# Impacts of dark matter on the $f$-mode oscillation of hyperon star

H. C. Das1,2<EMAIL_ADDRESS>Ankit Kumar1,2 S. K. Biswal3 S. K. Patra1,2
1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India 2Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India 3Department of Engineering Physics, DRIEMS Autonomous
Engineering College, Cuttack 754022, India

###### Abstract

We investigate the $f$-mode oscillation of the dark matter admixed hyperon
star within the relativistic Cowling approximation. The macroscopic properties
are calculated with the relativistic mean-field equation of states by assuming
that the dark matter particles are inside the star. The $f$-mode oscillation
frequencies (only for $l=2$) are calculated with four different neutron star
equation of states. We also check the effects of hyperons/dark matter and
hyperons with dark matter equation of states on the $f$-mode oscillations
varying with different astrophysical quantities such as mass ($M$), radius
($R$), compactness ($M/R$), surface red-shift ($Z_{s}$), average density
($\bar{\rho}$), dimensionless tidal deformability ($\Lambda$) of the neutron
star. Significant changes have been seen in the $f$-mode frequencies with and
without hyperons/dark matter or hyperons+dark matter. Substantial correlations
are observed between canonical frequencies and $\Lambda$
($f_{1.4}-\Lambda_{1.4}$) and maximum frequencies and canonical $\Lambda$ (
$f_{max}-\Lambda_{1.4}$).

##  I Introduction

The gravitational waves (GWs) coming from the coalescence of binary neutron
star (BNS) merger event decipher enough information that provides strong
constraints on the equation of state (EOS) and the internal composition of the
neutron star (NS) Abbott  _et al._ (2017, 2018); Bauswein  _et al._ (2017);
Annala  _et al._ (2018); Fattoyev  _et al._ (2018); Radice  _et al._ (2018);
Malik  _et al._ (2018); Most  _et al._ (2018); Tews  _et al._ (2018); Nandi
_et al._ (2019); Capano  _et al._ (2020). The detected GWs, including the
electromagnetic counterparts, open multi-messenger astronomy which helps us to
study the properties of compact objects in a different way. Till now, only two
BNS mergers were detected, which are GW170817 Abbott  _et al._ (2017) and
GW190425 Abbott  _et al._ (2020). In the future, terrestrial detectors such as
LIGO/Virgo/KAGRA may detect more BNS merger events, which can constraint the
properties of the compact stars more tightly. In addition to this, the latest
massive pulsar observation PSR J0640+6620 Cromartie  _et al._ (2020) and the
simultaneous observation of mass-radius ($M-R$) by Neutron star Interior
Composition Explorer (NICER) put a limit on the $M-R$ profile of the NS Miller
_et al._ (2019); Riley  _et al._ (2019).

Furthermore, the GWs coming from the oscillation of the compact objects is
another way to explore the stars’ micro and macroscopic properties. This is
because the oscillation frequency mainly depends on the internal structure of
the star. In Refs. Andersson and Kokkotas (1996, 1998), they claimed that one
could predict the $M$ and $R$ of the NS with the observation of such frequency
as an inverse problem. The formalism to describe the GWs from the oscillation
of the NS had been well established Lindblom and Detweiler (1983); Andersson
and Kokkotas (1996, 1998). NS oscillates with different quasi-normal modes,
mainly classified according to the restoring force that brings the system to
equilibrium positions. Such modes are $f$-mode, $p$-mode, $w$-mode, etc.,
which provide us various information about the internal structure and EOSs of
the NS. Sotani et al. Sotani  _et al._ (2011) claimed that one could find the
signature of hadron-quark phase transitions with the observation of $f$ and
$g$-modes frequencies. In Ref. Flores and Lugones (2014), they found that if a
compact object emits GWs in the range 0–1 kHz, it should be elucidated as
hybrid stars, and if the range of the frequency is more than 7 kHz, it should
be interpreted as strange stars. Similar type work by Igancio et al. Ranea-
Sandoval  _et al._ (2018), proposed that, if the frequencies of $g$-modes are
in the limit between 1–1.5 kHz, it should be interpreted as evidence of sharp
hadron-quark phase transitions. Hence, we conclude that there is still
uncertainty on different mode frequencies because the calculations are model-
dependent. Thus, for the first time, we calculate the $f$-mode frequency of
the dark matter (DM) admixed NS, including hyperons.

Different observations such as galaxy rotation curves, velocity dispersions,
galaxy clusters, gravitational lensing, cosmic microwave background, etc.,
give sufficient hints about the presence of DM in the Galaxies 111 See
wikipedia. If the DM exists, then it must be accreted inside compact objects
like NS. The accretion process may be different from one astrophysical object
to other, but the amount of accreted DM particles depend on the nature and age
of the compact objects Sandin and Ciarcelluti (2009). In this case, we take
weakly interacting massive particles (WIMPs) as a DM candidate, which are
already accreted inside the NS Panotopoulos and Lopes (2017); Das  _et al._
(2019, 2020a, 2021a, 2021b, 2021c). After accretion, the DM particles interact
with baryons which are discussed in the Sub-Sec. II.1.2. With the addition of
DM, the EOS of the NS becomes softer and $M$, $R$ and dimensionless tidal
deformability ($\Lambda$) decreases Quddus  _et al._ (2020); Das  _et al._
(2020a, 2021a, 2021b, 2021c). Different phenomena such as the cooling rate
become faster for DM admixed NS Bhat and Paul (2020). If the self-gravitating
DM particle having a mass more than the Chandrasekhar limit, it forms a mini
black hole and destroys the NS Sandin and Ciarcelluti (2009).

It is well known that hyperons appear in the NS interior at densities around
$2-3\rho_{0}$. The presence of the hyperons makes the EOS softer, and as a
consequence, the maximum mass is substantially reduced to be incompatible with
observation. In addition to DM, we also include hyperons inside the NS.
Hyperons appear at the higher density region or core of the NS. The study of
hyperons appearances inside the NS is not a brand-new idea. It is fast put
forwarded by Cameron et al. Cameron (1959). According to Ambartsumyan &
Saakyan Ambartsumyan and Saakyan (1960), the core of a massive NS consists of
an inner hyperon core and an outer nucleon shell. The detailed discussion on
hyperons productions and interactions with mesons and their coupling
parameters had been studied in Refs. Pais (1966); Millener  _et al._ (1988);
Glendenning (1992); Glendenning and Moszkowski (1991); Schaffner  _et al._
(1994); Schaffner and Mishustin (1996a); Batty  _et al._ (1997); Schaffner-
Bielich and Gal (2000); Harada and Hirabayashi (2005, 2006); Kohno  _et al._
(2006); Friedman and Gal (2007); Weissenborn  _et al._ (2012a);
*WeissenbornE_2014; Lopes and Menezes (2014); Bhuyan  _et al._ (2017); Biswal
_et al._ (2019); Biswal (2019). Although different approaches have been tried
to solve the issues such as (i) potential depths for each hyperon, (ii)
coupling between hyperons-mesons, and (iii) hyperon puzzles, these are still
open problems. This is because only some hypernuclei have been discovered in
the terrestrial laboratory Hayano  _et al._ (1989); Mareš  _et al._ (1995);
Nagae  _et al._ (1998); Saha  _et al._ (2004). More number of hypernuclei will
help us to constraint the coupling parameters by fitting the potential depth.
In this study, the hyperons–scalar mesons coupling constants are calculated by
fitting with hyperon potential depth, while for hyperons–vector mesons, SU(6)
method is used as mentioned in Refs. Lopes and Menezes (2018); Weissenborn
_et al._ (2012a, b). Another efficient approach to fix the hyperons-mesons
coupling parameters is SU(3) group theory method as given in Refs. Weissenborn
_et al._ (2012a); Miyatsu  _et al._ (2013); Lopes and Menezes (2014); Lopes
(2021). The coupling constants for both scalar and vector mesons are
constrained using SU(3) method. One can calculate nuclear matter (NM) and NS
properties by using both SU(3) and SU(6) methods and compared between these
methods as done in Refs. Weissenborn  _et al._ (2012a); Lopes and Menezes
(2014). We calculate the EOS for DM admixed hyperon star, which is the main
ingredient to find the $M$, $R$, $\Lambda$, and $f$-mode frequency.

We assume the DM particles interact with the baryon octet by the Higgs
exchange. In our previous studies Das  _et al._ (2020a, 2021a, 2021b, 2021c),
we have calculated the NS properties by considering that the DM only interacts
with nucleons via Higgs exchange. Hence, the total Lagrangian is the addition
of NS and DM. In addition to this, we include hyperons part with the NS
Lagrangian (see Sub-Sec. II.1.1). Till now, hyperons-mesons and hyperon-DM
coupling constants are uncertain. In Ref. Del Popolo  _et al._ (2020), they
have tried to solve the hyperon puzzle using DM. They have considered
different DM masses by changing the DM interaction strength parameter and
constrained the whole parameter space with recently observed NS mass. Although
they have parametrized the DM mass and interaction strength, they didn’t give
any satisfactory answers to hyperon-mesons coupling. We hope future
direct/indirect DM detection experiments such as DAMA, CDMS, GEDEON, LUX,
PANDA, and XENON may constrain the baryon-Higgs form factor. In this case, to
solve the $f$-mode frequency for DM admixed hyperon star, we take the values
of DM-Higgs coupling, and baryons-Higgs form factors are same as compared to
our previous calculations Das  _et al._ (2020a, 2021a, 2021b, 2021c).

