frac{2I}{R^{3}},\qquad\frac{d\bar{\omega}}{dr}\big{|}_{r=0}=0.$
(37)

The Keplerian frequency of the NS also calculated in these Refs. (Komatsu et
al., 1989a, b; Stergioulas, 2003; Dhiman et al., 2007; Jha et al., 2008;
Krastev et al., 2008; Worley et al., 2008; Haensel et al., 2009; Sharma & Jha,
2009; Koliogiannis & Moustakidis, 2020) and the expression is given as

$\nu_{k}(M)\approx\chi\Big{(}\frac{M}{M_{\odot}}\Big{)}^{1/2}\Big{(}\frac{R}{10\
\ km}\Big{)}^{-3/2},$ (38)

where $M$ and $R$ are the gravitational mass and radius of the RNS
respectively. The value of $\chi$ is calculated from the fitting or
empirically (Lattimer & Prakash, 2004; Haensel et al., 2009). In Ref. (Haensel
et al., 2009), they have found the value of $\chi=1.08$ kHz empirically. For
rotating NS, we calculate the $M$, $R$, $I$ and Keplerian frequency
($\nu_{K}$) using public-domain program, RNS code written by the Stergioulas
(Stergioulas & Morsink, 1999).

##  3 Results and Discussions

In this section, we present the calculated results for BE/A, $K$, $S$ and its
derivatives for NM varying the $k_{f}^{DM}$ at different $\alpha$. We extend
the calculations to NSs and find the $M$, $R$ and $I$.

Table 1: The 3-parameter sets NL3 (Lalazissis et al., 1997) , G3 (Kumar et
al., 2017) and IOPB-I (Kumar et al., 2018) are listed. All the coupling
constants are dimensionless.

Parameter | NL3 | G3 | IOPB-I
---|---|---|---
$m_{\sigma}/M_{nucl.}$ | 0.541 | 0.559 | 0.533
$m_{\omega}/M_{nucl.}$ | 0.833 | 0.832 | 0.833
$m_{\rho}/M_{nucl.}$ | 0.812 | 0.820 | 0.812
$m_{\delta}/M_{nucl.}$ | 0.0 | 1.043 | 0.0
$g_{\sigma}/4\pi$ | 0.813 | 0.782 | 0.827
$g_{\omega}/4\pi$ | 1.024 | 0.923 | 1.062
$g_{\rho}/4\pi$ | 0.712 | 0.962 | 0.885
$g_{\delta}/4\pi$ | 0.0 | 0.160 | 0.0
$k_{3}$ | 1.465 | 2.606 | 1.496
$k_{4}$ | -5.688 | 1.694 | -2.932
$\zeta_{0}$ | 0.0 | 1.010 | 3.103
$\eta_{1}$ | 0.0 | 0.424 | 0.0
$\eta_{2}$ | 0.0 | 0.114 | 0.0
$\eta_{\rho}$ | 0.0 | 0.645 | 0.0
$\Lambda_{\omega}$ | 0.0 | 0.038 | 0.024

###  3.1 NM Properties

The $\cal E$ and $P$ are obtained using RMF approaches with DM (Panotopoulos &
Lopes, 2017; Quddus et al., 2019; Das et al., 2019) in Eqs. 4 and 5. The
${\cal{E}}$ and $P$ as a function of baryon density $\rho$ are shown in Fig.
1. The RMF forces NL3 (Lalazissis et al., 1997), G3 (Kumar et al., 2017) and
IOPB-I (Kumar et al., 2018) are given in Table 1. Since, the $k_{f}^{DM}$ is
not yet settled, we change its values for $k_{f}^{DM}=$ 0.0, 0.03 and 0.06 GeV
and note down the variations as a function of $\rho$ in Fig. 1. We notice that
the value of ${\cal{E}}$ changes significantly without affecting the $P$ (see
Fig. 1).

Figure 1: (colour online) The energy density ( Eq. 4) and pressure ( Eq. 5)
for SNM with baryon density at $k_{f}^{DM}$= 0.0, 0.03 and 0.06 GeV.

NL3 gives the stiffest EoS as compare to IOPB-I and G3 for the SNM case. Here
also, G3 predicts the softest EoS, which is shown in the Fig. 1. Thus, the
qualitative nature of the EoS is similar with and without the presence of DM
as far as stiffness or softness is concerned. The BE/A is defined as
$\frac{\cal{E}}{\rho}$-$M_{nucl.}$, where the $\cal{E}$ is the total energy
density in Eq. (4) and $\rho$ is the baryon density. The BE/A as a function of
$\rho$ at different $k_{f}^{DM}$ is shown in Fig. 2 for both SNM and PNM. Here
the effect of DM on BE/A is significant with respect to $k_{f}^{DM}$ for both
SNM and PNM.

