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 a DM momentum of 0.04 GeV, the
binding energy for the canonical star becomes positive for different parameter
sets. However, the maximum mass NS still shows a bound system with negative
binding energy. From this, we conclude that one can constrain 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 [91, 195, 94, 95].
The cooling of NS is also faster with the increasing DM mass [193, 194]. That
means the positive $B$ may have a relation with the cooling properties of the
NS; in other words, it may accelerate the Urca process.

Figure 4.11: The variation of $B/M$ with the M/$M_{\odot}$ of the NS with and
without DM.

###  4.3 The secondary component in GW190814 event

The LIGO and Virgo collaboration detected the most enigmatic compact binary
coalescence event (GW190814) involving a black hole and a compact object with
a mass range of $22.2-24.3$ and $2.50-2.67\ M_{\odot}$ respectively [34].
Since the electromagnetic counterpart has not been found and has no measurable
signature of tidal deformations from the GW, the secondary component might be
the lightest black hole or the heaviest NS. Therefore, it is a big challenge
to explain the nature of such a compact object. There is a lot of debate about
understanding the mystery of the secondary component of the GW190814 event
[35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. The analysis of GW190814 data
implies the possible nature of such a compact object as a NS only when (i) the
EOS is very stiff [41] or (ii) it is a rapidly rotating compact object below
the mass shedding frequency [38, 45]. Thus, one can consider either of the two
facts, which can produce the mass of the secondary component of GW190814
around 2.50 $M_{\odot}$.

The binary NS merger event GW170817 [23] has evoked to put an upper bound on
the maximum mass of nonrotating NS. By combining the total binary mass of
GW170817 inferred from GW signal with electromagnetic observations, an upper
limit of $M_{{\rm max}}\leq 2.17\ M_{\odot}$ is predicted by Margalit et al.
[252]. Rezzolla et al. have put the upper bound by combining the GW
observations and quasi-universal relations as $M_{{\rm max}}\leq
2.16_{-0.15}^{+0.17}\ M_{\odot}$ [253]. Further analysis by employing energy
and momentum conservation laws and the numerical-relativity simulations show
that the maximum mass of a cold NS is bound to be less than $2.3\ M_{\odot}$
[254]. Also, various massive pulsar discoveries constrained the EOSs of the
supra-nuclear matter inside the core of the NS [14, 16, 17]. These
observational data also suggest strong constraints on the maximum mass of the
slowly rotating NS with a lower bound of $\sim 2\ M_{\odot}$, which discarded
many EOSs. Recently, the NICER data also imposed stringent constraints on the
mass and radius of a canonical star from the analysis of PSR J0030+0451 data
[20, 21, 218]. However, one cannot exclude the existence of a super-massive NS
as the secondary component of GW190814.

Many energy density functionals have been formulated to study the finite
nuclei, the nuclear matter, and the NS properties, out of which only a few
EOSs can reproduce the properties consistently with empirical/experimental
data [171, 255]. To achieve the mass of the secondary component of the
GW190814 event, one has to include stiff EOS, which may or may not satisfy
different constraints such as flow [169], GW170817 [23, 24], x-ray [16, 17],
and NICER data [20, 21]. For this, we search for stiff RMF EOSs and find that
the linear Walecka models (L-RMF) [248], the standard nonlinear $\sigma$-self-
coupling interactions (NL-RMF) [100] and the density-dependent RMF (DD-RMF)
[41, 83] sets generally reproduce the maximum mass of the NS more than 2.50
$M_{\odot}$. The BigApple force is designed to reproduce the maximum mass of
the NS more than 2.50 $M_{\odot}$ [43, 134]. Although the L-RMF model predicts
larger mass, it fails to reproduce the finite nuclei and nuclear matter
properties. The recent work of Huang et al. indicates the possibility of the
secondary object in the GW190814 event as pure hadronic NS with DD-RMF
parameter sets [41]. In the DD-RMF family, the DD-LZ1 and DD-MEX forces
reproduce the NS masses as 2.554 and 2.556 $M_{\odot}$, respectively. Thus,
the DD-RMF sets could only predict the secondary component’s lower limit mass
in the GW190814 event. Hence, it ruled out the possibility of the hybrid star
or any admixture of exotic components, such as hyperons and DM inside the NS.
Therefore, the L-RMF and DD-RMF forces are not very useful for the
investigation.

Figure 4.12: The EOS are shown for NL3 and NL-RA1 with DM Fermi momenta 0–0.06
GeV. The joint constraints from the gravitational wave data GW170817 and
GW190814 in the shaded region are adopted from Ref. [34].

In the present study, we consider the NL-RMF parameter sets as our primary
input to achieve the mass of the secondary component. Some of these forces are
NL3 [104], NL3* [256], NL1 [101], NL-SH [257], NL3-II [104] and NL-RA1 [258],
which are generally considered to be stiff EOSs. The maximum masses of all
these parametrizations fall in the $2.7-2.8$ $M_{\odot}$, which are slightly
more massive than the secondary component of the GW190814 event. Therefore, we
have two possible scenarios to reduce the maximum mass and radius of the NS
(i) with the addition of hyperons or (ii) with the addition of DM. With the
addition of exotic particles, the EOS becomes softer, which reduces the mass,
radius, and tidal deformability of the NS [50, 53, 259, 124, 201, 125, 134].
However, in the case of hyperons inside the NS, the predicted canonical radius
for NL-RMF type sets was found to be $\sim 15$ km, which does not lie in the
range given by the NICER ($11.96-14.26$ km) [20] and GW190814 ($12.2-13.7$ km)
[34] predictions. Therefore, we exclude the hyperons in this case and involve
the DM as an extra candidate inside the NS.