In the present study, we calculate the $f$-mode oscillation of the DM admixed
hyperon star with different macroscopic observables by employing the
relativistic Cowling approximation. The calculations are done for the $l=2$
case (quadrupole). In the future, we may extend our calculations to find
different modes frequencies such as $p$, $g$, and $w$-modes with higher values
of $l$ in the full general relativity (GR) method as done in Refs. Lindblom
and Detweiler (1983); Andersson and Kokkotas (1996); Wen  _et al._ (2019). The
paper is organized as; The formalism for the calculations of EOSs and $M-R$
with the addition of DM+hyperons are given in Sec. II. The Sub-Sec. II.1
corresponds to relativistic mean-field (RMF) formalism. Sub-Sec. II.2 is
devoted to the $M-R$ profile, and the $f$-mode formalism is given in Sub-Sec.
II.3. In Sec. III, we consistently present our numerical results. Finally, the
summary and conclusions are enumerated in Sec. IV

##  II Formalism

###  II.1 Calculation of EOSs of the NS

####  II.1.1 Interaction between Baryons-Mesons and Mesons-Mesons

In this section, we adopt the RMF model to calculate the NS properties. In RMF
model, the nucleons interact with each other by exchanging different mesons
such as isoscalar-scalar ($\sigma$), isoscalar-vector ($\omega$), isovector-
vector ($\rho$), isovector-scalar ($\delta$) Miller and Green (1972); Serot
and Walecka (1986); Furnstahl  _et al._ (1997); Reinhard  _et al._ (1986) and
the hyperon-hyperon interaction is mediated by the two additional strange
mesons, strange scalar ($\sigma^{*}$) and strange vector ($\phi$) Schaffner
and Mishustin (1996b); Biswal  _et al._ (2019); Pradhan and Chatterjee (2021).
Hence, the Lagrangian of the system includes the interaction from mesons-
nucleons, mesons-hyperons and their self and cross-couplings. The extended RMF
(E-RMF) model include the possible significant interactions between mesons up
to the fourth-order Kumar  _et al._ (2017a, 2018); Das  _et al._ (2020a);
Kumar  _et al._ (2020); Das  _et al._ (2021a, 2020b, b, c). We also add the
leptons ($e^{-}$ and $\mu^{-}$) contribution to the E-RMF Lagrangian because
they are required to maintain both $\beta$-equilibrium and charge neutrality
conditions which provide stability to the NS. Therefore, the E-RMF Lagrangian
of the NS system is given as Biswal  _et al._ (2019); Kumar  _et al._ (2020);
Das  _et al._ (2021a, b)

$\displaystyle{\cal L}_{NS}$ $\displaystyle=$
$\displaystyle\sum_{B}\bar{\psi}_{{}_{B}}\Bigg{\\{}i\gamma^{\mu}\partial_{\mu}-m_{{}_{B}}+g_{\sigma_{B}}\sigma-
g_{\omega_{B}}\gamma_{\mu}\omega^{\mu}$ (1) $\displaystyle-$
$\displaystyle\frac{1}{2}g_{\rho_{B}}\gamma_{\mu}\vec{\tau}_{{{}_{B}}}\\!\cdot\\!\vec{\rho}^{\,\mu}+g_{\delta_{B}}\vec{\tau}_{{}_{B}}\\!\cdot\\!\vec{\delta}\Bigg{\\}}\psi_{{}_{B}}+\frac{1}{2}\partial_{\mu}\sigma\,\partial^{\mu}\sigma$
$\displaystyle-$ $\displaystyle
m_{\sigma}^{2}\sigma^{2}\Big{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma_{B}}\sigma}{m_{{}_{B}}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma_{B}}^{2}\sigma^{2}}{m_{{}_{B}}^{2}}\Bigg{)}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}\Big{(}1+\eta_{1}\frac{g_{\sigma_{B}}\sigma}{m_{{}_{B}}}+\frac{\eta_{2}}{2}\frac{g_{\sigma_{B}}^{2}\sigma^{2}}{m_{{}_{B}}^{2}}\Big{)}-\frac{1}{4}\vec{R}_{\mu\nu}\\!\cdot\\!\vec{R}^{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{\zeta_{0}}{4!}g_{\omega_{B}}^{2}(\omega_{\mu}\omega^{\mu})^{2}+\frac{1}{2}m_{\rho}^{2}(\vec{\rho}_{\mu}\\!\cdot\\!\vec{\rho}\
^{\mu})\Big{(}1+\eta_{\rho}\frac{g_{\sigma_{B}}\sigma}{m_{{}_{B}}}\Big{)}$
$\displaystyle+$
$\displaystyle\Lambda_{\omega}g_{\omega_{B}}^{2}g_{\rho_{B}}^{2}(\omega_{\mu}\omega^{\mu}\times\vec{\rho}_{\mu}\\!\cdot\\!\vec{\rho}\
^{\mu})^{2}+\frac{1}{2}\Big{(}\partial_{\mu}\vec{\delta}\,\partial^{\mu}\vec{\delta}-m_{\delta}^{2}\vec{\delta}^{\,2}\Big{)},$
$\displaystyle+$ $\displaystyle{\cal L}_{YY}+{\cal L}_{l},$

where

$\displaystyle{\cal L}_{YY}$ $\displaystyle=$
$\displaystyle\sum_{Y}\bar{\psi}_{{}_{Y}}(g_{\sigma^{*}_{Y}}\sigma^{*}-g_{\phi_{Y}}\gamma_{\mu}\phi^{\mu})\psi_{{}_{Y}}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}$
(2) $\displaystyle-$
$\displaystyle\frac{1}{4}\phi_{\mu\nu}\phi^{\mu\nu}+\frac{1}{2}(\partial_{\mu}\sigma^{*}\partial^{\mu}\sigma^{*}-m_{\sigma^{*}}^{2}{\sigma^{*}}^{2}),$
$\displaystyle\mathrm{and}$ $\displaystyle{\cal L}_{l}$ $\displaystyle=$
$\displaystyle\sum_{l}\bar{\psi}_{l}\Big{(}i\gamma^{\mu}\partial_{\mu}-m_{l}\Big{)}\psi_{l}.$
(3)

The $m_{{}_{B}}$ represent the masses of the baryons. $m_{\sigma}$,
$m_{\omega}$, $m_{\rho}$, $m_{\delta}$, $m_{\sigma^{*}}$ and $m_{\phi}$ are
the masses and $g_{\sigma_{B}}$, $g_{\omega_{B}}$, $g_{\rho_{B}}$,
$g_{\delta_{B}}$, $g_{\sigma^{*}_{B}}$ and $g_{\phi_{B}}$ are the coupling
constants for the $\sigma$, $\omega$, $\rho$, $\delta$, $\sigma^{*}$ and
$\phi$ 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 cross-coupling constants as indicated in the Lagrangian. The quantities
$F^{\mu\nu}$, $\vec{R}^{\mu\nu}$, and $\Phi^{\mu\nu}$ are the field strength
tensors for the $\omega$, $\rho$ and $\phi$ mesons respectively, defined as
$F^{\mu\nu}$ = $\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$,
$\vec{R}^{\mu\nu}$ =
$\partial^{\mu}\vec{\rho}^{\,\nu}-\partial^{\nu}\vec{\rho}^{\,\mu}$, and
$\Phi^{\mu\nu}=\partial^{\mu}\phi^{\,\nu}-\partial^{\nu}\phi^{\,\mu}$. The
$\vec{\tau}_{{}_{B}}$ is the isospin operator, which carries the isospin
component of the baryons. $m_{l}$ represent the masses of leptons. $Y$
represents different hyperons such as $\Lambda$, $\Sigma^{+,-,0}$ and
$\Xi^{+,-}$.

Table 1: The ratio of masses of different mesons with nucleon mass ($M=939$
MeV), nucleons-mesons coupling constants, self and cross-couplings between
mesons are tabulated for considered parameter sets NL3 Lalazissis  _et al._
(1997), IOPB-I Kumar  _et al._ (2018), FSUGarnet Chen and Piekarewicz (2014)
and G3 Kumar  _et al._ (2017a). All coupling constants are dimensionless.

