# Impacts of dark matter on the curvature of the neutron star

H. C. Das,11footnotetext: Corresponding author. Ankit Kumar  Bharat Kumar  S.
K. Biswal  S. K. Patra

###### Abstract

The effects of dark matter (DM) on the curvatures of the neutron star (NS) are
examined by using the stiff and soft relativistic mean-field equation of
states. The curvatures of the NSs are calculated with the variation of baryon
density. It is found that the radial variation of different curvatures
significantly affected by DM inside the star. The surface curvature is found
to be more remarkable for the massive star. The effects of DM on the
compactness of the maximum NS mass is less as compared to canonical star. The
binding energy of the NS goes towards positive with the increase of DM
momentum and makes the system unstable.

##  1 Introduction

Neutron stars (NSs) are one of the most enigmatic stellar remnants with
incredibly dense core and sturdy crust, enough to hold up long-lived bulges
that could produce potentially large ripples in the space, known as
gravitational wave (GW). NSs can be considered as one of the best laboratory
in the Universe to appraise many astrophysical models in the strong
gravitational field regime [1, 2]. The observables of the NS are quite
complicated to explore due to the uncertainty in the equation of state (EoS)
at supra saturation density.

In August 2017, the LIGO and Virgo groups detect the gravitational waves
emerged from the collision of two NSs [3, 4]. They have considered the
Universe with no dark matter (DM) in their modelling [5]. But, the DM makes up
more than 80$\%$ mass in the Universe, and it is also believed that some of
the DM particles are weakly interacting, so it is customary that it will
impact the NS properties up to certain extent [6]. It has also been reported
that when a compact star rotates in the Galaxy through DM halo it captures
some of the DM particles [7]. The enormous gravitational force and the immense
baryonic density inside the NS are responsible for the incarceration of DM
Particles. The efficacy on the NS observational properties depends on the
amount of DM captured by it [7].

Theoretically several types of DM particles have been hypothesized and
reported till date, like, weakly interacting massive particle (WIMP), feebly
interacting massive particle (FIMP) etc. The WIMPs are the most abundant DM
particles in the early Universe due to its freeze-out mechanism [7, 8]. They
equilibrated with the environment at freeze-out temperature and annihilate to
form different standard model particles and leptons. Recently some approaches
have been dedicated on the heating of NS due to deposition of kinetic energy
by the DM [9, 10]. The annihilation of the DM particles enhance the cooling of
the NSs [11, 12, 13]. The resurrection of the DM develop its interaction with
baryon which affects the structure of the NS [14, 15]. The accumulation of the
DM inside the NS is constrained by the Chandrasekhar limit. If the accumulated
quantity of the DM exceeds this limit, it can evolve into a tiny black hole
and destroy the star. Different approaches have been used to calculate the NS
properties with the inclusion of DM inside the NS [16, 14, 17, 15, 18, 19, 20,
21, 13, 22, 23, 24, 25]. However, in the present scenario, we take the non-
annihilating WIMP as DM candidate inside the NS. The detailed discussions can
be found in our previous work [25]. We observed that the addition of DM
softens the EoS which results in the reduction of mass-radius of the NS.

According to Einstein’s theory, a massive body wraps the space-time and makes
curvature around it which we experience as gravity. The strength of the
gravitational field at a distance $r$ from an object of mass $M$ is measured
by the compactness parameter, which is defined as
$\eta\equiv\frac{GM}{rc^{2}}$ and its value lie in between 0 and 1 [26, 27].
The value of $\eta=0$ corresponds to the flat Minkowski space in special
theory of relativity, while $\eta=1$ is the event horizon limit of a black
hole, i.e. strongest gravitational field. Curvature is the crucial thing to
quantify gravity and in determining the strength of the gravitational field
[26]. A massive body has larger curvature than the lighter one. Some
experiments had already been done to measure the curvature of the space-time.
Recently, the direct detection of gravity-field curvature has also been
observed by using atom interferometers [28].

We describe the different curvature quantities briefly in the concept of
general theory of relativity (GR) from the Ref. [29]. The Riemann tensor is an
important quantity to measure the curvature and it has twenty independent
components in four dimensions. Kretschmann scalar is defined as the square
root of the full contraction of the Riemann tensor and has the same property
as Riemann tensor. The Ricci tensor is the contraction of the Riemann tensor
and the trace of the Ricci tensor is known as the Ricci scalar or curvature
scalar. The Ricci scalar and the Ricci tensor contain all the informations
about the Riemann tensor leaving only the trace-less part. Weyl tensor can be
formed by removing all the contraction terms of the Riemann tensor.
Physically, the Ricci scalar and the Ricci tensor measure the volumetric
change of the body in presence of the tidal effect and the Weyl tensor gives
information about the shape distortion of the body. However, the Riemann
tensor measures both the distortion of shape and the volumetric change of the
body in presence of the tidal force.

The strong field regime of the NS can be probed deeper with the help of modern
observational instruments. The core is around 15 times denser than its
surface, means most of the matter is concentrated in the core which makes the
accurate measurement of $M$-$R$ profile of the NS is more complicated. But in
case of the compactness and surface curvature within the star are increasing
radially towards the surface. This is probably the reason that the measurement
of the maximum mass-radius is more prominent for the EoS rather than the
gravity. We explain the results of some recent approaches on the study of the
curvature of the NS. In Ref. [30], it is quantified the unconstrained gravity
of the NS in the framework of GR. They have calculated the curvature of the NS
and noticed that GR is not well tested in the whole range of the star than
EoS. Also found that the radial variation of Weyl tensor follows the power law
for the large part of the star. Further, Xiao et al. [27] have taken both
relativistic mean-field (RMF) and Skyrme-Hartree-Fock approaches and concluded
that the symmetry energy affects the curvature of lighter NS significantly and
have minimal effects on the massive NS. Moreover, to quantify the deviations
come from GR in the strong-field regime, the detailed understanding required
to study the properties of DM in the Universe [26]. Therefore, in the present
work, we investigate the curvature of the NS in the presence of DM in
quantitative way by using RMF theory with three different parameter sets, NL3
[31], G3 [32] and IOPB-I [33].