Figure 4.13: The speed of sound for NL3 parameter set with DM Fermi momenta
$0-0.06$ GeV. The QCD conformal limit ($c_{s}^{2}=1/3$) and maximum speed
limit ($c_{s}^{2}=1$) are shown with a dashed magenta line. Figure 4.14: The
$M-R$ relations for NL3 and NL-RA1 with different DM Fermi momenta. The
horizontal bars represent the maximum mass constraints from the PSR J0740+6620
(light pink) and the GW190814 event (dark cyan). The NICER data are also shown
with two boxes from two different analyses [20, 21]. The double-headed green
line represents the radius constraints by the GW190814 [34] for 1.4
$M_{\odot}$ NS.

####  4.3.1 Results and Discussions

In this section, we calculate the NS properties by using NL-RMF-type EOSs. All
these sets predict the maximum NS mass of more than $2.67\ M_{\odot}$. In Fig.
4.12, the EOSs (such as NL3 and NL-RA1) are depicted with $k_{f}^{\rm DM}$ in
the range of $0-0.06$ GeV, which are shortly represented as DM0-DM6. The EOSs
become softer with the increase of $k_{f}^{\rm DM}$. The GW190814 discovery
provides a joint constraint for NS-black hole (NSBH) merger [34]. It is
assumed to be a NS only when the maximum mass is not less than the secondary
component of GW190814. The EOSs that correspond to DM3 and DM4 almost pass
through the joint constraints imposed by GW170817 and GW190814. This amount of
DM ($0.03-0.04$ GeV) may be available inside the NS. To test the validity of
the EOS, we calculate the speed of sound ($c_{s}^{2}=\partial
P/\partial\mathcal{E}$) for DM admixed NS. We find that our model respects the
causality, which is shown in Fig. 4.13. The value of $c_{s}^{2}$ increases up
to 0.4 fm-3, becoming constant beyond that.

Figure 4.15: The tidal deformability of the NS with the mass for NL3 and NL-
RA1 parameter set for different DM Fermi momenta. The dark cyan (green) line
implies the constraint given by GW190814 (GW170817) data for 1.4 $M_{\odot}$
NS.

The macroscopic properties such as mass ($M$), radius ($R$), and dimensionless
tidal deformability ($\Lambda$) are also calculated with different DM
fractions. By solving TOV equations [222, 223]; one can find the $M$-$R$
relation for each EOSs by assuming the $P(0)=P_{C}$ and $P(R)=0$ as the
boundary conditions. We find that the NS’s mass and radius reduce with DM’s
addition, as shown in Fig. 4.14. The maximum mass of the secondary object and
the canonical radius constraints are $2.50-2.67$ $M_{\odot}$ and $12.20-13.70$
km as given in GW190814 [34]. The $M_{{\rm max}}$ and $R_{1.4}$ range
correspond to DM3 are found to be $2.65-2.67$ $M_{\odot}$ and $12.38-12.75$ km
respectively for the considered parameter sets (see Table 4.2), which is
consistent with GW190814 data. Under the assumption of a stiff nuclear EOS,
the value of maximum mass with the assumed parameter set hints toward the
possibility of the secondary component of the GW190814 event being a DM
admixed NS. The NICER [20, 21] constraints are also respected by DM3 and DM4.
Thus we conclude that given a specific model of the nuclear EOS, one can fix
the fraction of DM inside the NS from the observational data.

When the NS presents in the tidal field created by its companion star, it is
deformed. The relation between quadruple deformation $Q_{ij}$ and tidal field
$\epsilon_{ij}$ is written as [260, 261]

$Q_{ij}=-\lambda\epsilon_{ij},$ (4.25)

where the $\lambda$ is defined as tidal deformability. The dimensionless tidal
deformability can be defined as $\Lambda=\lambda/M^{5}$. We calculate the
$\Lambda$ for different DM Fermi momenta as shown in Fig. 4.15. The values of
$\Lambda_{1.4}$ for DM3 are 842.12, 784.59, 932.99, 804.36, and 818.48 for
NL3, NL3*, NL-SH, NL3-II, and NL-RA1 respectively. The predictions of
$\Lambda_{1.4}$ from GW170817 is $190_{-120}^{+390}$ at 90% confidence level
[24]. The GW190814 also put a constraint on $\Lambda_{1.4}$ under NSBH
scenario, $\Lambda_{1.4}=616_{-158}^{+273}$ [34]. The calculated values of
$\Lambda_{1.4}$ for DM3 are well matched with GW190814 data, and, for the DM4
case, it is well consistent with GW170817 data. This also gives sufficient
hints to constrain the amount of DM inside the NS.

In summary, the mass gap of $2.5-5.0\ M_{\odot}$ in the secondary component of
GW190814 motivates us to explore its true nature, and we find the possibility
that the compact object could be a DM admixed NS. To know the mystery of DM
inside the NS, we modeled a Lagrangian density by assuming that the DM
particles interact with nucleons via SM Higgs. We adopt the RMF model with NL-
RMF parameter sets to calculate NS EOS. The DM admixed NS properties are
computed with different DM fractions inside it.

To achieve the mass of the secondary object, one should take stiff EOSs. In
RMF parametrizations, only NL-RMF and DD-RMF sets are stiff, which predicts
the maximum masses of more than 2.50 $M_{\odot}$. Huang et al. [41] also
suggested the possibility of the secondary object as a NS (with DD-RMF sets
without any admixture of DM or strange particles). However, the NL-RMF forces
with NS EOSs predict masses more than 2.67 $M_{\odot}$ are candidates for DM
admixed NS. This scenario is only seen for NL-RMF type sets because no DD-RMF
forces predict the mass of more than 2.67 $M_{\odot}$. Hence, we assume that
those stars corresponding to DD-RMF sets are only NS without DM inside.