Model | $\hskip 4.26773pt\frac{m_{\sigma}}{M}$ | $\hskip 4.26773pt\frac{m_{\omega}}{M}$ | $\hskip 4.26773pt\frac{m_{\rho}}{M}$ | $\hskip 4.26773pt\frac{m_{\delta}}{M}$ | $\hskip 4.26773pt\frac{g_{\sigma{{}_{N}}}}{4\pi}$ | $\hskip 4.26773pt\frac{g_{\omega}{{}_{N}}}{4\pi}$ | $\hskip 4.26773pt\frac{g_{\rho{{}_{N}}}}{4\pi}$ | $\hskip 4.26773pt\frac{g_{\delta{{}_{N}}}}{4\pi}$ | $\hskip 4.26773pt\kappa_{3}$ | $\hskip 4.26773pt\kappa_{4}$ | $\hskip 4.26773pt\zeta_{0}$ | $\hskip 4.26773pt\eta_{1}$ | $\hskip 4.26773pt\eta_{2}$ | $\hskip 4.26773pt\eta_{\rho}$ | $\hskip 4.26773pt\Lambda_{\omega}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
NL3 | 0.541 | 0.833 | 0.812 | 0.000 | 0.813 | 1.024 | 0.712 | 0.000 | 1.465 | -5.688 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
IOPB-I | 0.53 | 0.833 | 0.812 | 0.000 | 0.827 | 1.062 | 0.885 | 0.000 | 1.496 | -2.932 | 3.103 | 0.000 | 0.000 | 0.000 | 0.024
FSUGarnet | 0.529 | 0.833 | 0.812 | 0.000 | 0.837 | 1.091 | 1.105 | 0.000 | 1.368 | -1.397 | 4.410 | 0.000 | 0.000 | 0.000 | 0.043
G3 | 0.559 | 0.832 | 0.820 | 1.043 | 0.782 | 0.923 | 0.962 | 0.160 | 2.606 | +1.694 | 1.010 | 0.424 | 0.114 | 0.645 | 0.038

The equations of motion correspond to different mesons are calculated in the
Refs. Kumar  _et al._ (2018); Biswal  _et al._ (2019); Kumar  _et al._ (2020)
within the mean-field approximations. By applying the stress tensor technique,
the energy density and pressure of the system can be calculated as Kumar  _et
al._ (2018); Biswal  _et al._ (2019); Kumar  _et al._ (2020):

$\displaystyle{\cal E}_{NS}$ $\displaystyle=$
$\displaystyle\sum_{B}\frac{\gamma_{{}_{B}}}{2\pi^{2}}\int_{0}^{k_{F_{B}}}k^{2}\
dk\sqrt{k^{2}+m^{*2}_{{}_{B}}}+n_{{}_{B}}g_{\omega_{B}}\omega_{0}$ (4)
$\displaystyle+$
$\displaystyle\frac{n_{3B}}{2}g_{\rho{{}_{B}}}\rho_{0}-\frac{1}{3!}\zeta_{0}{g_{\omega_{B}}^{2}}\omega_{0}^{4}-\Lambda_{\omega}g_{\rho_{B}}^{2}g_{\omega_{B}}^{2}\rho_{03}^{2}\omega_{0}^{2}$
$\displaystyle+$ $\displaystyle
m_{\sigma}^{2}{\sigma_{0}}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\eta_{2}}{2}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}\Bigg{)}m_{\rho}^{2}\rho_{03}^{2}+\frac{1}{2}m_{\delta}^{2}\delta_{0}^{2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{\sigma^{*}}^{2}\sigma_{0}^{*2}+\sum_{l}\frac{\gamma_{{}_{l}}}{2\pi^{2}}\int_{0}^{k_{F{{}_{l}}}}dk\
\sqrt{k^{2}+m_{l}^{2}},$

and

$\displaystyle P_{NS}$ $\displaystyle=$
$\displaystyle\sum_{B}\frac{\gamma_{{}_{B}}}{6\pi^{2}}\int_{0}^{k_{F_{B}}}\frac{k^{4}\
dk}{\sqrt{k^{2}+m^{*2}_{{}_{B}}}}+\frac{1}{3!}\zeta_{0}{g_{\omega_{B}}^{2}}\omega_{0}^{4}$
(5) $\displaystyle-$ $\displaystyle
m_{\sigma}^{2}{\sigma_{0}}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}-\frac{1}{2}m_{\delta}^{2}\delta_{0}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\eta_{2}}{2}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}+\frac{1}{2}m_{\sigma^{*}}^{2}\sigma_{0}^{*2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}\Bigg{)}m_{\rho}^{2}\rho_{03}^{2}+\Lambda_{\omega}g_{\rho_{B}}^{2}g_{\omega_{B}}^{2}\rho_{03}^{2}\omega_{0}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}+\sum_{l}\frac{\gamma_{{}_{l}}}{6\pi^{2}}\int_{0}^{k_{F{{}_{l}}}}\frac{k^{2}dk}{\sqrt{k^{2}+m_{l}^{2}}},$

where $\gamma_{{}_{B}}$ and $\gamma_{l}$ are the spin degeneracy factor for
baryons and leptons, respectively. $k_{F_{B}}$ and $k_{F_{l}}$ are the baryons
and leptons Fermi momentum, respectively. $m^{*}_{{}_{B}}$ is the effective
masses of the baryons, which is written by

$\displaystyle m^{*}_{B}$ $\displaystyle=$ $\displaystyle
m_{B}-g_{\sigma_{B}}\sigma_{0}-g_{\delta_{B}}\tau_{B}\delta_{0}-g_{\sigma^{*}_{B}}\sigma^{*}.$
(6)

The last term in Eq. (6) doesn’t contribute to the $m^{*}_{B}$. This is
because the value of $g_{\sigma^{*}_{B}}$ and $g_{\phi_{B}}$is zero for the
nucleons.

Coupling constants for Nucleons-Mesons and Mesons-Mesons:-

To calculate the EOS for the NM and NS systems, one must know the coupling
constants. Many RMF parameter sets have been developed, including different
types of interaction between nucleons-mesons and mesons-mesons (both self and
cross). Therefore, to study how the coupling parameters affect both NM and NS
properties, we take four different types of parameter sets such as NL3
Lalazissis  _et al._ (1997), IOPB-I Kumar  _et al._ (2018), FSUGarnet Chen and
Piekarewicz (2014), and G3 Kumar  _et al._ (2017a). We tabulate the masses of
different mesons, coupling constants in Table 1 for these parameter sets. The
NM properties correspond to four parameter sets are also reported in Table 3
of the Ref. Kumar  _et al._ (2018). From Table 3 of the Ref. Kumar  _et al._
(2018), it is noticed that all three parameter sets reproduce the NM
properties very well except NL3. For example, the value of incompressibility,
$K=$ 271.38 MeV, is higher than other sets. This is because NL3 is a stiff EOS
as compared to G3, which is a softer EOS.

By spanning soft to stiff EOSs, one can have a better knowledge of the DM
impacts on the NS properties. Although the incompressibility of NL3 is quite
high as compared to other considered sets, the predicted properties of the
finite nuclei by NL3 are as good as the other sets. In case of G3, it is one
of the latest extended RMF (E-RMF) parameter set, which includes almost all
significant possible interaction terms between mesons and nucleons (either
self or cross) Kumar  _et al._ (2017a). Also, G3 predicts NS mass around $2\
M_{\odot}$, and it satisfies NICER data as well. By these forces, we calculate
the $f$-mode oscillations of the NS. The EOSs for these sets are shown in Fig.
1 to get a clear picture of their variation with energy density.

Coupling constants for Hyperons-Mesons:-

Till now, we don’t know exactly how the hyperons interact with each other as
compared to the nucleons-nucleons interaction Glendenning (2001), because of
the limitation of the hypernuclei data. Therefore, fixing the couplings
between hyperons-mesons is quite difficult. In general, the SU(6) symmetry
group is used to fix the couplings between hyperons and vector mesons, while
the couplings with scalar mesons are fixed through the hyperon potential depth
Pais (1966); Lopes and Menezes (2014); Biswal  _et al._ (2019); Biswal (2019).
The potential depth of the $\Lambda$-hyperon ($U_{\Lambda}$) is known to be
-28 MeV Millener  _et al._ (1988); Glendenning and Moszkowski (1991);
Schaffner  _et al._ (1994); Batty  _et al._ (1997); Torres  _et al._ (2017);
Lopes and Menezes (2018) but the $\Sigma$ and $\Xi$ potentials ($U_{\Sigma}$,
$U_{\Xi}$) provide a large uncertainty and not even the signs of the
potentials are well defined Dover and Gal (1984); Schaffner  _et al._ (1994).
But some hyper-nuclear experiments show that the $U_{\Sigma}$ and $U_{\Xi}$
potential depth close to +30 MeV and -18 MeV respectively Schaffner-Bielich
and Gal (2000); Harada and Hirabayashi (2005, 2006); Kohno  _et al._ (2006);
Friedman and Gal (2007).