Figure 2: (color online) The BE/A of NM in the presence of DM at $k_{f}^{DM}$=
0.0, 0.03 and 0.06 GeV.

The NS is mostly made of neutrons with a small fraction of protons, electrons
and muons in the charge-neutral and $\beta$-equilibrium system. Thus to get an
idea about the NM parameters, we check the variation of the effective mass
($M^{\star}$) with different $\alpha$. Here it is imperative to mention that
$k_{f}^{DM}$ = 0 GeV means $\rho_{\chi}$ is zero, but the effect on
$M^{\star}$ is very less due to non-zero Higgs-nucleon Yukawa coupling in Eq.
(6). The contribution of Higgs filed is very small
${\cal{O}}$($10^{-6}-10^{-8}$) even after varying $k_{f}^{DM}$ to its maximum
value. So that we plot the effective mass to mass ratio of the nucleon
($M^{\star}/M_{nucl.}$) as a function of $\rho$ for different $\alpha$, which
is shown in Fig. 3. In the presence of DM, the $M^{\star}/M_{nucl.}$ decreases
with baryon density $\rho$, similar to the normal nuclear medium. As far as
the neutron to proton ratio (N/Z) increases, the $M^{\star}/M_{nucl.}$ value
goes on increasing mostly in the high-density region. However, there is
practically no effect of $\alpha$ in the low-density region of the NM system.

Figure 3: (colour online) The variation of effective mass in Eq. (6) for
different $\alpha$ with baryon density with $k_{f}^{DM}$ = 0.06 GeV.

Another important NM parameter is $K$. This value tells us how much one can
compress the NM system. It is a standard quantity at the saturation point.
However, an astronomical object like the NS, its density varies from the
centre of the star to the crust with a variation of $\rho$ from
$\rho=10\rho_{0}$ to 0.1$\rho_{0}$ (Lattimer & Prakash, 2004). Thus, to
achieve better knowledge on the compression mode or monopole vibration mode,
we have to calculate the $K$ for all the density range of NM with different
$\alpha$ including 0 and 1. Since we see the earlier case, DM does not affect
on the pressure of either SNM or PNM also in NS (in sec. 3.2), so DM doesn’t
affect the $K$ of NM. The variation of $K$ with baryon density for different
$\alpha$ displayed in Fig. 4. One can see in Table 2, for SNM system the
incompressibility at saturation $K_{\infty}$ are 271.38, 243.96 and 222.65 MeV
for NL3, G3 and IOPB-I respectively. It is worthy of mentioning that the DM
does not affect on the incompressibility. That means, the $K$ values remain
unaffected with the variation of $k_{f}^{DM}$. On the other hand, substantial
variation is seen with the different $\alpha$. We found that the value of $K$
increases initially up to a maximum and then gradually decreases, as shown in
Fig. 4. The calculations also show that with increasing $\alpha$, the
incompressibility decreases irrespective of the parameter sets. The values of
$K$ for G3 and IOPB-I parameter sets lie in the region (except for NL3) given
by the experimental value in Table 2. Since the NL3 gives very stiff EOS so
that it’s all NM parameters like $K$, $J$, $L$ etc. provide very large values
and do not lie the region given in Table 2.

Figure 4: (colour online) The variation of incompressibility $K$ with
different $\alpha$ as a function of baryon density $\rho$ with $k_{f}^{DM}$ =
0.06 GeV.

The recent gravitational wave observation from the merger of two NSs, the
GW170817 (Abbott et al., 2017, 2018), constraints the upper limit on the tidal
deformability $\Lambda$ and predicts a small radius. Also, the recent
discovery of the three highly massive stars $\sim$ 2 $M_{\odot}$ (Antoniadis
et al., 2013; Fonseca et al., 2016; Cromartie et al., 2019) predicts that the
pressure in the inner core of the star is large, where the typical baryon
number density quite high $\rho$ $>$ 3$\rho_{0}$ in this region. The pressure
in the outer core of the massive NS is considered to be small in the density
range 1 to 3 $\rho_{0}$ (McLerran & Reddy, 2019). Combining these observations
of large masses and the smaller radii of the massive NSs, one can infer that
the causality (Rhoades & Ruffini, 1974; Bedaque & Steiner, 2015; Kojo et al.,
2015; Moustakidis et al., 2017; McLerran & Reddy, 2019) of the NM inside the
inner core of the NS can violate (McLerran & Reddy, 2019). It is conjectured
that the speed of the sound $C_{s}\leq c/\sqrt{3}$, where
$C_{s}^{2}=\frac{\partial P}{\partial{\cal{E}}}$ with $C_{s}$ and $c$ are the
speed of the sound and light, respectively.