Table 4.2: The quantities such as $M_{{\rm max}}$, $R_{{\rm max}}$, $R_{1.4}$
and $\Lambda_{1.4}$ are given with $k_{f}^{\rm DM}=$ 0, 0.03 and 0.04 GeV for
NL3, NL3*, NL1, NL-SH, NL3-II and NL-RA1 respectively.

$k_{f}^{\rm DM}$ (GeV) |  NL3 |  NL3* |  NL1 |  NL-SH |  NL3-II |  NL-RA1
---|---|---|---|---|---|---
0.00 | 0.03 | 0.04 | 0.00 | 0.03 | 0.04 | 0.00 | 0.03 | 0.04 | 0.00 | 0.03 | 0.04 | 0.00 | 0.03 | 0.04 | 0.00 | 0.03 | 0.04
| $M_{{\rm max}}$
---
$(M_{\odot})$
2.78 | 2.65 | 2.50 | 2.76 | 2.64 | 2.49 | 2.84 | 2.71 | 2.55 | 2.79 | 2.66 | 2.51 | 2.77 | 2.65 | 2.50 | 2.78 | 2.66 | 2.51
| $R_{{\rm max}}$
---
(km)
13.40 | 12.46 | 11.53 | 13.10 | 12.38 | 11.60 | 13.63 | 12.75 | 11.67 | 13.53 | 12.63 | 11.60 | 13.14 | 12.42 | 11.52 | 13.19 | 12.47 | 11.57
| $R_{1.4}$
---
(km)
14.08 | 13.16 | 11.82 | 14.03 | 13.13 | 12.13 | 14.73 | 13.46 | 11.90 | 14.35 | 13.41 | 11.90 | 14.05 | 13.16 | 11.81 | 14.11 | 13.22 | 11.85
$\Lambda_{1.4}$ | 1311.16 | 842.12 | 499.04 | 1250.45 | 784.59 | 492.40 | 1503.77 | 912.65 | 548.83 | 1528.42 | 932.99 | 563.90 | 1282.90 | 804.36 | 503.89 | 1338.98 | 818.48 | 516.96

With the addition of DM, the EOS becomes softer, $M_{{\rm max}}$, $R_{{\rm
max}}$, $R_{1.4}$ and $\Lambda_{1.4}$ reduce with increase of DM percentage as
given in Table 4.2. The EOS corresponds to DM3, and DM4 well passes through
the joint constraint inferred from GW170817+GW190814 by assuming that the
secondary component is a NS only when its maximum mass is not less than the
2.50 $M_{\odot}$. This model also respects the causality for all values of DM
Fermi momenta. Other properties such as $M_{{\rm max}}$ and $R_{1.4}$ are well
consistent with the GW190814 data for the DM3 case. The values of $R_{1.4}$
for DM3 and DM4 are well constrained with the NICER results in almost all
assumed sets. The calculated $\Lambda_{1.4}$ for DM3 and DM4 lie in the range
of GW190814 and GW170817 data. Thus, one can constrain the DM percentage
inside the NS from these observational data.

Therefore, the possibility of the secondary object in GW190814 as a DM admixed
NS only when the maximum mass should be more than 2.67 $M_{\odot}$. We hope
future GW and x-ray observations may answer the presence of DM inside the NS
and constrain its properties. Thus, we suggest that the LIGO/Virgo include DM
inside the compact objects when they infer some properties, such as mass,
radius, and tidal deformability of the binary NS.

### Conclusion

In conclusion, we have calculated the properties of isolated, static, and
rotating DM admixed NS. The simple DM model is taken as discussed in
Chapter-3. It has been noticed that DM has significant effects on both static
and rotating NS. For example, the mass and radius decrease by increasing the
DM percentage. When we compared the static and rotating cases, the change was
found to be $\sim 20\%$ for different macroscopic properties. The change of
curvature with and without DM is obtained as $\sim 33\%$ for the canonical
star, and the value increases with mass. The magnitude of surface curvature
and compactness are increased. The binding energy for the DM admixed NS
increased towards a positive value with the increase of DM momentum, which
makes the NS unstable. From this study, we concluded that a tiny amount of DM
can accumulate inside the NS. If a star has more DM contains, it heats the NS,
and results in the acceleration of the Urca process. Due to this phenomenon,
the cooling of NS enhances and makes the NS unstable. With the DM admixed
EOSs, we have also suggested that the secondary component might be a NS with
DM content if the underlying nuclear EOS is sufficiently stiff.

##  Chapter 5 Dark matter effects on properties of the oscillating
neutron/hyperon stars

The NS oscillates due to vibrant explosion in the type-II supernovae. Due to
its complex internal compositions, it oscillates with different ranges of
frequencies and emits gravitational waves. In this chapter, we explore the
fundamental mode frequency of the NS since it emits a large amount of energy
through this mode. We use the relativistic mean-field EOS for the DM admixed
hyperons star within the relativistic Cowling approximations to get the eigen
frequencies of the star. Various oscillating properties of hyperon stars are
calculated with different fractions of DM for well-known EOSs. We enumerate
some empirical relations, which can be used to constrain the internal
properties of the NS if we detect the emitted GWs in the future.