The hyperon potential depth is defined as Glendenning (1997)

$U_{Y}=-g_{\sigma{{}_{N}}}x_{\sigma{{}_{Y}}}\sigma_{0}+g_{\omega{{}_{N}}}x_{\omega{{}_{Y}}}\omega_{0},$
(7)

where $x_{\sigma{{}_{Y}}}$ and $x_{\omega{{}_{Y}}}$ is defined as
$g_{\sigma{{}_{Y}}}/g_{\sigma{{}_{N}}}$ and
$g_{\omega{{}_{Y}}}/g_{\omega{{}_{N}}}$ respectively. For symmetric NM the
quantities $g_{\sigma{{}_{Y}}}/g_{\sigma{{}_{N}}}$ and
$g_{\omega{{}_{Y}}}/g_{\omega{{}_{N}}}$ are given in Refs. Glendenning (1997);
Biswal  _et al._ (2019)

$\displaystyle
g_{\sigma{{}_{N}}}\sigma_{0}=M-M^{*}=M\Big{(}1-\frac{M^{*}}{M}\Big{)},$
$\displaystyle
g_{\omega{{}_{N}}}\omega_{0}=\Big{(}\frac{g_{\omega{{}_{N}}}}{m_{\omega}}\Big{)}^{2}\rho_{0},$
(8)

Rewriting Eq. (7) by inserting Eq. (8), we get

$\displaystyle
U_{Y}=M\Big{(}\frac{M^{*}}{M}-1\Big{)}x_{\sigma{{}_{Y}}}+\Big{(}\frac{g_{\omega{{}_{N}}}}{m_{\omega}}\Big{)}^{2}\rho_{0}x_{\omega{{}_{Y}}}.$
(9)

In this case, we choose the values of $U_{\Lambda}=-28$ MeV, $U_{\Sigma}=+30$
MeV and $U_{\Xi}=-18$ MeV. To get the desire potential one has to choose the
values of $x_{\sigma{{}_{Y}}}$ and $x_{\omega{{}_{Y}}}$ for a fix parameter
set. For example NL3 case, we fix, $x_{\sigma{{}_{\Lambda}}}=0.8$, and the
value of $x_{\omega{{}_{\Lambda}}}$ is found to be 0.8982. Similarly for
IOPB-I set, we fix, $x_{\sigma{{}_{\Lambda}}}$ = 0.8 which gives
$x_{\omega{{}_{\Lambda}}}=0.8338$. For $\Sigma$ and $\Xi$-hyperons the values
are $x_{\sigma_{\Sigma}}=0.7$, $x_{\omega_{\Sigma}}=0.8932$,
$x_{\sigma_{\Xi}}=0.8$, and $x_{\omega_{\Xi}}=0.8639$ for IOPB-I set. Hence,
one can get different combinations to fit the potential depth. The hyperon
interactions with $\rho$-meson are fitted according to SU(6) symmetry method
Dover and Gal (1984); Schaffner  _et al._ (1994) and their couplings with
hyperons are given as: $x_{\rho{{}_{\Lambda}}}=0.0$,
$x_{\rho{{}_{\Sigma}}}=2.0$ and $x_{\rho{{}_{\Xi}}}=1.0$.

The couplings constants for hyperons and $\sigma^{*}$-meson are
$x_{\sigma^{*}\Lambda}=0.69$, $x_{\sigma^{*}\Sigma}=0.69$, and
$x_{\sigma^{*}\Xi}=1.25$. Similarly, for hyperons and $\phi$-mesons coupling
constants are $x_{\phi\Lambda}=-\frac{\sqrt{2}}{3}g_{\omega_{N}}$,
$x_{\phi\Sigma}=-\frac{\sqrt{2}}{3}g_{\omega_{N}}$, and
$x_{\phi\Xi}=-\frac{2\sqrt{2}}{3}g_{\omega_{N}}$ respectively for $\Lambda$,
$\Sigma$ and $\Xi$-hyperons Biswal  _et al._ (2019). In this calculation we
neglects the $\sigma^{*}$ and $\phi$ mesons-nucleons interaction for the
numerical simplicity.

####  II.1.2 Interaction between Baryons-Dark Matter

When NS evolves in the Universe, the DM particles are accreted inside it due
to its huge gravitational potential and immense baryonic density. In this
work, we assume that the DM particles are already inside the NS. With this
assumption, we model the interaction Lagrangian for DM and baryons. In this
case, the Neutralino is the DM particle having a mass of 200 GeV as considered
in our earlier works Das  _et al._ (2020a, 2021a).

The DM particles are interacted with baryons by exchanging standard model (SM)
Higgs. The interacting Lagrangian is given in the form Panotopoulos and Lopes
(2017); Das  _et al._ (2019); Quddus  _et al._ (2020); Das  _et al._ (2020a,
2021a, 2021b, 2021c):

$\displaystyle{\cal{L}}_{DM}$ $\displaystyle=$
$\displaystyle\bar{\chi}\Big{[}i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\Big{]}\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(10) $\displaystyle-$
$\displaystyle\frac{1}{2}M_{h}^{2}h^{2}+\sum_{B}f_{{}_{B}}\frac{m_{{}_{B}}}{v}\bar{\psi}_{{}_{B}}h\psi_{{}_{B}},$

$\psi_{{}_{B}}$ and $\chi$ are the baryons and DM wave functions respectively.
The parameters $y$ is DM-Higgs coupling, $f_{{}_{B}}$ is the baryons-Higgs
form factors and $v$ is the vacuum expectation value of Higgs field. The
values of $y$ and $v$ are 0.07 and 246 GeV respectively taken from the Refs.
Das  _et al._ (2020a, 2021a). The value of $f_{{}_{B}}(=$ 0.35) for all
baryons taken to be same, because we don’t know the form factor for hyperons
except nucleons with Higgs.

We calculate the spin-independent scattering cross-section of the nucleons
with DM using the relation Cline  _et al._ (2013)

$\displaystyle\sigma_{SI}=\frac{y^{2}f^{2}M_{nucl.}^{2}}{4\pi}\frac{\mu_{r}}{v^{2}M_{h}^{2}},$
(11)

where $\mu_{r}$ is the reduced mass. The calculated cross-sections for
$M_{\chi}=200$ GeV is found to be $9.70\times 10^{-46}$ cm2, which is well
consistent with XENON-1T Aprile  _et al._ (2016), PandaX-II Tan  _et al._
(2016), PandaX-4T Meng  _et al._ (2021) and LUX Akerib  _et al._ (2017) within
90% confidence level. The LHC had also given a limit on the WIMP-nucleon
scattering cross-section in the range from $10^{-40}$ to $10^{-50}$ cm2
Djouadi  _et al._ (2012). For hyperons cases, we find the $\sigma_{SI}$ are
1.37, 1.56, 1.91 $\times$ $10^{-45}$ cm2 for $\Lambda$, $\Sigma$ and $\Xi$
respectively. 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.

Nucleon-Higgs form factor ($f$) had been calculated in Ref. Djouadi  _et al._
(2012) using the implication of both lattice QCD Czarnecki  _et al._ (2010)
and MILC results Toussaint and Freeman (2009) whose value is
$0.33_{-0.07}^{+0.30}$ Aad  _et al._ (2015). The taken value of $f$ (= 0.35 )
in this calculation lies in the region. Thus, we also constrain the values of
$f$ with available data.

The Euler Lagrange equation of motion for DM particle ($\chi$) and Higgs boson
($h$) can be derived from Lagrangian in Eq. (10) as,

$\displaystyle\bigg{(}i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\bigg{)}\chi{}=0,$
$\displaystyle\partial_{\mu}\partial^{\mu}h+M_{h}^{2}h=y\bar{\chi}\chi+\sum_{B}\frac{f_{{}_{B}}m_{{}_{B}}}{v}\bar{\psi_{{}_{B}}}\psi_{{}_{B}},$
(12)

respectively. Applying RMF approximation, we get Das  _et al._ (2019),

$\displaystyle
h_{0}=\frac{y\langle\bar{\chi}\chi\rangle+\sum_{B}f\frac{m_{{}_{B}}}{v}\langle\bar{\psi_{{}_{B}}}\psi_{{}_{B}}\rangle}{M_{h}^{2}},$
$\displaystyle\bigg{(}i\gamma^{\mu}\partial_{\mu}-M^{\star}_{\chi}\bigg{)}\chi{}=0,$
(13)

where $M_{\chi}^{\star}$ is the dark matter effective mass can be given as,

$M_{\chi}^{\star}{}=M_{\chi}-yh_{0}.$ (14)

The dark matter scalar density ($\rho_{s}^{DM}$) is

$\rho_{s}^{DM}=\langle\bar{\chi}\chi\rangle=\frac{\gamma}{2\pi^{2}}\int_{0}^{k_{f}^{DM}}dk\
\frac{M_{\chi}^{\star}}{\sqrt{M_{\chi}^{\star}{{}^{2}}+k^{2}}},$ (15)

where $k_{f}^{DM}$ is the Fermi momentum for dark matter. $\gamma$ is the spin
degeneracy factor which has value 2 for neutron and proton individually.