Figure 5: (colour online) The variation of speed of the sound with baryon
density for NL3, G3 and IOPB-I parametrizations at different $\alpha$ with
$k_{f}^{DM}$ = 0.06 GeV.

To see the causality condition for the NM case with an admixture of DM, we
plot $C_{s}^{2}$ as a function of baryon number density $\rho$ in Fig. 5 at
different $\alpha$ for NL3, G3 and IOPB-I parameter sets. We find that the
$C_{s}^{2}$ increases approximately up to 0.8 fm-3, then it is constant for
high density regions. It is clear from the values of $C_{s}^{2}$ that the
causality remains intact for a wide range of density for all three parameter
sets, as shown in Fig. 5.

The symmetry energy $S$ and its coefficients $L$, $K_{sym}$ and $Q_{sym}$ are
defined in Eqs. (12 – 14), play a crucial role in the EoS for symmetric and
asymmetric NM.

Figure 6: (colour online) The symmetry energy $S$ and its slope parameter $L$
are plotted with the varying $k_{f}^{DM}$ as a function of with $\rho$. We
found almost the same results ( Table 2 ) with and without DM for all the
three-parameter sets.

As we have mentioned earlier, these parameters are important quantities to
determine the nature of the EoS. Just after the supernova explosion, its
remnants, which lead towards the formation of a NS, is in a high temperature
($\sim 200$ MeV) state (Gnedin et al., 2001; Page et al., 2004; Yakovlev &
Pethick, 2004; Yakovlev et al., 2005, 2010). Soon after the formation of the
NS, it starts cooling via the direct urca processes to attain the stable
charge neutrality and $\beta$-equilibrium condition. The dynamical process of
NS cooling is affected heavily by these NM parameters. Thus, it is very much
intuitive to study these parameters in more rigorously.

Table 2: The nuclear matter properties such as BE/A, symmetry energy and its derivatives etc. are given at saturation density for three-parameters sets with k${}_{f}^{DM}$ = 0 GeV (without DM), 0.03 GeV and 0.06 GeV respectively for SNM. Similarly the effective mass ($M^{\star}/M_{nucl}$), incompressibility ($K$) varying with $\alpha$ = 0, 0.6, 1.0 at $\rho_{0}$ (not in the variations of k${}_{f}^{DM}$) in last three-rows. The empirical/expt. value for $\rho_{0}$, BE/A, $J$, $L$, $K_{sym}$ and $K$ are also given at saturation density. Parameters |  NL3 |  G3 |  IOPB-I |  Empirical/Expt. value
---|---|---|---|---
| 0.0 | 0.03 | 0.06 | 0.0 | 0.03 | 0.06 | 0.0 | 0.03 | 0.06 |
$\rho_{0}$ (fm-3) | 0.148 | 0.148 | 0.148 | 0.148 | 0.148 | 0.148 | 0.149 | 0.149 | 0.149 |  0.148 – 0.185 (Bethe, 1971)
BE/A (MeV) | -16.35 | 143.95 | 1266.11 | -16.02 | 143.28 | 1266.44 | -16.10 | 143.09 | 1257.51 |  -15 – -17 MeV (Bethe, 1971)
$J$ (MeV) | 37.43 | 38.36 | 38.36 | 31.84 | 31.62 | 31.62 | 33.30 | 34.45 | 34.45 |  30.20 – 33.70 MeV (Danielewicz & Lee, 2014)
$L$ (MeV) | 118.65 | 121.44 | 121.45 | 49.31 | 49.64 | 49.64 | 63.58 | 67.16 | 67.67 |  35.00 – 70.00 MeV (Danielewicz & Lee, 2014)
$K_{sym}$ (MeV) | 101.34 | 101.05 | 100.32 | -106.07 | -110.38 | -111.10 | -37.09 | -45.94 | -46.67 |  -174 – -31 MeV (Zimmerman et al., 2020)
$Q_{sym}$ (MeV) | 177.90 | 115.56 | 531.30 | 915.47 | 929.67 | 1345.40 | 868.45 | 927.84 | 1343.58 |  —————-
$\alpha$ = | 0 | 0.6 | 1.0 | 0 | 0.6 | 1.0 | 0 | 0.6 | 1.0 |
$M^{\star}/M_{nucl.}$ | 0.595 | 0.596 | 0.606 | 0.699 | 0.700 | 0.704 | 0.593 | 0.599 | 0.604 |  —————
$K$ (MeV) | 271.38 | 312.45 | 372.13 | 243.96 | 206.88 | 133.04 | 222.65 | 204.00 | 176.28 |  220 – 260 MeV (Stone et al., 2014)