###  5.1 Hyperons inside the neutron star

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;
consequently, the magnitude of the maximum mass substantially reduces, which
is incompatible with the observation known as the “hyperon puzzle”. In
addition to DM, we also include hyperons production inside the NS. Hyperons
appear at the higher-density region or core of the NS. The idea behind the
appearance of hyperons inside the NS is not brand-new. In 1959, Cameron et al.
first put forward the concept of hyperon puzzle [262]. According to
Ambartsumyan & Saakyan, the core of a massive NS consists of an inner hyperon
core and an outer nucleon shell [263]. The detailed discussion on hyperons
production and interaction with mesons and their coupling parameters have been
studied in Refs. [264, 265, 266, 267, 268, 47, 269, 49, 270, 271, 272, 273,
50, 274, 275, 276, 53, 259]. 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, however, these are still
open problems. This is because only some hypernuclei have been discovered in
the terrestrial laboratory [277, 278, 279]. More hypernuclei will help us to
constrain 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. [280, 50, 51]. Another efficient approach to
fix the hyperons-mesons coupling parameters is SU(3) group theory method, as
given in Refs. [50, 281, 275, 282]. The coupling constants for scalar and
vector mesons are constrained using the SU(3) method. One can calculate NM and
NS properties by using both SU(3) and SU(6) methods and compare their results
as done in Refs. [50, 275]. In this chapter, our aim is to calculate the
$f$-mode frequency of the DM admixed hyperon star as the function of different
astrophysical quantities.

####  5.1.1 Interaction between baryons-mesons and mesons-mesons

In this chapter, we have taken baryon octets
($p,n,\Lambda,\Sigma^{+,-,0},\,{\rm and}\,\Xi^{+,-}$) along with other two
strange mesons such as $\sigma^{*}$, and $\phi$ [47, 53, 283]. Hence, the
Lagrangian density in Eq. (2.49) is almost the same, but an extra part is
added for the strange meson. Hence the Lagrangian density is modified as
follows [183]

$\displaystyle{\cal L}_{\rm NS}={\cal L}_{\rm NM}+{\cal L}_{\rm
YY}+\sum_{l}\bar{\psi}_{l}\Big{(}i\gamma^{\mu}\partial_{\mu}-m_{l}\Big{)}\psi_{l},$
(5.1)

where

$\displaystyle{\cal L}_{\rm YY}$ $\displaystyle=$
$\displaystyle\sum_{Y}\bar{\psi}_{{}_{Y}}(g_{\sigma^{*}_{Y}}\sigma^{*}-g_{\phi_{Y}}\gamma_{\mu}\phi^{\mu})\psi_{{}_{Y}}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\frac{1}{4}\phi_{\mu\nu}\phi^{\mu\nu}$
(5.2) $\displaystyle+$
$\displaystyle\frac{1}{2}(\partial_{\mu}\sigma^{*}\partial^{\mu}\sigma^{*}-m_{\sigma^{*}}^{2}{\sigma^{*}}^{2}).$

Also, it is imperative to mention that the summation over $\Sigma_{p,n}$ in
Eq. (2.20) is now modified as $\Sigma_{B}$, where $B$ represents the baryon
octets. The energy density and pressure are also modified with the addition of
other baryons and strange mesons and given as [119, 53, 144]:

$\displaystyle{\cal E}_{\rm NS}=$
$\displaystyle\sum_{B}\frac{\gamma_{{}_{B}}}{2\pi^{2}}\int_{0}^{k_{F_{B}}}k^{2}\
dk\sqrt{k^{2}+m^{*2}_{{}_{B}}}+n_{{}_{B}}g_{\omega_{B}}\omega_{0}\frac{n_{3B}}{2}g_{\rho{{}_{B}}}\rho_{0}$
$\displaystyle-$
$\displaystyle\frac{1}{3!}\zeta_{0}{g_{\omega_{B}}^{2}}\omega_{0}^{4}-\Lambda_{\omega}g_{\rho_{B}}^{2}g_{\omega_{B}}^{2}\rho_{03}^{2}\omega_{0}^{2}+m_{\sigma}^{2}{\sigma_{0}}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\eta_{2}}{2}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}-\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}\Bigg{)}m_{\rho}^{2}\rho_{03}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{\delta}^{2}\delta_{0}^{2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}+\frac{1}{2}m_{\sigma^{*}}^{2}\sigma_{0}^{*2}+\sum_{l}\frac{\gamma_{{}_{l}}}{2\pi^{2}}\int_{0}^{k_{F{{}_{l}}}}dk\
\sqrt{k^{2}+m_{l}^{2}},$ (5.3)

and

$\displaystyle P_{\rm NS}=$
$\displaystyle\sum_{B}\frac{\gamma_{{}_{B}}}{6\pi^{2}}\int_{0}^{k_{F_{B}}}\frac{k^{4}\
dk}{\sqrt{k^{2}+m^{*2}_{{}_{B}}}}+\frac{1}{3!}\zeta_{0}{g_{\omega_{B}}^{2}}\omega_{0}^{4}+\Lambda_{\omega}g_{\rho_{B}}^{2}g_{\omega_{B}}^{2}\rho_{03}^{2}\omega_{0}^{2}$
$\displaystyle-$ $\displaystyle
m_{\sigma}^{2}{\sigma_{0}}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}+\frac{\eta_{2}}{2}\frac{g_{\sigma_{B}}^{2}\sigma_{0}^{2}}{m_{{}_{B}}^{2}}\Bigg{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma_{B}}\sigma_{0}}{m_{{}_{B}}}\Bigg{)}m_{\rho}^{2}\rho_{03}^{2}-\frac{1}{2}m_{\delta}^{2}\delta_{0}^{2}+\frac{1}{2}m_{\sigma^{*}}^{2}\sigma_{0}^{*2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}$
$\displaystyle+$
$\displaystyle\sum_{l}\frac{\gamma_{{}_{l}}}{6\pi^{2}}\int_{0}^{k_{F{{}_{l}}}}\frac{k^{2}dk}{\sqrt{k^{2}+m_{l}^{2}}},$
(5.4)

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

$\displaystyle m^{*}_{B}$ $\displaystyle=$ $\displaystyle
m_{B}-g_{\sigma_{B}}\sigma_{0}-g_{\delta_{B}}\tau_{B}\delta_{0}-g_{\sigma^{*}_{B}}\sigma^{*}.$
(5.5)