Assuming the average number density of nucleons ($n_{b}$) is $10^{3}$ times
larger than the average dark matter density ($n_{DM}$), which implies the
ratio of the dark matter and the neutron star mass to be $\sim\frac{1}{6}$
Panotopoulos and Lopes (2017). The nuclear saturation density $n_{0}\sim 0.16$
fm-3, therefore, the DM number density becomes $n_{DM}\sim 10^{-3}n_{0}\sim
0.16\times 10^{-3}$ fm-3. Using the $n_{DM}$, the $k_{f}^{DM}$ is obtained
from the equation $k_{f}^{DM}=(3\pi^{2}n_{DM})^{1/3}$. Hence the value of
$k_{f}^{DM}$ is $\sim 0.033$ GeV. Therefore, in our case, we vary the DM
momenta from 0 to 0.06 GeV. For IOPB-I+DM3 222DM3 means DM Fermi momentum
$k_{f}^{DM}=$ 0.03 GeV. case, the predicted mass and radius is consistent with
Cromartie et al. Cromartie  _et al._ (2020) and NICER Miller  _et al._ (2019);
Riley  _et al._ (2019); Miller  _et al._ (2021) data respectively (see Fig.
2). Hence, we take IOPB-I EOS with DM momentum 0.03 GeV in this calculation.

The energy density (${\cal{E}}_{DM}$) and pressure ($P_{DM}$) for NS with DM
can be obtained by solving the Eq. (10)

$\displaystyle{\cal{E}}_{DM}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int_{0}^{k_{f}^{DM}}k^{2}\
dk\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2},$ (16)

and

$\displaystyle P_{DM}$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}\int_{0}^{k_{f}^{DM}}\frac{k^{4}\
dk}{\sqrt{k^{2}+(M_{\chi}^{\star})^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2},$ (17)

$M_{h}$ is the mass of the Higgs equal to 125 GeV, and $h_{0}$ is the Higgs
field calculated by applying the mean-field approximation Das  _et al._
(2019). The contribution of the Higgs field in both energy density and
pressure is minimal.

Figure 1: Left: EOSs for different RMF parameter sets. Right: EOSs are for
IOPB-I with yellow line (only nucleons), IOPB-I+DM3 with blue dashed line
(nucleons+DM3), IOPB-I+Y with purple dashed line (nucleons+hyperons),
IOPB-I+Y+DM with dashed cyan line (nucleons+hyperons+DM3).

Therefore, the total energy density (${\cal E}$) and pressure ($P$) for the DM
admixed hyperonic NS are as follow:

$\displaystyle{\cal{E}}={\cal{E}}_{NS}+{\cal{E}}_{DM},\ {\rm and}\
P=P_{NS}+P_{DM}.$ (18)

In Fig. 1, we plot P with ${\cal E}$ for four considered parameter sets. It is
clear from the figure that the NL3 is the stiffest EOS as compared to others,
while G3 is the softer EOS. On the right side of the plot, we show the EOS for
IOPB-I parameter set for nucleons (IOPB-I), nucleons with DM (IOPB-I+DM3),
nucleons with hyperons (IOPB-I+Y), and nucleons with both hyperons and DM
(IOPB-I+Y+DM3). The EOS becomes softer with the addition of DM/hyperons or
DM+hyperons, but the softness is more for DM+hyperons (dashed cyan line) than
others, which is clearly visible in the figure.

In the presence of hyperons/DM particles, the EOS of neutron star becomes
softer. This is because every system would like to minimize its energy. As the
density increases, the Fermi momenta or the Fermi energy increases. Because
the nucleons are fermions and need to place them in a higher orbit with
increasing density. Also, it is well known that the density is proportional to
the cube of the Fermi momenta. At sufficiently high density, the energy of the
nucleon increases, which is the rest mass energy plus the kinetic energy or
$E=\sqrt{k_{f}^{2}+M^{2}}$. When this energy exceeds the mass of the
hyperons/kaons/DM, the nucleons decay to these particles. In other words, it
is economical energy-wise for the system to have the hyperons/kaons/DM in the
lower energy states rather than nucleons at higher Fermi energy at higher
density, though the hyperons are heavier than the nucleons. In other words,
the nucleons are replaced by hyperons/kaons/DM depending on the system’s
density. As a result, a fraction of the gravitational mass is converted to
kinetic energy and decreases the gravitational mass. With these EOSs, we
calculate the $M-R$ profiles and the $f$-mode frequencies of the NS.

###  II.2 NS Macroscopic Properties

The metric for a spherically symmetric and non-rotating relativistic NS is
expressed as

$ds^{2}=-e^{2\Phi(r)}dt^{2}+e^{2\Lambda(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2},$
(19)

where $\Phi(r)$ and $\Lambda(r)$ are the metric functions. The spherical
symmetric solutions are given by Tolman-Oppenheimer-Volkoff (TOV) coupled
equations Tolman (1939); Oppenheimer and Volkoff (1939) as follows:

$\displaystyle\frac{dm(r)}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}{\cal E}(r),$ $\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle-\Big{[}P(r)+{\cal E}(r)\Big{]}\frac{d\Phi}{dr},$
$\displaystyle\frac{d\Phi(r)}{dr}$ $\displaystyle=$
$\displaystyle\frac{m(r)+4\pi r^{3}P(r)}{r[r-2m(r)]},$ (20)

where $m(r)$ represents the enclosing mass at some radius $r$. The hydrostatic
equations are solved for a given central density which gives mass and radius
of the NS.

Figure 2: Left: $M$–$R$ relations for four different RMF parameter sets.
Right: $M$–$R$ relations for IOPB-I, IOPB-I+DM3, IOPB-I+Y and IOPB-I+Y+DM3 .
The red and blue band represents the masses of the massive pulsars observed by
Cromartie et al. Cromartie  _et al._ (2020) and Antoniadis et al. Antoniadis
_et al._ (2013). The old NICER results are shown with two green boxes from two
different analyses Miller  _et al._ (2019); Riley  _et al._ (2019). The new
NICER result is shown for canonical radius (blue horizontal line) given by
Miller et al. Miller  _et al._ (2021). Reed et al. radius range is also shown
with dark red line for canonical star Reed  _et al._ (2021).

We plot the $M-R$ relations for the EOSs on the left side of Fig. 2. We
observe that the NL3 EOS predicts the maximum mass around 2.77 $M_{\odot}$ and
radius $\sim$14 km. Since it is a stiffer EOS compared to others, its maximum
mass is more than the other three sets. We put maximum mass constraints
measured from the different pulsars such as PSR J0740+6620 Cromartie  _et al._
(2020), and PSR J0348+0432 Antoniadis  _et al._ (2013). Except for NL3, the
other three EOSs can support maximum NS mass 2.0 $M_{\odot}$, which are
compatible with the precise measured NS masses by Cromartie et al.
($2.14_{-0.09}^{+0.10}\ M_{\odot}$ with 68.3% credibility interval) Cromartie
_et al._ (2020) and Antoniadis et al. ($2.01\pm 0.04\ M_{\odot}$) Antoniadis
_et al._ (2013). The simultaneous measurement of mass and radius by the NICER
give the $M=1.44_{-0.14}^{+0.15}\ M_{\odot}$ ($M=1.34_{-0.16}^{+0.15}\
M_{\odot}$) and $R=13.02_{-1.06}^{+1.24}$ km ($R=12.71_{-1.19}^{+1.14}$ km)
respectively from the analysis of PSR J0030+0451 by Miller et al. (Riley et
al.) Miller  _et al._ (2019); Riley  _et al._ (2019). We called it as old
NICER data. We put the old NICER data with two green boxes from two different
analyses Miller  _et al._ (2019); Riley  _et al._ (2019). The predicted radius
by all the EOSs are compatible with old NICER data as shown in Fig. 2. On the
left side of Fig. 2, we noticed that IOPB-I and IOPB-I+DM3 could able to
produce the maximum mass constraints. The maximum mass and radius correspond
to IOPB-I EOS are 2.15 $M_{\odot}$ and 11.75 km respectively. The maximum
masses and radii are (2.05 $M_{\odot}$, 11.02 km), (1.70 $M_{\odot}$, 11.39
km), and (1.61 $M_{\odot}$, 10.40 km) for IOPB-I+DM3, IOPB-I+Y, and
IOPB-I+Y+DM3 respectively. These EOSs satisfy the old NICER data. Recently,
Miller et al. put another radius constraints on both for the canonical
($R_{1.4}=12.45\pm 0.65$ km) and maximum NS ($R_{2.08}=12.35\pm 0.75$ km) from
the NICER and X-ray Multi-Mirror (XMM) Newton data are termed as new NICER
data Miller  _et al._ (2021). We depicted the new NICER data as a horizontal
blue line in Fig. 2. All considered EOSs satisfy the new NICER data except NL3
and IOPB-I+Y sets.