The $S$ and its $L$-coefficient for the whole density range for all three
parameter sets NL3, G3 and IOPB-I with different $k_{f}^{DM}$ are displayed in
Fig. 6. The effect of DM on $J$ and $L$ is very small, and it is difficult to
notice in the figure. To have the knowledge, we summarised the results at the
saturation point in Table 2. For example, with NL3 set, $S(\rho_{0}$) = $J$ =
37.43 MeV for PNM and it increases to a value $J$ = 38.36 MeV with DM.
Similarly, $L$ = 118.65 MeV for without DM, $L$ = 121.44 and 121.45 MeV in the
presence of DM for various DM momentum $k_{f}^{DM}$. For other sets, one can
see the variation in Table 2. This is due to the fact that the effect of DM
does not change NM asymmetry to a significant extent. A careful inspection of
Fig. 6 and Table 2, it is clear that $S$ and $L$ are force-dependent. It is
maximum for NL3 and minimum for G3 sets. The experimental value of $J$ and $L$
is given in the Table 2 at $\rho_{0}$ is in between 30.2 – 33.7 MeV and 35.0 –
70.0 MeV respectively. With the addition of DM, the values of $J$ and $L$ lie
(except for NL3) in this region as given in the Table 2.

The other higher-order derivatives of symmetry energy like $K_{sym}$ and
$Q_{sym}$ are also calculated in this section. The results are displayed in
Fig. 7 and their numerical values are tabulated in Table 2. The $K_{sym}$ is a
parameter, which tells a lot not only about the surface properties of the
astrophysical object (such as NS and white dwarf), but also the surface
properties of finite nuclei. The whole density range of $K_{sym}$ and
$Q_{sym}$ for G3 and IOPB-I sets are shown. The $K_{sym}$ affected marginally,
but the parameter $Q_{sym}$ influenced significantly by DM (see Table 2) for
these values at saturation. At low density, $K_{sym}$ initially decreases
slightly, then it increases up to $\rho$ ($\sim$ 0.2 fm-3) and after that
decreases the value almost like an exponential function. Recently the value of
$K_{sym}$ is calculated by Zimmerman et al. (Zimmerman et al., 2020) with the
help of NICER (Miller et al., 2019; Riley et al., 2019; Bogdanov et al., 2019;
Bilous et al., 2019; Raaijmakers et al., 2019; Guillot et al., 2019) and
GW170817 data and its values lies in the range -174 to -31 MeV as given in
Table 2. Our $K_{sym}$ values lie in this region at the saturation density.

Figure 7: (colour online) The variation of $K_{sym}$ and $Q_{sym}$ with baryon
density is plotted for G3 and IOPB-I at different $k_{f}^{DM}$. The $K_{sym}$
and $Q_{sym}$ are opposite to each other both for G3 and IOPB-I.

###  3.2 Neutron Star Matter

As an application of our mixed EoS, i.e. DM and hadron matter, we used the EoS
to calculate the M, R and I of RNS. For this, we constructed the star EoS
maintaining the charge neutrality and $\beta$-equilibrium condition varying
the DM momentum $k_{f}^{DM}$. The EoS for NL3, G3 and IOPB-I at $k_{f}^{DM}$=
0.0, 0.03 and 0.06 GeV are shown in Fig. 8. As we have mentioned earlier in
the PNM and SNM, the EoS is very sensitive to $k_{f}^{DM}$, here also the
energy density ${\cal{E}}$ is equally sensitive to the DM. We find softer EoS
as in (Li et al., 2012; Panotopoulos & Lopes, 2017; Bhat & Paul, 2019; Das et
al., 2019; Quddus et al., 2019) with $k_{f}^{DM}$. Similar to the SNM and PNM,
NL3 predicts the stiffest and G3 the softest EoS for different $k_{f}^{DM}$.
The calculations of $M$, $R$ and $I$ of the SNS and RNS need the entire EoS of
the NS. In crust region, which has very low density ( $<5\times 10^{14}$ g
cm-3 ), the inter nucleon distances is higher than the core region of the NS
and the clusterisation of the nucleon happens. In the core region, due to the
high density the NS system behave like a incompressible fluid of the nucleons.
For the core part, we use the RMF equation of state. In crust part, we use BPS
(Baym et al., 1971) EoS. Construction of the BPS EoS is based on the
minimisation of the Gibbs function and effects of the Coulomb lattice, which
gives a suitable combination of the A and Z. Once we know the entire EoS, we
can calculate the properties of the SNS solving TOV equations (Tolman, 1939;
Oppenheimer & Volkoff, 1939) and RNS using RNS code (Stergioulas & Morsink,
1999). The brief formalism for RNS can be found in Sub-Sec. 2.5.