The last term in Eq. (5.5) doesn’t contribute to nucleons. This is because the
value of $g_{\sigma^{*}_{B}}$ and $g_{\phi_{B}}$is zero for the nucleons. The
mass, charge, isospin, and strangeness of each baryon are taken in these
calculations are given in Table 5.1

Table 5.1: Mass, charge, isospin, and strangeness of all baryon octet.

| Baryons
---
Name
$p$ | $n$ | $\Lambda$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Sigma^{-}$ | $\Xi^{0}$ | $\Xi^{-}$
| Mass
---
(MeV)
939 | 939 | 1116 | 1193 | 1193 | 1193 | 1318 | 1318
Charge ($q$) | +1 | 0 | 0 | +1 | 0 | -1 | 0 | -1
Isospin ($I_{3}$) | 1/2 | -1/2 | 0 | +1 | 0 | -1 | 1/2 | -1/2
Strangeness ($s$) | 0 | 0 | -1 | -1 | -1 | -1 | -2 | -2

##### 1\. Coupling constants for Nucleons-Mesons and Mesons-Mesons:-

The coupling between nucleons-mesons and meson-mesons differs and depends on
the parameter sets. For example, the G3 parameter set has different nucleons-
mesons and mesons-mesons coupling compared to IOPB-I forces (see Table 2.1).
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 [104],
IOPB-I [119], FSUGarnet [162], and G3 [118]. We tabulate the masses of
different mesons and coupling constants in Table 2.1 for these parameter sets.
The NM properties corresponding to four parameter sets are also reported in
Table 3.1 [119, 134]. Table 3.1 shows that all three parameter sets reproduce
the NM properties 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 compared to G3, which is a softer one.

##### 2\. Coupling constants for Hyperons-Mesons:-

Until now, we do not know exactly how the hyperons interact with each other
compared to the nucleons-nucleons interaction because of the limitation of the
hyper-nuclei data [284]. Therefore, fixing the couplings between hyperons-
mesons is quite difficult. Generally, 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 [264, 275, 53,
259]. The potential depth of the $\Lambda$-hyperon ($U_{\Lambda}$) is known to
be -28 MeV [265, 267, 268, 269, 285, 280] 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 [48, 268]. But some hyper-nuclear
experiments show that the $U_{\Sigma}$ and $U_{\Xi}$ potential depth close to
+30 MeV and -18 MeV respectively [49, 270, 271, 272, 273].

The hyperon potential depth is defined as [136]

$U_{Y}=-g_{\sigma{{}_{N}}}x_{\sigma{{}_{Y}}}\sigma_{0}+g_{\omega{{}_{N}}}x_{\omega{{}_{Y}}}\omega_{0},$
(5.6)

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. [136, 53]

$\displaystyle
g_{\sigma{{}_{N}}}\sigma_{0}=M-M^{*}=M\Big{(}1-\frac{M^{*}}{M}\Big{)},\,g_{\omega{{}_{N}}}\omega_{0}=\Big{(}\frac{g_{\omega{{}_{N}}}}{m_{\omega}}\Big{)}^{2}\rho_{0},$
(5.7)

Rewriting Eq. (5.6) by inserting Eq. (5.7), we get

$\displaystyle
U_{Y}=M\Big{(}\frac{M^{*}}{M}-1\Big{)}x_{\sigma{{}_{Y}}}+\Big{(}\frac{g_{\omega{{}_{N}}}}{m_{\omega}}\Big{)}^{2}\rho_{0}x_{\omega{{}_{Y}}}.$
(5.8)

In this case, we choose the values of $U_{\Lambda}=-28$ MeV, $U_{\Sigma}=+30$
MeV, and $U_{\Xi}=-18$ MeV. To get the desired potential, one has to choose
the values of $x_{\sigma{{}_{Y}}}$ and $x_{\omega{{}_{Y}}}$ for a fixed
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
[48, 268] 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 [53]. In this calculation, we neglected the
$\sigma^{*}$ and $\phi$ mesons-nucleons interaction for numerical simplicity.

###  5.2 Dark matter admixed hyperon star

####  5.2.1 Interaction between Baryons-DM

In this sub-sec, our aim is to calculate the DM interactions with all baryons.
We have already formulated the DM-nucleons interaction in chapter-2 (see sub-
section 3.2). With the addition of a baryon octet, the interaction Lagrangian
can be modified as

$\displaystyle{\cal{L}}_{\rm
DM}=\bar{\chi}\Big{[}i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\Big{]}\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}M_{h}^{2}h^{2}+\sum_{B}f_{{}_{B}}\frac{m_{{}_{B}}}{v}\bar{\psi}_{{}_{B}}h\psi_{{}_{B}},$
(5.9)

where $\psi_{{}_{B}}$ 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 the
Higgs field. The values of $y$ and $v$ are 0.07 and 246 GeV, respectively,
taken from the Refs. [125, 128]. The value of $f_{{}_{B}}(=$ 0.35) for all
baryons is taken to be the same because we do not know the form factor for
hyperons except nucleons with Higgs.

Figure 5.1: Left: EOSs for different RMF parameter sets. Right: EOSs are for
IOPB-I with a yellow line (only nucleons), IOPB-I+DM3 with a blue dashed line
(nucleons+DM3), IOPB-I+Y with a purple dashed line (nucleons+hyperons),
IOPB-I+Y+DM with dashed cyan line (nucleons+hyperons+DM3).

####  5.2.2 EOSs, $M-R$ profiles

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

$\displaystyle{\cal{E}}={\cal{E}}_{\rm NS}+{\cal{E}}_{\rm DM},\ {\rm and}\
P=P_{\rm NS}+P_{\rm DM}.$ (5.10)

In Fig. 5.1, 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 the 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 visible in the figure.