In the present analysis, we take the RMF formalism to calculate the EoS. The
RMF model reproduces the experimental value for exotic and super-heavy nuclei
[34, 35] precisely and grant a good description of the finite nuclei up to the
drip line. The extended RMF (E-RMF) formalism [32, 33] is in excellent
agreement with all the nuclear matter (NM) properties and also satisfy most of
the NS observable constraints allocated till now. In Sec. 2, we calculate the
EoS of NS with the addition of DM. The curvature calculations are discussed in
2.5. The detailed procedures are given in [30]. In Sec. 3, we present the
variation of curvatures with the different quantities like baryon density,
mass and radius with the addition of DM. The compactness and binding energy of
the NS is also calculated with the inclusion of DM. Finally, we give a brief
conclusion on the curvatures of the NS in Sec. 4.

##  2 Formalism

In this section, we provide the formalism required to compute the curvature of
the NS in the presence of DM. First, we briefly sketch the E-RMF formalism
along with DM by presenting model Lagrangian [31, 32, 33]. All the parameters
used in the E-RMF approach are fitted to reproduce the observables of finite
nuclei and infinite NM. The NS matter EoS is computed in the presence of DM
and hence on the NS properties to solve the Tolman-Openheimer-Volkoff (TOV)
equations and its different curvatures inside on it.

###  2.1 Construction of EoS using RMF approach

The RMF Lagrangian is built from the interaction of mesons-nucleons and their
self ( $\sigma^{2}$, $\sigma^{3}$, $\sigma^{4}$, $\omega^{2}$, $\omega^{4}$,
$\rho^{2}$ and $\rho^{4}$ ) and cross-couplings ($\sigma^{2}-\omega^{2}$,
$\omega^{2}-\rho^{2}$, $\sigma-\omega^{2}$ and $\sigma-\rho^{2}$) up to fourth
order. The RMF Lagrangian is discussed in these Refs. [36, 37, 38, 39, 40, 41,
32, 33]. The RMF Lagrangian for NM system is

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

where $M_{nucl}$ (= 939 MeV) is the mass of the nucleon. $m_{\sigma}$,
$m_{\omega}$, $m_{\rho}$ and $m_{\delta}$ are the masses and $g_{\sigma}$,
$g_{\omega}$, $g_{\rho}$ and $g_{\delta}$ are the coupling constants for the
$\sigma$, $\omega$, $\rho$ and $\delta$ mesons respectively. $\kappa_{3}$ (or
$\kappa_{4}$) and $\zeta_{0}$ are the self-interacting coupling constants of
the $\sigma$ and $\omega$ mesons respectively. $\eta_{1}$, $\eta_{2}$,
$\eta_{\rho}$ and $\Lambda_{\omega}$ are the coupling constants of non-linear
cross-coupled terms. The quantities $W^{\mu\nu}$ and $\vec{R}^{\mu\nu}$ being
the field strength tensors for the $\omega$ and $\rho$ mesons respectively,
defined as $W^{\mu\nu}$ =
$\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$ and $\vec{R}^{\mu\nu}$
= $\partial^{\mu}\vec{\rho}^{\,\nu}-\partial^{\nu}\vec{\rho}^{\,\mu}$. The
$\vec{\tau}$ are the Pauli matrices and behave as the isospin operator.
Parameters and saturation properties for NL3 [31], G3 [32] and IOPB-I [33]
along with the empirical/experimental values are given in Table 1.

The meson fields for the NM system are calculated by solving the mean-field
equation of motions [32, 33, 23] in a self-consistent way. The energy density
(${\cal{E}}_{nucl.}$) and pressure ( $P_{nucl.}$) are calculated using the
energy-momentum stress-tensor technique which is given by [42, 43]

$\displaystyle{\cal{E}}_{nucl.}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{(2\pi)^{3}}\sum_{i=p,n}\int_{0}^{k_{i}}d^{3}kE_{i}^{\star}(k_{i})+\rho_{b}W+\frac{1}{2}\rho_{3}R+\frac{m_{s}^{2}\Phi^{2}}{g_{s}^{2}}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi}{M_{nucl.}}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}$
(2.2)
$\displaystyle-\frac{1}{4!}\frac{\zeta_{0}W^{4}}{g_{\omega}^{2}}-\frac{1}{2}m_{\omega}^{2}\frac{W^{2}}{g_{\omega}^{2}}\Bigg{(}1+\eta_{1}\frac{\Phi}{M_{nucl.}}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}-\Lambda_{\omega}(R^{2}\times
W^{2})-\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}\Phi}{M_{nucl.}}\Bigg{)}$
$\displaystyle\times\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}+\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}D^{2},$
$\displaystyle P_{nucl.}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{3(2\pi)^{3}}\sum_{i=p,n}\int_{0}^{k_{i}}d^{3}k\frac{k^{2}}{E_{i}^{\star}(k_{i})}+\frac{1}{4!}\frac{\zeta_{0}W^{4}}{g_{\omega}^{2}}-\frac{m_{s}^{2}\Phi^{2}}{g_{s}^{2}}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi}{M_{nucl.}}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\frac{W^{2}}{g_{\omega}^{2}}\Bigg{(}1+\eta_{1}\frac{\Phi}{M_{nucl.}}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M_{nucl.}^{2}}\Bigg{)}+\Lambda_{\omega}(R^{2}\times
W^{2})+\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}\Phi}{M_{nucl.}}\Bigg{)}\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}$
$\displaystyle-\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}D^{2}.$