Figure 8: (colour online) The variation of energy density and pressure in Eq.
(23) with different $k_{f}^{DM}$. We find NL3 is the stiffest and G3 is the
softest.

Before calculating the $M$ and $R$, we tested the causality condition (Rhoades
& Ruffini, 1974; Bedaque & Steiner, 2015; Kojo et al., 2015; Moustakidis et
al., 2017; McLerran & Reddy, 2019) in the NS medium with an admixture of DM.
We have seen that both in NM and NS matter case, the causality is not violated
throughout the region, as shown in Fig. 5 and 9 respectively. The dashed
horizontal line is the conjectured $C_{s}^{2}=1/3$ value given in Fig. 9. The
NL3 predicts the stiff rise in $C_{s}^{2}$ as compared to G3 and IOPB-I. But
in all the cases $C_{s}^{2}$ is less than 1/3 for very low density region
($\rho<0.4$ $fm^{-3}$). As compared to the NM, the NS contains nucleons,
electrons and muons, which is completely a different stable system survived by
the balancing force due to the attractive gravitation and the repulsive
degenerate neutrons also with short-range repulsive nuclear force.

Figure 9: (colour online) The speed of the sound is plotted with the baryon
density for 3-parameters sets with $k_{f}^{DM}$ = 0.06 GeV. The orange dashed
line represents the conjecture (C${}_{S}^{2}$ = 1/3) and magenta dashed line
represents the C${}_{S}^{2}$ is equal to c. Figure 10: (colour online) The
variation of mass with radius with different $k_{f}^{DM}$ is shown. The orange
colour dot shows the maximum mass of the corresponding $k_{f}^{DM}$ for G3
parameter set. The dotted-dashed line represents for the SNS, and the bold
line represents the RNS. The maximum rotational frequencies of NS also shown.
The recent observational constraints on NS masses (Demorest et al., 2010;
Cromartie et al., 2019) are also shown.

Now, we calculate the M-R relations and $I$ for G3 as a representative case.
The M-R relation for different k${}_{f}^{DM}$ are shown in Fig. 10. The
results from the precisely measured NSs masses, such as PSR J1614-2230
(Demorest et al., 2010) and PSR J0740+6620 (Cromartie et al., 2019) are shown
in the horizontal lines in pink colours. These observations suggest that the
maximum mass predicted by any theoretical model should reach the limit $\sim$
2.0 $M_{\odot}$, and this condition is satisfied in all of the EoSs, which are
taken into consideration. We noticed that the increase in $k_{f}^{DM}$, higher
the energy density at a given $\rho$, which yield the lower the maximum mass
and radius of the SNS and RNS. For RNS case the maximum mass is increased by
$\sim$ 20%, and radius increased by $\sim$ 26% for the given EoS, which is
approximately equal to the increase in mass due to the rapid rotation of the
NS (Stergioulas, 2003; Worley et al., 2008).

Figure 11: (colour online) The variation NS mass with Keplerian frequency
$\nu_{K}$ is shown for G3 parameter set. The two vertical magenta line
represents the frequencies of the fastest NSs J1748-2446ad (Hessels et al.,
2006) and XTE J1739-285 (Kaaret et al., 2007).