In the presence of hyperons/DM particles, the EOS of the NS 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 be placed in a higher orbit with increasing
density, 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 of 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 decay of the nucleon 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.

####  5.2.3 Mass-Radius profile

Figure 5.2: 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. [17] and Antoniadis et al. [16]. The old NICER results are
shown with two green boxes from two different analyses [20, 21]. The new NICER
result is shown for canonical radius (blue horizontal line) given by Miller et
al. [22]. Reed et al. radius range is also shown with a dark red line for
canonical star [149].

We calculate the mass-radius profile of the DM admixed hyperon star by solving
the TOV equations (see Eq. 4.12) and shown in Fig. 5.2. 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 than others, its maximum mass is higher than the
other three sets. We put maximum mass constraints measured from the different
pulsars such as PSR J0740+6620 [17], and PSR J0348+0432 [16]. 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) [17] and
Antoniadis et al. ($2.01\pm 0.04\ M_{\odot}$) [16]. 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.) [20, 21]. We
called it old NICER data. We put the old NICER data with two green boxes from
two different analyses in the same figure [20, 21]. The predicted radius by
all the EOSs is compatible with old NICER data, as shown in Fig. 5.2. On the
left side of Fig. 5.2, we noticed that IOPB-I and IOPB-I+DM3 could able to
reproduce the maximum mass constraints. The maximum mass and radius
corresponding 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. Hence, these models satisfy the old
NICER data. Recently, Miller et al. put another radius constraint 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 [22]. We depicted the new NICER data as a horizontal blue
line in Fig. 5.2. 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 [148].
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 limited relativistic mean-field forces (using PREX-2 data) [149].
These $J$ and $L$ values are larger in magnitude than the old PREX data
(PREX-I). The latest $J$ and $L$ can be reproduced mostly from the stiff EOS.
That means the PREX-2 result allows a stiff EOS. Reed et al. also predicted
the radius for canonical NS is $13.25<R_{1.4}<14.26$ km. In this case, only
NL3 satisfies the Reed et al. data as shown in Fig. 5.2. 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 [22]. The improved radius estimate for
a 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.

###  5.3 $f$-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 ($g$) [78, 286, 283]. 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 [287] 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 [288]. To
solve the non-radial oscillations, one can also use the Cowling approximation,
which is simpler than the Lindblom and Detweiler methods. In Cowling
approximations, the space perturbations are neglected [289]. 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 [290, 291, 286, 283]

$\displaystyle\frac{dW(r)}{dr}$ $\displaystyle=$ $\displaystyle\frac{d{\cal
E}}{dP}\left[\omega^{2}r^{2}e^{\Lambda(r)-2\Phi(r)}V(r)+\frac{d\Phi(r)}{dr}W(r)\right]-l(l+1)e^{\Lambda(r)}V(r)$
$\displaystyle\frac{dV(r)}{dr}$ $\displaystyle=$ $\displaystyle
2\frac{d\Phi(r)}{dr}V(r)-\frac{1}{r^{2}}e^{\Lambda(r)}W(r).$ (5.11)

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

$\displaystyle\eta=\frac{1}{r^{2}}\Big{(}e^{-\Lambda(r)}W(r),-V(r)\partial_{\theta},-\frac{V(r)}{\sin^{2}{\theta}}\
\partial_{\phi}\Big{)}Y_{lm},$ (5.12)

where $Y_{lm}$ is the spherical harmonic, which is the function of $\theta$
and $\phi$. Solution of Eq. (5.11) with the fixed background metric in Eq.
(4.3) near origin will behave as:

$W(r)=Br^{l+1},\ V(r)=-\frac{B}{l}r^{l},$ (5.13)

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

$\omega^{2}e^{\Lambda(R)-2\Phi(R)}V(R)+\frac{1}{R^{2}}\frac{d\Phi(r)}{dr}\Big{|}_{r=R}W(R)=0.$
(5.14)

After solving Eq. (5.11) with two boundary conditions (Eqs. (5.13) & (5.14)),
one can get eigenfrequencies of the NS. In this work, we solve the
oscillations equations Eq. (5.11) by using the shooting method with some
initial guess for $\omega^{2}$. The equations are integrated from the center
to the surface and try to match the surface boundary conditions. After each
integration, the initial guess of $\omega^{2}$ is corrected through the Secant
method to get the desired precision, improving the initial guess.

####  5.3.1 Calculation of $f$-mode frequency as functions of different
observables

The $f$-mode frequencies (only for $l=2$)for different RMF EOSs are shown in
Fig. 5.3. 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 as shown in the lower panel of Fig. 5.3.

Figure 5.3: 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 the middle one but for
FSUGarnet. Figure 5.4: Upper: $f$-mode frequencies are shown for four
different RMF parameter sets as a function of compactness. Lower: $f$-mode
frequencies for IOPB-I set with DM3/hyperons and hyperons+DM3.

The $f$-mode frequency corresponds to maximum mass ($f_{\rm max}$) for the
four EOSs are shown in the upper panel of Fig. 5.3. The $f_{\rm max}$ are
2.16, 2.32, 2.38, and 2.55 kHz for NL3, IOPB-I, FSUGarnet, and G3,
respectively. The magnitude of $f_{\rm max}$ is maximum for G3 and minimum for
NL3, which are in-between for IOPB-I and FSUGarnet sets. This is because G3 is
a soft EOS, and NL3 is the stiffest EOS. In the middle panel of Fig. 5.3, we
observe a minor 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. 5.3. 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 in the dense region of the NS, which
generally occurs in the central region. It is well known that the EOS is
model-dependent, mainly in the core region. Also, the appearance of
hyperons/DM is not possible in the lower-density region. Therefore, the change
in $f-$mode frequency due to hyperon/DM particle is seen mainly at the core
part rather than the crust.