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
$E_{i}^{\star}(k_{i})$=$\sqrt{k_{i}^{2}+{M_{i}^{\star}}^{2}}$, where
$M_{i}^{\star}$ is the effective mass and $k_{i}$ is the momentum of the
nucleon and $\gamma$ is the spin degeneracy factor which is equal to 2 for
individual nucleons.

Table 1: The parameter sets NL3 [31], G3 [32] and IOPB-I [33] are listed. All the coupling constants are dimensionless. The NM parameters are given in the lower panel including with empirical/experimental values at the saturation density. The references are $[a]$,$[b]$, $[c]$ $\&$ $[d]$ [44], $[e]\&[f]$ [45], $[g]$ [46], $[h]\&[i]$ [47], and $[j]$ [48]. Parameter | NL3 | G3 | IOPB-I | Empirical/Expt. Value
---|---|---|---|---
$m_{\sigma}/M_{nucl.}$ | 0.541 | 0.559 | 0.533 | 0.426 – 0.745 $[a]$
$m_{\omega}/M_{nucl.}$ | 0.833 | 0.832 | 0.833 | 0.833 – 0.834 $[b]$
$m_{\rho}/M_{nucl.}$ | 0.812 | 0.820 | 0.812 | 0.825 – 0.826 $[c]$
$m_{\delta}/M_{nucl.}$ | 0.0 | 1.043 | 0.0 | 1.022 – 1.064 $[d]$
$g_{\sigma}/4\pi$ | 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 |
$\rho_{0}$ ($fm^{-3}$) | 0.148 | 0.148 | 0.149 | 0.148 – 0.185 $[e]$
$BE(MeV)$ | -16.29 | -16.02 | -16.10 | -15.00 – 17.00 $[f]$
$K(MeV)$ | 271.38 | 243.96 | 222.65 | 220 – 260 $[g]$
$J(MeV)$ | 37.43 | 31.84 | 33.30 | 30.20 – 33.70 $[h]$
$L(MeV)$ | 118.65 | 49.31 | 63.58 | 35.00 – 70.00 $[i]$
$K_{sym}(MeV)$ | 101.34 | -106.07 | -37.09 | -174 – -31 $[j]$
$Q_{sym}$ (MeV) | 177.90 | 915.47 | 862.70 | ———–

Inside the NS, many particles like hyperons, nucleons and leptons are present.
The neutron decays to proton, electron and anti-neutrino inside the NS [43].
This process is called as $\beta$–decay. To maintain the charge neutrality
condition, the inverse $\beta$–decay process occurred. The process can be
expressed as

$\displaystyle n\rightarrow p+e^{-}+\bar{\nu},$ $\displaystyle
p+e^{-}\rightarrow n+\nu.$ (2.4)

To maintain the stability of NSs, there must have both $\beta$-equilibrium and
charge-neutrality conditions, which are expressed in term of chemical
potential

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e},$
$\displaystyle\mu_{e}$ $\displaystyle=$ $\displaystyle\mu_{\mu},$
$\displaystyle\rho_{p}$ $\displaystyle=$ $\displaystyle\rho_{e}+\rho_{\mu}.$
(2.5)

Where $\mu_{n}$, $\mu_{p}$, $\mu_{e}$, and $\mu_{\mu}$ are the chemical
potentials of neutrons, protons, electrons, and muons, respectively. When the
chemical potential of electron is equal to the muon rest mass, then muon
appear inside the NS. The chemical potentials $\mu_{n}$, $\mu_{p}$, $\mu_{e}$,
and $\mu_{\mu}$ are given by [25]

$\mu_{n,p}=g_{\omega}\omega_{0}\pm
g_{\rho}\rho_{0}+\sqrt{k_{n,p}^{2}+(M_{n,p}^{\star})^{2}},$ (2.6)
$\mu_{e,\mu}=\sqrt{k_{e,\mu}^{2}+m_{e,\mu}^{2}},$ (2.7)

where $M_{n}^{\star}$ and $M_{p}^{\star}$ are the effective masses of neutron
and proton respectively. The particle fraction inside the NS is calculated by
solving Eqs. (2.5) using the Eqs. (2.6 – 2.7) for a given baryon density by
self-consistently. The energy density and pressure of NS are given by,

$\displaystyle{\cal{E}}_{NS}$ $\displaystyle=$
$\displaystyle{\cal{E}}_{nucl.}+{\cal{E}}_{l},$ $\displaystyle and\hskip
28.45274ptP_{NS}$ $\displaystyle=$ $\displaystyle P_{nucl.}+P_{l},$ (2.8)

with,

${\cal{E}}_{l}=\sum_{l=e,\mu}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{l}}d^{3}k\sqrt{k^{2}+m_{l}^{2}},$
(2.9)

and

$P_{l}=\sum_{l=e,\mu}\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{l}}\frac{d^{3}k\hskip
5.69046ptk^{2}}{\sqrt{k^{2}+m_{l}^{2}}}.$ (2.10)

Where ${\cal{E}}_{l}$, $P_{l}$ and $k_{l}$ are the energy density, pressure
and Fermi momentum for leptons respectively. The Eq. (2.8) gives the total
energy and pressure of the NS.