Recently, the PREX-2 experiment has given the updated neutron skin thickness
of 208Pb as $R_{\rm skin}=0.284\pm 0.071$ fm, within an $1\%$ error Adhikari
_et al._ (2021). Based on this data, Reed et al. inferred the symmetry energy
and slope parameter as $J=38.1\pm 4.7$ MeV and $L=106\pm 37$ MeV respectively,
with the help of a limited relativistic mean-field forces (using PREX-2 data)
Reed  _et al._ (2021). These values of J and L are larger as compared to the
old PREX data (PREX-I). The latest $J$ and $L$ can be reproduced mostly from
the stiff equation of state. That means the PREX-2 result allows a stiff
equation of state. Reed et al. also predicted the radius for canonical NS as
$13.25<R_{1.4}<14.26$ km. In this case, only NL3 satisfies the Reed et al.
data as shown in Fig. 2. Recently, Miller et al. also gave new NICER
constraints both for canonical and maximum mass NS from the X-ray study of PSR
J0030+0451 Miller  _et al._ (2021). The improved radius estimate for canonical
star is $11.8<R_{1.4}<13.1$ km. This new NICER constraint allows a narrow
radius range as compared to old NICER data ($11.52<R_{1.4}<14.26$ km). From
the radii constraint, we find that the new NICER data allows a narrow radius
range contrary to a large range of PREX-2 and the old NICER data, leaving us
an inconclusive determination of the NS radius.

###  II.3 Calculation of $f$-mode oscillation of the NS

NS oscillates when it is disturbed by an external/internal event. Hence, it
emits gravitational waves with different modes of frequencies. The most
important modes are fundamental mode ($f$), first and second pressure modes
($p$), and the first gravitational mode ($g$) Andersson and Kokkotas (1996);
Ranea-Sandoval  _et al._ (2018); Pradhan and Chatterjee (2021). Almost all the
energy of the NS is emitted as GWs radiations with these modes. To study
different modes of oscillations, one has to solve perturbed fluid equations in
the vicinity of GR. Throne and Campollataro Thorne and Campolattaro (1967)
first calculated the non-radial oscillation of NS in the framework of general
relativity. Lindblom and Detweiler gave the first integrated numerical
solution of the NS Lindblom and Detweiler (1983). To solve the non-radial
oscillations, one can also use the Cowling approximation, which is simpler
than the Lindblom and Detweiler method. In Cowling approximations, the metric
perturbations are neglected Cowling (1941). The obtained frequencies within
Cowling approximations differ by 10-30% as compared to full linearized
equations of GR.

In this work, we want to calculate the non-radial oscillations of the NS using
the Cowling approximations for a DM admixed hyperon star. To find different
oscillations mode frequencies, one has to solve the following coupled
differential equations Sotani  _et al._ (2011); Flores and Lugones (2014);
Ranea-Sandoval  _et al._ (2018); Pradhan and Chatterjee (2021)

$\displaystyle\frac{dW(r)}{dr}$ $\displaystyle=$ $\displaystyle\frac{d{\cal
E}}{dP}\left[\omega^{2}r^{2}e^{\Lambda(r)-2\Phi(r)}V(r)+\frac{d\Phi(r)}{dr}W(r)\right]$
$\displaystyle-l(l+1)e^{\Lambda(r)}V(r)$ $\displaystyle\frac{dV(r)}{dr}$
$\displaystyle=$ $\displaystyle
2\frac{d\Phi(r)}{dr}V(r)-\frac{1}{r^{2}}e^{\Lambda(r)}W(r).$ (21)

The functions $V(r)$ and $W(r)$ along with frequency $\omega$, characterize
the Lagrange displacement vector ($\eta$) associate to perturbed fluid,

$\displaystyle\eta=\frac{1}{r^{2}}\Big{(}e^{-\Lambda(r)}W(r),-V(r)\partial_{\theta},-\frac{V(r)}{\sin^{2}{\theta}}\
\partial_{\phi}\Big{)}Y_{lm},$ (22)

where $Y_{lm}$ is the spherical harmonic which is function of $\theta$ and
$\phi$. Solution of Eq. (21) with the fixed background metric Eq. (19) near
origin will behave as:

$W(r)=Br^{l+1},\ V(r)=-\frac{B}{l}r^{l},$ (23)

where $B$ is an arbitrary constant. At the surface of the star, the
perturbation pressure must vanish which provides another boundary condition as
follows:

$\omega^{2}e^{\Lambda(R)-2\Phi(R)}V(R)+\frac{1}{R^{2}}\frac{d\Phi(r)}{dr}\Big{|}_{r=R}W(R)=0.$
(24)

After solving Eq. (21) with two boundary conditions (Eqs. (23) & (24)), one
can get eigenfrequencies of the NS. In this work, we solve the oscillations
equations Eq. (21) by using the shooting method with some initial guess for
$\omega^{2}$. The equations are integrated from the center to the surface and
trying to match the surface boundary conditions. After each integration, the
initial guess of $\omega^{2}$ is corrected through the Secant method to get
the desire precision which improves the initial guess.

Figure 3: Upper: $f$-mode frequencies as a function of mass for four different
RMF parameter sets. Middle: $f$-mode frequencies for IOPB-I set with
DM3/hyperons and hyperons+DM3. Lower: Same as middle one but for FSUGarnet.

##  III Results

In this Sec. we calculate the $f$-mode oscillation frequencies as the
functions of $M$, compactness ($C$), average density ($\bar{\rho}$), surface
red-shift ($Z_{s}$), and $\Lambda$ using the RMF EOSs for DM admixed hyperon
star. The EOSs correspond to nucleons, and nucleons+DM3+Y are given in Sub.
Sec. II.1. The formalism for the calculation of $f$-mode oscillation
frequencies is given in Sub. Sec. II.3.

###  III.1 Calculation of $f$-mode frequency as functions of different
observables

The $f$-mode frequencies (only for $l=2$) with different RMF EOSs are shown in
Fig. 3. In addition to this, the $f$-mode frequencies for IOPB+DM3, IOPB+Y,
and IOPB-I+Y+DM are also shown in the middle panel of the figure. To see the
parametric dependence of either DM/Y or DM+Y, we repeat the calculations with
FSUGarnet and shown in the lower panel of Fig. 3.

The $f$-mode frequency corresponds to maximum mass ($f_{max}$) for the four
EOSs are shown in the upper panel of Fig. 3. The $f_{max}$ are 2.16, 2.32,
2.38, and 2.55 kHz for NL3, IOPB-I, FSUGarnet, and G3, respectively. The
$f_{max}$ is maximum for G3 and minimum for NL3, and these are in-between for
IOPB-I, and FSUGarnet sets. This is because G3 is softer, and NL3 is the
stiffest EOS in our calculations. In the middle panel of Fig. 3, we observe
that there is a negligible change of $f$-mode frequencies up to 1.3
$M_{\odot}$ for the IOPB-I case. There is no change in the $f$-mode
frequencies up to $1\ M_{\odot}$ for FSUGarnet as shown in the lower panel of
Fig. 3. The change in the $f$-mode frequencies is seen mainly at the core
part. This is due to the presence of hyperons and DM particles at the dense
region of the NS, which generally occurs in the central region. It is well
known that the EOS is model-dependent, mostly at the core region. Also, the
appearance of hyperons/DM is not possible in the lower density region. The
change in $f-$mode frequency due to hyperon/DM particle needed a minimum mass
of the neutron star, which is seen in Ref. Flores, Cesar V.  _et al._ (2020).
Therefore, the change in the $f$-mode frequency varies with forces.

The $f$-mode frequencies differ considerably with force parametrizations. The
values of $f_{max}$ are 2.32, 2.57, 2.58, and 2.85 kHz for IOPB-I, IOPB-I+DM3,
IOPB-I+Y, and IOPB-I+Y+DM3, respectively. The EOSs are softer with the
addition of either DM3/hyperons or DM3+hyperons compared to the original
IOPB-I EOS. Therefore, the $f_{max}$ value is more for IOPB-I+Y+DM3 as
compared to others. We tabulated the value of $f_{max}$ and $f_{1.4}$ for four
different RMF forces with different NS observables in Table 3. The mass
variation of frequency changes within interval 1.75-2.55 kHz for $l=2$, which
is almost consistent with Pradhan et al. Pradhan and Chatterjee (2021) for the
considered parameter sets. For hyperons/DM3 and hyperons+DM3 cases, the
frequency range within interval 1.8-2.85 kHz. This range is a little higher as
compared with no hyperons/DM case.

Figure 4: Upper: $f$-mode frequencies as a function of compactness are shown
for four different RMF parameter sets. Lower: $f$-mode frequencies for IOPB-I
set with DM3/hyperons and hyperons+DM3.

The compactness of a star is defined as ($C=M/R$), where $M$ and $R$ are the
mass and radius of the star Glendenning (1997); Schaffner-Bielich (2020). Like
$C$, another significant quantity is surface red-shift $Z_{s}$ is defined as
Glendenning (1997)

$Z_{s}=\frac{1}{\sqrt{1-\frac{2M}{R}}}-1=\frac{1}{\sqrt{1-2C}}-1.$ (25)

If we find the value of $Z_{s}$, one can constraints the mass and radius of
the star. Till now, only one value of $Z_{s}=0.35$ is reported in the Ref.
Cottam  _et al._ (2002) from the analysis of stacked bursts in the low-mass
x-ray binary EXO 0748-676, which is also discarded by the subsequent
observation Cottam  _et al._ (2008).