The $f$-mode frequencies differ considerably with force parametrizations. The
values of $f_{\rm max}$ 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 than the original IOPB-I
EOS. Therefore, the $f_{\rm max}$ value is more for IOPB-I+Y+DM3 in comparison
to others. We tabulate the value of $f_{\rm max}$ and $f_{1.4}$ for four
different RMF forces along with different NS observables in Table 5.3. The
mass variation of frequency changes within interval $1.75-2.55$ kHz for $l=2$,
which is almost consistent with Pradhan et al. [283] for the considered
parameter sets. For hyperons/DM3 and hyperons+DM3 cases, the frequency range
is within the interval of $1.8-2.85$ kHz. This range is higher in magnitude as
compared without hyperons/DM cases.

The compactness of a star is defined as ($C=M/R$), where $M$ and $R$ are the
mass and radius of the star [136, 292]. Like $C$, another significant quantity
is surface red-shift $Z_{s}$ is defined as [136]

$Z_{s}=\frac{1}{\sqrt{1-\frac{2M}{R}}}-1=\frac{1}{\sqrt{1-2C}}-1.$ (5.15)

If we find the value of $Z_{s}$, one can constrain the mass and radius of the
star. Till now, only one value of $Z_{s}=0.35$ is reported in the Ref. [293]
from the analysis of stacked bursts in the low-mass x-ray binary EXO 0748-676,
which is also discarded by the subsequent observation [294].

We calculate $C$ and $Z_{s}$, which are shown in Figs. 5.4 and 5.5 with
$f$-mode frequencies. We observed that the variation of $f$-mode frequencies
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 in Table 5.3. In our case, we
find the range of $C_{\rm max}$ and $Z_{s_{\rm max}}$ are $0.22-0.31$ and
$0.33-0.62$, respectively, for the considered EOSs. The values of $C_{\rm
max}$ and $Z_{s_{\rm max}}$ decrease 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 5.2) from previous calculation [79, 295,
283] including our fitting results.

Figure 5.5: Same as Fig. 5.4, but for $f$ as a function of $Z_{s}$.

##### Fitting formula:-

Andersson and Kokkotas (AK) first calculated the relation between $f$-mode
frequency as a function of average density [78]. They got an empirical
relation by taking the polytropic EOSs as follows:

$f(\mathrm{kHz})\approx 0.17+2.30\sqrt{\left(\frac{M}{1.4\
M_{\odot}}\right)\left(\frac{10\ \mathrm{km}}{R}\right)^{3}}.$ (5.16)

This empirical relation is again modified by their subsequent paper for the
realistic EOSs [79], which is given as

$f(\mathrm{kHz})\approx 0.78+1.635\sqrt{\frac{\bar{M}}{\bar{R^{3}}}}.$ (5.17)

Figure 5.6: Upper: $f$-mode frequency as a function of average density for
four parameter sets. The dashed lines with different colors 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 5.7:
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 a yellow line, IOPB-I+DM3 with a blue dashed line, IOPB-I+Y with a purple
dashed line, IOPB-I+Y+DM with a cyan dashed line.

Later, these empirical relations were modified by Benhar, Ferrari, and
Gualtieri (BFG) [295] by using hybrid EOSs. Pradhan and Chatterjee (PC) [283]
have put an empirical relation using hyperonic EOSs. We tabulate the
coefficients of the empirical relations from different analyses in Table 5.2.
We also find the 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 5.2 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 constrain
the NS EOS [78]. Several approaches have already tried to find the universal
relations between $f$-mode frequency and compactness/average density [78, 79,
295, 296, 297, 298]. However, in this study, our aim is to explore those
relations for DM admixed hyperon stars.

Table 5.2: 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 coefficients.

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

##### Universal relations:-

Different correlations have been seen between different modes, such as $f$,
$p$, and $g$, with compactness/average density. All these relations are
relatively independent of the EOSs. In Ref. [290], 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 a function of compactness for $p$ and $w$ modes have
been proposed in Ref. [299] and for $f$-modes see Ref. [298]. Thus, we have
checked these universal relations as a function of compactness, as shown in
Fig. 5.7 and Fig. 5.8. We find a universal relation between $\omega M$ and
$M/R$. However, the correlation is weaker in the case of $\omega R$ with
$M/R$. The Universal relations are given as follows:

$\displaystyle\omega M(\mathrm{kHz\ km})$ $\displaystyle=$ $\displaystyle
a_{i}\left(\frac{M}{R}\right)-b_{i},$ (5.18) $\displaystyle\omega
R(\mathrm{kHz\ km})$ $\displaystyle=$ $\displaystyle
c_{j}\left(\frac{M}{R}\right)+d_{j},$ (5.19)

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 $\omega R\sim M/R$.

Figure 5.8: Same as Fig. 5.7, but for $\omega R$ with $M/R$. Figure 5.9:
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 limited the tidal
deformability of the canonical NS, which is used to constrain the EOS of
neutron-rich matter at 2–3 times the nuclear saturation densities [23, 24].
Due to a strong dependence on the tidal deformability with radius
($\Lambda\sim R^{5}$), it can put stringent constraints on the EOS. Several
approaches [25, 26, 27, 28, 29, 30, 31, 32, 33] have been tried to constraint
the EOS on the tidal deformability bound given by the GW170817. The
dimensionless tidal deformability $\Lambda$ is defined as [260, 300, 301]

$\displaystyle\Lambda=\frac{2}{3}k_{2}\ C^{-5},$ (5.20)

where $k_{2}$ is the second Love number, which depends on the internal
structure and mass and radius of a star [260, 300, 261, 134].