###  2.2 Interaction of DM candidates in NS

DM particles accreted inside the NS core due to its high gravitational field
[49, 11, 50, 23] and the amount of accreted DM depends directly on its
evolving life time. In this scenario, we consider the Neutralino [51, 20, 23,
25] as a fermionic DM candidate which interacts with nucleon via SM Higgs. The
detailed formalism has been taken from our previous analysis [25] and the
total Lagrangian is written as:

$\displaystyle{\cal{L}}_{tot.}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{NS}+\bar{\chi}\left[i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}M_{h}^{2}h^{2}+f\frac{M_{nucl.}}{v}\bar{\varphi}h\varphi,$
(2.11)

where ${\cal{L}}_{NS}$ is the NS Lagrangian and $\varphi$ and $\chi$ are the
nucleonic and DM wave functions respectively. $h$ is the Higgs field. The
values of the parameters like $y(=0.07)$, $f(=0.35)$ and $v(=246$ GeV) are
given in [25]. From the Lagrangian (Eq. (2.11), we get the total energy
density (${\cal{E}}_{tot.}$) and pressure ($P_{tot.}$) for NS with DM given as
[25]

$\displaystyle{\cal{E}}_{tot.}$ $\displaystyle=$
$\displaystyle{\cal{E}}_{NS}+\frac{2}{(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}d^{3}k\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2},$
(2.12)

and

$\displaystyle P_{tot.}$ $\displaystyle=$ $\displaystyle
P_{NS}+\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{f}^{DM}}\frac{d^{3}k\hskip
2.84526ptk^{2}}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2},$
(2.13)

where $k_{f}^{DM}$ is the DM Fermi momentum. The last term of Eqs. (2.12) and
(2.13) represents the potential of the Higgs field, where $M_{h}$ is the mass
of the Higgs which is 125 GeV and $h_{0}$ is the Higgs field calculated by
applying the mean-field approximation ( see Eq. (8c) of [23]). We find that
the Higgs field contribution to both energy density and pressure are very
small ($10^{-12}$ MeV). Since the interaction between nucleon and Higgs is
very small, the vacuum expectation value $v$ (= 246 GeV) don’t change inside
the neutron star. The $M_{n,p}^{\star}$ and $M_{\chi}^{\star}$ are the
effective masses of nucleon and DM, which are given as

$\displaystyle M_{n,p}^{\star}$ $\displaystyle=$ $\displaystyle
M_{nucl.}+g_{\sigma}\sigma_{0}\mp
g_{\delta}\delta_{0}-\frac{fM_{nucl.}}{v}h_{0},$ $\displaystyle
M_{\chi}^{\star}$ $\displaystyle=$ $\displaystyle M_{\chi}-yh_{0},$ (2.14)

where the $\sigma_{0}$ and $\delta_{0}$ are the mean-field for $\sigma$ and
$\delta$ mesons respectively.

###  2.3 Experimental evidences

In the present formalism, two coupling constants has major significance (i)
DM-Higgs coupling ($y$), and (ii) nucleon-Higgs form factor ($f$). The detail
prescription given as follows
(i) The direct detection experiment don’t show any collision events till now
but they gave an upper bounds on the WIMP-nucleon scattering cross-section
which is function of the DM mass. The WIMP undergoes elastic collision with
the detector nucleus ( in quark level) by the Higgs exchange. Therefore the
interaction Lagrangian which contains both DM wave function ($\chi$) and quark
wave function ($q$) can be written as [13]

${\cal{L}}_{int}=\alpha_{q}\bar{\chi}\chi\bar{q}q,$ (2.15)

where $\alpha_{q}=\frac{yfm_{q}}{vM_{h}^{2}}$. $q$ is the valence quark, $f$
is nucleon-Higgs form factor and $m_{q}$ is the mass of the quark. In this
calculations, the values of $y$ and $f$ are taken as 0.07 and 0.35. The spin
independent cross-section for the fermionic dark matter can be written as [13]

$\sigma_{SI}=\frac{y^{2}f^{2}M_{n}^{2}}{4\pi}\frac{\mu_{r}}{v^{2}M_{h}^{2}},$
(2.16)

where $M_{n}$ (= 939 MeV) is the nucleon mass and $\mu_{r}$ is the reduced
mass $\frac{M_{n}M_{\chi}}{M_{n}+M_{\chi}}$, $M_{\chi}$ is the mass of the DM
particle. We calculated the $\sigma_{SI}$ for three different masses of dark
matter 50, 100, and 200 GeV and their corresponding cross-section found to be
9.43, 9.60 and 9.70 of the order of ($10^{-46}$ cm2) respectively. That means
the predicted values are consistent with the direct detection experiment like
XENON-1T [52], PandaX-II [53] and LUX [54] with 90% confidence level. In case
of LHC, which produced various WIMP-nucleon cross section limit in the range
from $10^{-40}$ to $10^{−50}$ cm2 depending on the dark matter production
models [53]. Thus our model also satisfies the LHC limit. Therefore in the
present calculations, we constrained the value of $y$ from both the direct
detection experiments and the LHC results.
(ii) Nucleon-Higgs form factor ($f$) had been calculated in Ref. [55] using
the implication of both lattice QCD [56] and MILC results [57] whose value is
$0.33_{-0.07}^{+0.30}$ [58]. The taken value of $f$ (= 0.35 ) in this
calculation lies in the region.