We calculate $C$ and $Z_{s}$, which are shown in Figs. 4 and 5 with $f$-mode
frequencies. We observed that the variation of $f$-mode frequencies both for
$C$ and $Z_{s}$ looks almost identical. This is because the $Z_{s}$ is the
function of stellar compactness. The numerical values of $C$ and $Z_{s}$
correspond to both canonical and maximum mass NS are given in Table 3. In our
case, we find the range of $C_{max}$ and $Z_{s_{max}}$ are 0.22-0.31 and
0.33-0.62 respectively for the considered EOSs. The values of $C_{max}$ and
$Z_{s_{max}}$ decreases as compared to only nucleonic EOSs.

The average density of a star is defined as
$\bar{\rho}=\sqrt{\bar{M}/\bar{R^{3}}}$, where $\bar{M}=\frac{M}{1.4\
M_{\odot}}$ and $\bar{R}=\frac{R}{10\ \mathrm{km}}$. We plot the $f$-mode
frequencies with the variation of $\bar{\rho}$. On top of that, we plot some
empirical fit relations (see Table 2) from previous calculation Andersson and
Kokkotas (1998); Benhar  _et al._ (2004); Pradhan and Chatterjee (2021)
including our fitting results.

Figure 5: Same as Fig. 4, but for $f$ as a function of $Z_{s}$.

###  III.2 Fitting formula

The relation between $f$-mode frequency as a function of average density was
first calculated by Andersson and Kokkotas (AK) Andersson and Kokkotas (1996).
They got an empirical relation by using some polytropic EOSs as follow:

$f(\mathrm{kHz})\approx 0.17+2.30\sqrt{\left(\frac{M}{1.4\
M_{\odot}}\right)\left(\frac{10\ \mathrm{km}}{R}\right)^{3}}.$ (26)

This empirical relations is again modified by their subsequent paper using
some realistic EOSs Andersson and Kokkotas (1998), which is given as

$f(\mathrm{kHz})\approx 0.78+1.635\sqrt{\frac{\bar{M}}{\bar{R^{3}}}}.$ (27)

Figure 6: Upper: $f$-mode frequency as a function of average density for four
parameter sets. The dashed line with different colours are the fitted
relations taken from different analyses including ours. Lower: The same
relations but for nucleonic EOSs along with DM3/Y and DM3+Y. Figure 7: Upper:
Angular frequencies ($\omega=2\pi f$) scaled by mass ($\omega M$) are shown
for four different RMF parameter sets. Lower: $\omega M$ for only IOPB-I with
yellow line, IOPB-I+DM3 with blue dashed line, IOPB-I+Y with purple dashed
line, IOPB-I+Y+DM with cyan dashed line. Table 2: Empirical relations,
$f(\mathrm{kHz})=a+b\sqrt{\frac{\bar{M}}{\bar{R^{3}}}}$ between $f$-mode
frequency and average density from different works for $l=2$ mode, where $a$
and $b$ are fitting co-efficients.

| Different
---
works
a(kHz) | b(kHz)
Our fit (N) | 1.185 | 1.246
Our fit (N+Y+DM) | 1.256 | 1.311
AK fit | 0.78 | 1.635
BFG fit | 0.79 | 1.500
PC fit | 1.075 | 1.412

Latter, these empirical relations were modified by Benhar, Ferrari, and
Gualtieri (BFG) Benhar  _et al._ (2004) by using hybrid EOSs. Recently,
Pradhan and Chatterjee (PC) Pradhan and Chatterjee (2021) have put an
empirical relation using hyperonic EOSs. We tabulate the coefficients of the
empirical relations from different analyses in Table 2. We also find fitting
relations (i) for only nucleon EOSs (N) and (ii) for nucleon EOSs with DM/Y
and Y+DM (N+Y+DM). The fitting coefficients $a$ and $b$ for different analyses
are given in Table 2 for comparison.

If we measure the $f$-mode frequency of a star, then one can infer the mass
and radius of the NS using the fit relations, which can be used to constraint
the NS EOS Andersson and Kokkotas (1996). Some works have been tried to find
some universal relations between $f$-mode frequency and compactness, or
average density Andersson and Kokkotas (1996, 1998); Benhar  _et al._ (2004);
Tsui and Leung (2005); Lau  _et al._ (2010); Wen  _et al._ (2019). Thus, the
fitting relations are significant to constrain the NS macroscopic properties.

###  III.3 Universal relations

Different correlations have been seen between different modes, such as $f$,
$p$, and $g$ with compactness/average density. All these relations are quite
independent of the EOSs. In Ref. Sotani  _et al._ (2011), it is observed that
the mass scaled angular frequency ($\omega M$) as a function of compactness
was found to be universal for $g$-mode frequency. The same type of correlation
between $\omega M$ as function of compactness for $p$ and $w$ modes had been
proposed in Ref. Blázquez-Salcedo  _et al._ (2014) and for $f$-modes see Ref.
Wen  _et al._ (2019). Thus, we want to check these universal relations as a
function of compactness as shown in Fig. 7 and Fig. 8. We find a universal
relation between $\omega M$ and $M/R$. However, the correlation is a little
weaker in the case of $\omega R$ with $M/R$. The Universal relations are given
as:

$\displaystyle\omega M(\mathrm{kHz\ km})$ $\displaystyle=$ $\displaystyle
a_{i}\left(\frac{M}{R}\right)-b_{i},$ (28) $\displaystyle\omega R(\mathrm{kHz\
km})$ $\displaystyle=$ $\displaystyle c_{j}\left(\frac{M}{R}\right)+d_{j},$
(29)

where $a_{i}$, $b_{i}$, $c_{j}$ and $d_{j}$ are the fitting co-efficients in
kHz km. The value of $i=1$ for nucleon (N) and $i=2$ for N+Y+DM3. The
coefficients are $a_{1}=190.447$, $b_{1}=4.538$ and $a_{2}=190.475$,
$b_{2}=2.984$ for $\omega M\sim M/R$. The values of $c_{1}=182.585$,
$d_{1}=127.1$ and $c_{2}=233.596$, $d_{2}=119.18$ for $\omega R\sim M/R$.

Table 3: The observables such as $M_{max}$ ($M_{\odot}$), $R_{max}$ (km),
$R_{1.4}$ (km), $f_{max}$ (kHz), $f_{1.4}$ (kHz), $C_{max}$, $C_{1.4}$,
$Z_{s_{max}}$, $Z_{s_{1.4}}$, $\Lambda_{max}$, $\Lambda_{1.4}$ are given for
NL3, G3, FSUGarnet, IOPB-I, IOPB-I+DM3, IOPB-I+Y, IOPB-I+Y+DM3 both for
maximum mass and canonical NS.

Observable | NL3 | G3 | FSUGarnet | IOPB-I | IOPB-I+DM3 | IOPB-I+Y | IOPB-I+Y+DM3
---|---|---|---|---|---|---|---
$M_{max}$ | 2.77 | 1.99 | 2.07 | 2.15 | 2.05 | 1.70 | 1.61
$R_{max}$ | 13.17 | 10.79 | 11.57 | 11.76 | 11.04 | 11.41 | 10.38
$R_{1.4}$ | 14.08 | 12.11 | 12.59 | 12.78 | 11.76 | 12.81 | 11.57
$f_{max}$ | 2.16 | 2.55 | 2.38 | 2.32 | 2.57 | 2.58 | 2.85
$f_{1.4}$ | 1.78 | 2.06 | 1.94 | 1.87 | 2.14 | 1.92 | 2.22
$C_{max}$ | 0.31 | 0.27 | 0.26 | 0.27 | 0.27 | 0.22 | 0.23
$C_{1.4}$ | 0.15 | 0.17 | 0.16 | 0.16 | 0.18 | 0.16 | 0.18
$Z_{s_{max}}$ | 0.62 | 0.48 | 0.45 | 0.47 | 0.49 | 0.33 | 0.36
$Z_{s_{1.4}}$ | 0.18 | 0.23 | 0.22 | 0.21 | 0.24 | 0.21 | 0.25
$\Lambda_{max}$ | 4.48 | 12.12 | 17.67 | 14.78 | 14.39 | 59.58 | 50.58
$\Lambda_{1.4}$ | 1267.79 | 461.28 | 624.81 | 681.27 | 471.06 | 650.55 | 391.68

Figure 8: Same as Fig. 7, but for $\omega R$ with $M/R$. Figure 9: Upper:
$f$-mode frequencies are shown with the variation of $\Lambda$ for four
different RMF parameter sets. Lower: $f$-mode frequencies with $\Lambda$ for
IOPB-I, IOPB-I+DM3, IOPB-I+Y, and IOPB-I+Y+DM.