Here, we examine the effect of Kepler frequency ($\nu_{K}$) on the mass of the
NS. In Fig. 10, the mass of the RNS is increased due to the rapid rotation of
the NS due to its high $\nu_{K}$, i.e. the mass of NS directly depends on the
$\nu_{K}$. Theoretical calculations allows the value of $\nu_{K}$ is more than
2000 $Hz$ (Koliogiannis & Moustakidis, 2020), but till now two fastest pulsar
was detected having frequency 716 $Hz$ (Hessels et al., 2006) and 1122 $Hz$
(Kaaret et al., 2007). From Fig. 11, one can conclude that the NS mass
approximately more than 1.7 $M_{\odot}$ is rotated more than the fastest
pulsar as of today. In our case, we find the $\nu_{K}$s are 1553, 1671 and
2079 $Hz$ for the DM momentum 0, 0.03 and 0.06 GeV respectively at the maximum
mass. The spherical NS is leading to deformed shape with the increasing of
mass (or frequency)

A measurement of the $I$ of PSR J0737-3039A is expected via optical
observation of the orbital precision in the double pulsar system in the near
future (Burgay et al., 2003). As the $I$ depends on the internal structure of
the NS, it’s measurement will constrain the unknown EoS of supra-nuclear
matter, which is believed to be universal for all NS (Landry & Kumar, 2018;
Kumar & Landry, 2019). Here we show the variation of $I$ with $M(M_{\odot}$)
in Fig. 12. The change of $I$ with $M$ is almost linear for different values
of $k_{f}^{DM}$ up to the maximum mass of the star. Then there is a drop of
$I$, as shown in Fig. 12.

Figure 12: (colour online) The variation of $I$ with the mass of NS for
different $k_{f}^{DM}$ using G3 parameter set. The orange colour dot shows the
maximum mass of the corresponding $k_{f}^{DM}$. The dotted-dashed line
represents for the SNS, and the bold line represents the RNS. The overlaid
arrows represent the constraints on $I$, of PSR J0737-3039A set (Landry &
Kumar, 2018; Kumar & Landry, 2019) from the analysis of GW170817 data (Abbott
et al., 2017, 2018)

Since the increase of DM momentum ($k_{f}^{DM}$) leads to softer EoS, and
hence the decrease of $I$. This is because $I$ ($\sim$ MR2) is directly
proportional to the mass and square of the radius of the rotating object. The
moment of inertia is larger for the stiffer EoS as it predicts a larger radius
and vice-versa.

##  4 Summary and Conclusions

In summary, we studied the effects of DM on the NM parameters, such as nuclear
incompressibility ($K$), symmetry energy ($S$) and its higher-order
derivatives like $L$-slope parameter, $K_{sym}$-isovector incompressibility
and $Q_{sym}$-skewness parameter for different asymmetric. These are the
significant quantities responsible for the behaviour of nuclear EoS. The EoS
becomes softer or stiffer depending on the values of these parameters. We
calculated these quantities taking different admixture of DM fraction in the
NM with varying neutron-proton asymmetry. We take Neutralino as a DM candidate
which is trapped inside the NS and its interaction to nucleons through SM
Higgs via Yukawa potential. The RMF Lagrangian with NL3, G3 and IOPB-I forces
are used to get the hadron EoS, and the DM part is added on top of it.

We find softer EoS with the increasing DM momentum, i.e. the energy density
increases with $k_{f}^{DM}$ without adding much to the pressure. The influence
of DM on effective mass, symmetry energy, $L$-coefficient and $K_{sym}$ are
not much change with the variation of $k_{f}^{DM}$ due to the small
contributions of the Higgs field. However, some other derivatives of $S$
(Qsym) and $\cal{E}$ ($K$) affected significantly by DM. These effects
contribute to the mass, radius and moment of inertia of stellar object like a
NS. Also, the variation of ${\cal{E}}$, $K$ and $Q_{sym}$ due to DM not only
affect the structure of NS but also significantly influence on the cooling
effect of newly born NS after a supernova explosion. Thus, a detailed study of
the temperature-dependent EoS is due, which will be published somewhere (Kumar
et al., 2020).

To check the influence of DM on the accreting object, we constructed the NS
EoS at various momenta of DM admixture. The mass, radius and the moment of
inertia are evaluated for static as well as rotating cases using the TOV and
RNS equations. The mass and radius are significantly reduced with the increase
of DM momentum as we know that the stiffer EoS gives higher mass, radius and
$I$ of a RNS. The mass of the NS is significantly changed due to the rapid
rotation of the NS. Quantitatively, the mass of NS approximately more than 1.7
$M_{\odot}$ are rotating having the frequency more than the fastest pulsar
ever detected.

ACKNOWLEDGEMENTS

H. C. Das would like to thank T. K. Jha for the fruitful discussion about the
RNS code and Abhishek Roy for the necessary discussion on the DM. This work is
supported in part by JSPS KAKENHI Grant No.18H01209, and also S. K. Biswal and
Ang Li are supported by the National Natural Science Foundation of China Grant
No. 11873040.

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