###  2.4 Mass and Radius of the NS

Here we calculate the NS observables like $M$ and $R$ etc. using TOV
equations. Hence, we take the EoSs of NS with DM and input to the TOV
equations [59, 60] are given as

$\displaystyle\frac{dP_{tot.}(r)}{dr}=-\frac{(P_{tot.}(r)+{\cal{E}}_{tot.}(r))(m(r)+4\pi
r^{3}P_{tot.}(r))}{r(r-2m(r))},$ (2.17)

and

$\displaystyle\frac{dm(r)}{dr}=4\pi r^{2}{\cal{E}}_{tot.}(r),$ (2.18)

where ${\cal{E}}_{tot.}(r)$ and $P_{tot.}(r)$ are the total energy and
pressure density as a function of radial distance. $m(r)$ is the gravitational
mass, and $r$ is the radial parameter. These two coupled equations are solved
to get the mass and radius of the NS at certain central density.

###  2.5 Mathematical formulation for different curvatures

We adopt the mathematical form of different curvature quantities from [30],
which measure the curvatures for both inside and outside the star. The
curvatures are Ricci scalar, Ricci tensor, Riemann tensor and Weyl tensor,
which are formulated as

The Ricci scalar

${\cal R}(r)=8\pi\bigg{[}{\cal{E}}_{tot.}(r)-3P_{tot.}(r)\bigg{]},$ (2.19)

the full contraction of the Ricci tensor

${\cal J}(r)\equiv\sqrt{{\cal R}_{\mu\nu}{\cal
R}^{\mu\nu}}=\bigg{[}(8\pi)^{2}\left[{\cal{E}}_{tot.}^{2}(r)+3P_{tot.}^{2}(r)\right]\bigg{]}^{1/2},$
(2.20)

the Kretschmann scalar (full contraction of the Riemann tensor)

$\displaystyle{\cal{K}}(r)$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\cal{R}}^{\mu\nu\rho\sigma}{\cal{R}}_{\mu\nu\rho\sigma}}$
(2.21)
$\displaystyle=\bigg{[}(8\pi)^{2}[3{\cal{E}}_{tot.}^{2}(r)+3P_{tot.}^{2}(r)+2P_{tot.}(r){\cal{E}}_{tot.}(r)]-\frac{128{\cal{E}}_{tot.}(r)m(r)}{r^{3}}+\frac{48m^{2}(r)}{r^{6}}\bigg{]}^{1/2},$

and the full contraction of the Weyl tensor

${\cal W}(r)\equiv\sqrt{{\cal C}^{\mu\nu\rho\sigma}{\cal
C}_{\mu\nu\rho\sigma}}=\bigg{[}\frac{4}{3}\left(\frac{6m(r)}{r^{3}}-8\pi{\cal{E}}_{tot.}(r)\right)^{2}\bigg{]}^{1/2}.$
(2.22)

Where ${\cal{E}}_{tot.}$, $P_{tot.}$, $m(r)$ and $r$ are the energy density,
pressure, mass and radius of the NS respectively. At the surface $m\rightarrow
M$ due to $r\rightarrow R$. The Ricci tensor and Ricci scalar vanish outside
the star because they depends on the ${\cal{E}}_{tot.}(r)$, $P_{tot.}(r)$
which are zero outside the star. But, there is a non-vanishing component of
the Riemann tensor which does not vanish; $\tensor{{\cal
R}}{{}^{1}_{010}}=-\frac{2M}{R^{3}}=-\xi$, even in the outside of the star
[30, 27]. So the Riemann tensor is the more relevant quantity to measure the
curvature of the stars than others. Kretschmann scalar is the square root of
the full contraction of the Riemann tensor. The vacuum value for both
$\cal{K}$ and $\cal{W}$ is $\frac{4\sqrt{3M}}{R^{3}}$ as easily can see Eqs.
(2.21) and (2.22). There-fore, one can take $\cal K$ and $\cal W$ as two
reasonable measures the curvature within the star.

##  3 Results and Discussion

###  3.1 Choice of parameter sets

In the present work, three sets of parameter are chosen, namely NL3 [31],
IOPB-I [33] and G3 [32]. The NL3 parameter set corresponds to the standard RMF
model, which contains non-linear interactions (self-interaction of the sigma
mesons). It has relatively high incompressibility $K$ in comparison to the
other two parameter sets (see Table 1). The NM properties and their
empirical/experimental values of all the three-parameter sets are given in
Table 1. In case of IOPB-I, there are two extra coupling parameters,
$\Lambda_{\omega}$ and $\zeta_{0}$ on top of the NL3 set. These two coupling
parameters play a vital role in both finite and infinite NM system [61, 62,
33]. The parameter $\Lambda_{\omega}$ controls the symmetry energy (or neutron
skin thickness of finite nuclei) as well as the maximum mass of the NS [63].
$\zeta_{0}$, which is the coupling constants for the self-interaction of the
vector-meson, affects the EoS at higher density [64, 65]. So it is imperative
to include IOPB-I parameter set for the study of the NS physics. G3
interaction holds all the coupling parameters present in the Lagrangian
discussed in Sec. 2.1. One can note that G3 parameter has six extra couplings
constants ($\eta_{1}$, $\eta_{2}$, $\eta_{\rho}$, $g_{\delta}$,
$\Lambda_{\omega}$ and $\zeta_{0}$) in comparison to the NL3 parameter set.
All the couplings constants of the Lagrangian in Eq. (2.1) are obtained by
fitting the several properties of finite nuclei and infinite NM at saturation
density. The NL3 being the stiffest EoS enrich with the higher NS mass (2.774
$M_{\odot}$) in comparison to other parameter sets. The maximum mass,
canonical radius and tidal deformability are calculated by using IOPB-I
parameter set are in excellent agreement with the data suggested by GW170817
observational event [3, 4]. We give the results with these parameter sets, for
the comparative study and better understanding of the parametric dependency of
the DM effects on the curvatures of NS.