The tidal deformability is the most important quantity, which gives important
information about the NS EOSs. The GW170817 event has put a limit on the tidal
deformability of the canonical NS, which is used to constraint the EOS of
neutron-rich matter at 2–3 times the nuclear saturation densities Abbott  _et
al._ (2017, 2018). Due to a strong dependence of the tidal deformability with
radius ($\Lambda\sim R^{5}$), it can put stringent constraints on the EOS.
Several approaches Bauswein  _et al._ (2017); Annala  _et al._ (2018);
Fattoyev  _et al._ (2018); Radice  _et al._ (2018); Malik  _et al._ (2018);
Most  _et al._ (2018); Tews  _et al._ (2018); Nandi  _et al._ (2019); Capano
_et al._ (2020) have been tried to constraint the EOS on the tidal
deformability bound given by the GW170817. The dimensionless tidal
deformability $\Lambda$ is defined as Hinderer (2008, 2009); Hinderer  _et
al._ (2010)

$\displaystyle\Lambda=\frac{2}{3}k_{2}\ C^{-5},$ (30)

where $k_{2}$ is the second Love number, which depends on the internal
structure as well as mass and radius of a star Hinderer (2008, 2009); Kumar
_et al._ (2017b); Das  _et al._ (2020b).

We calculate the $f$-mode frequencies as a function of $\Lambda$ for different
RMF EOSs, including hyperons and DM. The value of $\Lambda$ decreases when one
goes from stiff to soft EOSs and the corresponding $f$-mode frequencies
increase contrary to $\Lambda$. The numerical values of $\Lambda_{1.4}$ are
given in Table 3 for different EOSs. The GW170817 event put constraint on
$\Lambda_{1.4}=190_{-120}^{+390}$ Abbott  _et al._ (2018). In Ref. Wen  _et
al._ (2019), they have put a limit on the value of $f$-mode frequencies for
1.4 $M_{\odot}$ are 1.67–2.18 kHz by combining two constraints: the EOS
parameter space allowed by terrestrial nuclear experiments and the tidal
deformability data from GW170817. Our predicted results of $\Lambda$ are
consistent with Wen et al. Wen  _et al._ (2019), except for the case of
IOPB-I+Y+DM3. More NS mergers are expected to be measured in the future, which
may constraint the $f$-mode frequency more tightly.

Figure 10: $f_{max}$ are shown with different DM momenta for IOPB-I parameter
sets. The colour scheme represents the maximum masses correspond to different
$k_{f}^{DM}$.

Figure 11: Left: Correlation between $f_{1.4}$ and $\Lambda_{1.4}$ are shown
for 41 parameter sets including (RMF, SHF, and DDRMF). The colour bar
represents the maximum masses of the corresponding parameter sets. Right: For
$f_{max}$ with $\Lambda_{1.4}$.

In Fig. 10, we plot the values of $f_{max}$ with different DM Fermi momenta.
The $f$-mode frequencies increase with increasing $k_{f}^{DM}$. The range of
$f_{max}$ is 2.3–2.4 kHz corresponds to $k_{f}^{DM}=0.00-0.025$ GeV after that
it increases with more proportion beyond $k_{f}^{DM}=0.03$ GeV. The colour bar
represents the maximum masses ($M_{max}$) of the NS for different
$k_{f}^{DM}$. The area of $M_{max}$ at lower $k_{f}^{DM}$ (green colour) is
found to be more as compared to the higher $k_{f}^{DM}$ and the area slightly
reduces from lower $k_{f}^{DM}$ to higher one. This is because the slight
increase of DM percentage reduces the value $M_{max}$ a little bit. When we
increases the DM percentage more than 0.03 GeV, the maximum mass reduces a
lot. For example, for $k_{f}^{DM}=$ 0.00, 0.01, and 0.02 GeV, the values of
$M_{max}$ are 2.149, 2.146, and 2.119 $M_{\odot}$ respectively. If we increase
the DM Fermi momenta $k_{f}^{DM}=$ 0.03, 0.04, 0.05, and 0.06 GeV, the values
of $M_{max}$ are 2.051, 1.938, 1.788, and 1.614 $M_{\odot}$ respectively.

We also find some correlations between $f_{1.4}$–$\Lambda_{1.4}$, and
$f_{max}$–$\Lambda_{1.4}$ which are shown in Fig. 11. This correlation is
reported first calculated by Wen et al. Wen  _et al._ (2019) by taking 23000
phenomenological EOSs, 11 microscopic EOSs, and two quark EOSs. Here, in our
calculation, we use 23 RMF EOSs, 14 Skyrme-Hartree-Fock (SHF) forces, and four
density-dependent (DDRMF) sets are given in Ref. Biswal  _et al._ (2020)
having mass more than $\sim 2\ M_{\odot}$. We get a correlation between
$f_{1.4}$–$\Lambda_{1.4}$ and $f_{max}$–$\Lambda_{1.4}$ as shown in figure.
The accurately measured tidal deformability of the star from the GW analysis
can be used as a constraint on the $f$-mode frequency of a star. We expect
that the future GW observation may answer for this type of correlation.

##  IV SUMMARY AND CONCLUSIONS

Gravitational waves emitted by both the isolated and binary neutron stars are
the key quantities that give us enough information about the internal
structure and EOSs of the star. If we detect the frequencies of the oscillated
star, one can infer the mass and radius of the star from the empirical formula
given in the Ref. Andersson and Kokkotas (1996) with 90% accuracy. Therefore,
the oscillation frequencies are the most promising quantity to constraints the
EOS of the NS. The main problem arises to make a Universal empirical relation
due to the lack of knowledge on the internal structure of the NS. Several
attempts have already been initiated with a little success leaving us an open
challenge of the problem. The oscillations frequencies are in the range 1–5
kHz for different modes such as $f$, $p$, $w$, etc. This range of frequencies
can’t be detected by our present terrestrial detectors, such as LIGO/Virgo. We
may be sufficiently advanced to detect that range of frequencies coming from
the stars in the future. Prior to this, we have to be equipped with enough
possible theoretical solutions to study oscillating stars. Therefore, in this
study, we explore the $f$-mode frequencies of the DM admixed NS, and we hope
that the results may help the community to explore more about the oscillations
frequencies emitted by stars.

In this study, we calculate $f$-mode oscillation frequencies for quadrupole
mode within the relativistic Cowling approximations. To compute $f$-mode
frequencies, we take RMF EOSs by assuming that the hyperons are present inside
the NS. Four well-known RMF forces, such as NL3, G3, IOPB-I, and FSUGarnet,
are taken for this calculation. The $f$-mode frequencies decrease with mass
for softer EOSs as compared to stiffer ones. We also include DM as an extra
candidate inside the hyperonic star to see their effects on the $f$-mode
frequency. The coupling constants for the hyperons– vectors mesons are
calculated using the SU(6) method, while hyperons–scalar mesons are computed
by fitting with hyperon potential depth. The baryons-Higgs form factor is
assumed to be the same, which is consistent with available theoretical data.
The DM-Higgs coupling is constrained with the help of a direct detection
experiment and LHC searches.

The $f$-mode frequencies as a function of different astrophysical observables
such as $M$, $R$, $\Lambda$, $C$, $Z_{s}$, etc., are calculated and compared
with other theoretical results. The numerical values of $f_{max}$ and
$f_{1.4}$ as functions of different quantities are evaluated. The value of
$f_{max}$ increases with the addition of either DM/hyperons or DM+hyperons.
This is because the EOSs become softer with the addition of DM and hyperons.
Softer EOS predicts higher $f$-mode frequency as compared to stiffer one. One
can get clear pictures about the variation of $f_{max}$ with different DM
Fermi momenta from the Fig. 10. We fit the $f$-mode frequencies with the
average density, and the coefficients are in harmony with previous models. The
angular frequencies scaled by mass and radius are calculated with compactness.
It is found that both $\omega M$ and $\omega R$ follow a linear relationship.

The $f$-mode frequencies with tidal deformability are calculated using the 23
RMF, 14 SHF, and 6 DDRMF equation of states. We find the correlations between
$f_{1.4}$–$\Lambda_{1.4}$ and $f_{max}$–$\Lambda_{1.4}$ are almost consistent
with the prediction of Wen et al. Wen  _et al._ (2019). The correlation is
slightly weaker for $f_{max}$–$\Lambda_{1.4}$ as compared to
$f_{1.4}$–$\Lambda_{1.4}$. Wen et al. have given a range for $f_{1.4}$ as
1.67–2.18 kHz. Our results correspond to $f_{1.4}$ is in the range 1.78–2.22
kHz for the considered EOSs with DM and hyperons. We hope the discovery of
more BNS merger events will open up tight constraints on the $f$-mode
frequencies in the future. Thus, it is evident that the effects of either
DM/hyperons or DM+hyperons on the $f$-mode frequencies are significant. One
can take DM EOSs to calculate $f$-mode frequency in full GR method as done by
Lindblom and Detweiler Lindblom and Detweiler (1983), which may provide more
accurate results.

## Acknowledgement

HCD like to thank Prof. H. Sotani, Prof. I. F. Ranea Sandoval, and Prof. C. V.
Flores for their fruitful discussions about the numerical methods for
calculating eigenfrequencies of the star.

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