###  3.2 Equation of State

EoS is the most vital equipment to understand the properties of NS. We use EoS
emerged from different parameter sets as mentioned above, NL3 [31], G3 [32]
and IOPB-I [33] to explore the curvature of NS. The calculated EoS in Eqs.
(2.8) is the only for the core part of the NS and it is different for NL3, G3
and IOPB-I parameter sets. However, for the crust part (both inner and outer
crust), we adopt the Sharma et. al. EoS [66] which is added to form unified
EoS for the whole density range. The entire EoS is depicted in Fig. 1.
Moreover, we have found in our recent work [25] that the EoS becomes softer
with the addition of DM and reduces the $M$ and $R$ of the NS. The maximum
mass, radii and its corresponding central densities are given in Table 2 for
different DM momenta.

Figure 1:  (colour online) The EoS of the NS is shown for different parameter
sets for core part ($<5\rho_{0}$), where $\rho_{0}$ is the NM saturation
density. The red and green shaded line represents the inner crust ( 3$\times
10^{-4}$ – 8$\times 10^{-2}$ fm-3) and outer crust (6$\times 10^{-12}$ –
2.61$\times 10^{-4}$ fm-3) of Ref. [66] respectively.

###  3.3 Contribution to curvatures by various parameters

We calculate various curvatures like $\cal{K},\cal{J},\cal{R}$ and $\cal{W}$
of the NS in the presence of DM. The curvature of the NS with the variation of
baryon density with different DM momentum for all the assumed parameter sets
are shown in Fig. 2. At the low density region (near to the surface), the
curvatures $\cal{J}$ and $\cal{R}$ almost vanish due to their zero vacuum
value (see Eq. 2.19 and 2.20) but the curvatures $\cal{K}$ and $\cal{W}$
approach each other at the local maximum $\frac{4\sqrt{3}M}{R^{3}}$. In the
high dense portion the $\cal K$ and $\cal J$ procure larger curvature than
others. All the curvatures increase with the $k_{f}^{DM}$ as shown in Fig. 2.
This is due to the fact that the EoS becomes more softer with $k_{f}^{DM}$,
which gives more curvature as compared to stiffer EoS. G3 gives large
curvature as compare to IOPB-I and NL3 due to its soft EoS.

Figure 2: (colour online) The variation of different curvatures $\cal{K}$
(red), $\cal{J}$ (green), $\cal{R}$ (blue) and $\cal{W}$ (magenta) with baryon
density for NL3 (left), G3 (middle) and IOPB-I (right) in the presence of DM
for corresponding maximum mass.

The radial variation of the curvatures with the addition of DM is shown in Fig
3. All the curvatures are maximum at the centre of the star except Weyl
tensor. However, the Ricci scalar is negative within the star (for maximum
mass), as shown in Fig. 3. At the surface of the star, ${\cal{E}}_{tot.}=0$
and $P_{tot.}=0$, so $\cal{K}$ and $\cal{W}$ are equal (see Eqs. (2.21) and
(2.22)). Near the surface of the NS, the $\cal{J}$ and $\cal{R}$ approach to
zero. If we assume that the NS has uniform density, i.e. $m=\frac{4}{3}\pi
r^{3}\rho$, then the Eq. (2.22) is equal to zero. Therefore, the $\cal{W}$
tends to zero at the core. As we go from outer crust to the surface, the
density in this region is like diffuse state, so that $\cal{W}$ is maximum at
the surface. Thus, it can be concluded that $\cal K$ and $\cal W$ attain
different values within the star and approach each other at the crust and give
identical value in vacuum as shown in Fig. 3. The radial variation of
curvatures follow the same trends in presence of the DM, but the magnitude of
the curvature is more.

Figure 3: (colour online) The radial variation of all the curvatures $\cal{K}$
(red), $\cal{J}$ (green), $\cal{R}$ (blue) and $\cal{W}$ (magenta) for NL3
(left), G3 (middle) and IOPB-I (right) in the presence of DM. The yellow, grey
and cyan colour regions represents the core, the crust and the vacuum of the
NS respectively. The curvatures $\cal R$ become negative for maximum mass star
without DM. This negative value is about 6.5 km, 2 km and 1.2 km for NL3, G3
and IOPB-I respectively. This is due to the limitations of EoSs obtained from
the assumed parameter sets which are not fully compatible with special
relativity. The EoS violates ${\cal{E}}\geq 3P$ in higher densities limit
[30].

Here, we calculate the ${\cal{K}}(r)$ within the NS. To see the parametric
dependence of the curvature with radius, we fix the DM momentum at 0.04 GeV as
shown in Fig. 4 for different masses of the star. If one see carefully, the
change of ${\cal{K}}(r)$ is not significant up to the canonical mass as
compare to the maximum mass of the star. We calculate the change of
${\cal{K}}(r)$ with and without DM is $\approx$ 33 %, and the change increases
for maximum star mass. Hence, we inferred that the DM affects all the
curvature parameters $\cal{K}$, $\cal{J}$, $\cal{R}$ and $\cal{W}$ of the NS
considerably.

Figure 4: (colour online) The radial variation of ${\cal{K}}(r)$ without and
with DM having momentum 0.04 GeV. The corresponding maximum mass is shown bold
line (without DM) and dashed-line (with DM). Table 2: The central density
${\cal{E}}_{c}$, mass $M$, radius $R$, Surface curvature ${\cal{K}}(R)$,
binding energy $B/M$ of the NS are tabulated with the variation of
$k_{f}^{DM}$ both for canonical (1.4 $M_{\odot}$) and maximum mass star for
NL3, G3 and IOPB-I parameter sets.

$k_{f}^{DM}$ (GeV) |  Star type |  |  ${\cal{E}}_{c}$
---
(MeV fm-3)
|  $M$
---
($M_{\odot}$)
|  $R$
---
(km)
|  ${\cal{K}}(R)$
---
($10^{14}\cal{K}_{\odot}$)
$B/M$
NL3 | G3 | IOPB-I | NL3 | G3 | IOPB-I | NL3 | G3 | IOPB-I | NL3 | G3 | IOPB-I | NL3 | G3 | IOPB-I
0.00 |  Cano. | 270 | 460 | 366 | 1.400 | 1.400 | 1.400 | 14.08 | 12.11 | 12.78 | 1.477 | 2.320 | 1.977 | -0.084 | -0.098 | -0.092
Max. | 870 | 1340 | 1100 | 2.774 | 1.997 | 2.149 | 13.16 | 10.78 | 11.76 | 3.584 | 4.695 | 3.894 | -0.207 | -0.162 | -0.165
0.02 |  Cano. | 286 | 482 | 385 | 1.400 | 1.400 | 1.400 | 13.63 | 11.77 | 12.42 | 1.626 | 2.534 | 2.153 | -0.038 | -0.066 | -0.057
Max. | 890 | 1400 | 1120 | 2.734 | 1.974 | 2.118 | 12.91 | 10.55 | 11.54 | 3.741 | 4.957 | 4.061 | -0.178 | -0.141 | -0.139
0.03 |  Cano. | 320 | 530 | 430 | 1.400 | 1.400 | 1.400 | 12.78 | 11.09 | 11.75 | 1.976 | 3.034 | 2.546 | 0.045 | -0.006 | 0.016
Max. | 940 | 1480 | 1190 | 2.646 | 1.923 | 2.050 | 12.39 | 10.13 | 11.06 | 4.097 | 5.468 | 4.492 | -0.116 | -0.096 | -0.009
0.04 |  Cano. | 383 | 640 | 518 | 1.400 | 1.400 | 1.400 | 11.60 | 10.27 | 10.76 | 2.638 | 3.839 | 3.317 | 0.016 | 0.076 | 0.105
Max. | 1100 | 1600 | 1390 | 2.502 | 1.837 | 1.937 | 11.46 | 9.53 | 10.23 | 4.093 | 6.274 | 5.341 | -0.023 | -0.026 | -0.002

Figure 5: (colour online) The ratio of the surface curvature of NS and the Sun
with the variation of NS mass with DM for NL3, G3 and IOPB-I.

The curvature at the surface of the NS, which is more prominent to quantify
the space-time wrap in the Universe. The variation of
${\cal{K}}(R)$/$\cal{K}_{\odot}$ with the mass of the NS is shown in Fig. 5.
In our calculations, we find that the surface curvature of NS w.r.t the Sun
for NL3, G3 and IOPB-I parameter sets are 3.584, 4.695 and 3.894 respectively
without including the DM and the values are 4.093, 6.478 and 5.341 with DM
momentum (0.04 GeV). The surface curvature of the Sun ($\cal{K}_{\odot}$) is
$3.06\times 10^{-27}$ cm-2 [30]. The comparison of our results with the Sun
gives the ${\cal{K}}(R)$/$\cal{K}_{\odot}$ $\approx$ 1014. The ratio
${\cal{K}}(R)$/$\cal{K}_{\odot}$ increases with the inclusion of DM. G3
parameter set provides softer EoS than IOPB-I, which indicates that the softer
EoS facilitate us with larger surface curvature. The numerical values for both
the canonical and maximum mass star are given in Table 2. If the DM density is
very high inside the NS, then the EoS becomes softer, which affects the
curvatures significantly at the surface. More curvature means space-time is
more curved and it depends on the amount of DM percentage inside the NS.

From the above discussions, we led to conclude that the prediction of the
strength of gravity within the NS is $\approx$ $10^{15}$ times more than the
Sun. It increases few times with the addition of DM. In our calculations, the
curvature $\cal K$ within the star decreases a few times towards the crust of
the star and that order is $\approx$ 15 times than ${\cal{K}}_{\odot}$ without
DM (shown in Fig. 3). The $\cal K$ increases with increase of DM density. The
value of $\cal W$ is zero at the core and almost radially increases towards
the crust. On the other hand, $\cal K$ is maximum at the core which comes
mainly from the unconstrained parts of the NS. But the $\cal W$ has maximum
value almost at the crust where GR play significant role. That means GR breaks
down in the strong-field gravity while it retains its nobleness at the
surface. From this analysis, one can say that GR is not well tested on the
whole part of the NS than the EoS.

The compactness ($\eta=\frac{GM}{rc^{2}}$) measures the degree of denseness of
a star. The NS has larger mass and smaller radius as compared to the Sun, so
its compactness is $10^{5}$ times larger than our Sun. Therefore to study the
compactness of the NS, we plot the radial variation of the compactness of the
NS in the presence of the DM, which is depicted in Fig. 6. The compactness
within the star increases with the increase of the DM momentum for different
parameter sets, which is shown in Fig. 6. It has a larger value for the
maximum mass NS as compare to the canonical star. With the increase of the DM
percentage, the EoS becomes softer, which has less compactness as compared to
the stiff EoS. The compactness is maximum at the surface of the star.

Figure 6: (colour online) The radial variation of compactness ($\eta$) for NL3
(left), G3 (middle) and IOPB-I (right) in the presence of DM.

###  3.4 Binding energy of the NS

The gravitational binding energy ($B$) is defined as the difference between
the gravitational mass ($M$) and baryonic mass ($M_{B}$) of the NS;
$B=M-M_{B}$, where $M$ is calculated as [43, 27]

$M=\int_{0}^{R}dr\ 4\pi r^{2}{\cal E}(r),$ (3.1)

and $M_{B}=Nm_{b}$, where $m_{b}$ is the mass of baryons (931.5 MeV) and $N$
is the number of baryons calculated by integrating over the whole volume in
the Schwarzchild limit as

$\displaystyle N=\int_{0}^{R}dr\ 4\pi
r^{2}\Big{[}1-\frac{2m(r)}{r}\Big{]}^{-1/2}.$ (3.2)

The $N$ is found to be $\approx 10^{57}$ same as given in the Ref. [43]. In
our work binding energy $B$ originally corresponds to $B/M$, which is more
convenient for comparison purpose. The binding energy per particle of the
symmetric NM is $\approx$ -16 MeV, i.e. it needs 16 MeV to make the system
unbound. For pure neutron matter (PNM) system, it is positive [37]. That means
the PNM system is already unstable. It is well acknowledged that the nuclear
force is state dependent and the nucleon-nucleon interaction is divided into
three categories- singlet-singlet, triplet-triplet and singlet-triplet. The
singlet-singlet and the triplet-triplet interaction are repulsive in nature
while the singlet-triplet interaction is attractive [67, 68, 69]. Due to the
excess number of neutrons, the repulsive part adds instability to NS. However,
its enormous gravitational force balances the repulsive nuclear force. Thus
for the whole NS, the $B$ is negative.

With the addition of DM inside the NS, the $B$ step up towards positive, that
means it is going to be unstable. However, the instability of NS depends on
the DM percentage. The variation of $B/M$ with $k_{f}^{DM}$ is depicted in
Fig. 7 for assumed parameter sets. The numerical values given in Table 2. A
careful inspection of the Table 2 shows that up to 0.02 GeV the $B/M$ of the
canonical and maximum mass NS are negative. It indicates that both the
canonical and maximum mass NS are bound systems with this amount of DM.
However, if we increase the DM momentum, the canonical NS system becomes
unbound with positive $B$. For example, with DM momentum 0.04 GeV the binding
energy for the canonical star becomes positive for different parameter sets.
But still, the maximum mass NS shows a bound system with negative binding
energy. From this, we conclude that one can constraint the DM percentage
inside the NS. If the DM contained is more than the canonical star, it forms a
mini black hole at the core and destroys the NS [49, 14, 7, 70]. The cooling
of NS is also faster with the increasing of DM mass [12, 13]. That means the
positive $B$ may have a relation with the cooling properties of the NS or in
other words; it may accelerate the Urca process.

Figure 7: (colour online) The variation of $B/M$ with the M/$M_{\odot}$ of the
NS with and without DM.

##  4 Conclusion

In the present work, we study the impacts of DM on the curvatures of the NS.
The calculations are done with some well tested RMF parameters sets like NL3,
the E-RMF parameter IOPB-I (two additional couplings to RMF) and G3 parameter
(six extra couplings to RMF). The E-RMF formalism well suited to both finite
nuclei properties as well as NM in both normal and extreme conditions. The
parameter sets G3 and IOPB-I establish the recently measured the maximum mass
and the radius of the NS. The EoSs of the NS are calculated by assuming that
the DM particles inside the star. We find that the DM plays a significant role
in the NS curvatures, even in the E-RMF model, which yield a softer EoS.

We calculate various curvatures with the variations of the baryon density,
mass and radius of the NS in the presence of the DM. The curvature increases
or decreases with the baryon density. It is observed that at lower density the
quantities $\cal{K}$ and $\cal{W}$ gives more curvature than $\cal{J}$,
$\cal{R}$. At the crust region, the curvatures $\cal{J}$, and $\cal{R}$ almost
vanish. The $\cal{K}$ and $\cal{W}$ approach each other within the crust and
have a local maximum in that region. Moreover, the radial variation of
${\cal{K}}(r)$ increases with the increasing of DM momentum for maximum mass
and it has very small effects up to canonical star. The percental change of
${\cal{K}}(r)$ with and without DM is approximately 33% for 1.4 $M_{\odot}$,
and it increases for the maximum mass. From the surface curvature study, we
conclude that the softer EoS gives large curvature than the stiffer one. Both
curvature ($\cal W$) and compactness ($\eta$) approach each other at the
crust. Assuming the EoS of the crust is not well defined and hence the crust
is the best site to measure the deviation of GR in the strong-field gravity
limit, activity of pulsar glitch etc. The binding energy increases towards
positive with the increase of DM momentum which makes the NS unstable. From
this, we conclude that a tiny amount of DM can accumulate inside the NS. The
more percentage of the DM heat the NS, and it accelerates the Urca process,
which enhance the cooling of the NS, and it makes the NS unstable.

## Acknowledgement

SKB is supported by the National Natural Science Foundation of China Grant No.
11873040

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