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 5.3 for different EOSs. The GW170817 event put constraint on
$\Lambda_{1.4}=190_{-120}^{+390}$ [24]. In Ref. [298], they have put a limit
on the value of $f$-mode frequencies for $1.4M_{\odot}$ 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. [298], except
for the case of IOPB-I+Y+DM3. More NS mergers are expected to be measured in
the future, which may constrain the $f$-mode frequency more tightly.

Table 5.3: The observables such as $M_{\rm max}$ ($M_{\odot}$), $R_{\rm max}$
(km), $R_{1.4}$ (km), $f_{\rm max}$ (kHz), $f_{1.4}$ (kHz), $C_{\rm max}$,
$C_{1.4}$, $Z_{s_{\rm max}}$, $Z_{s_{1.4}}$, $\Lambda_{\rm max}$,
$\Lambda_{1.4}$ 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
---|---|---|---|---|---|---|---
$M_{\rm max}$ | 2.77 | 1.99 | 2.07 | 2.15 | 2.05 | 1.70 | 1.61
$R_{\rm max}$ | 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
$f_{\rm max}$ | 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
$C_{\rm max}$ | 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
$Z_{s_{\rm max}}$ | 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
$\Lambda_{\rm max}$ | 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 5.10: $f_{\rm max}$ are shown with different DM momenta for IOPB-I
parameter sets. The color scheme represents the maximum masses corresponding
to different $k_{f}^{\rm DM}$.  Figure 5.11: Correlation between $f_{1.4}$ and
$\Lambda_{1.4}$ are shown for 41 parameter sets, including (RMF, SHF, and
DDRMF). The color bar represents the maximum masses of the corresponding
parameter sets. Figure 5.12: Same as Fig. 5.11, but for $f_{\rm max}$ and
$\Lambda_{1.4}$.

In Fig. 5.10, we plot the values of $f_{\rm max}$ with different DM Fermi
momenta. The $f$-mode frequencies increase with increasing $k_{f}^{\rm DM}$.
The range of $f_{\rm max}$ is 2.3–2.4 kHz corresponds to $k_{f}^{\rm
DM}=0.00-0.025$ GeV; after that, it increases with more proportion beyond
$k_{f}^{\rm DM}=0.03$ GeV. The color bar represents the maximum masses
($M_{\rm max}$) of the NS for different $k_{f}^{\rm DM}$. The area of $M_{\rm
max}$ at lower $k_{f}^{\rm DM}$ (green color) is found to be more as compared
to the higher $k_{f}^{\rm DM}$, and the area slightly reduces from lower
$k_{f}^{\rm DM}$ to a higher one. This is because the slight increase in DM
percentage reduces the value $M_{\rm max}$ a little bit. When we increase the
DM percentage by more than 0.03 GeV, the maximum mass reduces significantly.
For example, for $k_{f}^{\rm DM}=$ 0.00, 0.01, and 0.02 GeV, the values of
$M_{\rm max}$ are 2.149, 2.146, and 2.119 $M_{\odot}$ respectively. If we
increase the DM Fermi momenta $k_{f}^{\rm DM}=$ 0.03, 0.04, 0.05, and 0.06
GeV, the values of $M_{\rm max}$ are 2.051, 1.938, 1.788, and 1.614
$M_{\odot}$ respectively.

####  5.3.2 Correlations between $f$-mode and tidal deformability

We also find some correlations between $f_{1.4}$–$\Lambda_{1.4}$, and $f_{\rm
max}$–$\Lambda_{1.4}$ which are shown in Figs. 5.11 and 5.12. This correlation
is first calculated by Wen et al. [298] 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. [302] having mass more than $\sim 2\
M_{\odot}$. We get a correlation between $f_{1.4}$–$\Lambda_{1.4}$ and $f_{\rm
max}$–$\Lambda_{1.4}$ 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 future GW observation may
answer this type of correlation.

### Conclusion

In this chapter, we have calculated the $f$-mode oscillations of the DM
admixed hyperon stars within the relativistic Cowling approximations. Hyperons
are appeared inside the NS, mainly at the core part. To compute the hyperon
interactions mediated by the mesons, we used SU(6) model to obtain the
coupling between hyperons-vector mesons. However, the coupling between
hyperons-scalar mesons has been obtained by fitting with the depth of the
hyperon potential. For DM admixed hyperons stars, the DM particles interact
with nucleons as well as hyperons by exchanging the Higgs. The hyperons-Higgs
form factor is assumed to be the same as nucleons, and the available
theoretical data constrain its value.

The $f$-mode frequencies of the DM admixed hyperon star for various EOSs are
computed. The magnitude of $f$-mode frequencies decreases for softer EOSs
because it predicts smaller mass as compared to stiffer ones. The angular
frequencies scaled by mass and radius are also calculated as the function of
compactness. It has been found that both quantities follow a linear
relationship. The variation of $f$-mode frequencies as a function of tidal
deformability is calculated for a large variety of EOSs. We obtained the
correlations between $f_{1.4}–\Lambda_{1.4}$ and $f_{\rm max}–\Lambda_{\rm
max}$ which are almost consistent with the earlier prediction [298]. The
predicted range of $f_{1.4}$ is $1.78-2.22$ kHz for the considered EOSs. We
hope the discovery of more BNS merger events will open up tight constraints on
the $\Lambda$ in the future, which will put more stringent constraints on
$f$-mode frequency.

##  Chapter 6 Effects of dark matter on binary neutron star properties

Neutron stars (NSs) in binary systems can interact through their gravitational
fields, resulting in tidal deformability, a measure of the degree to which the
tidal forces distort the shape of the NS. In this study, we calculate the
tidal Love numbers and corresponding deformabilities for NSs containing dark
matter (DM) for different levels of deformation. During the late inspiral
stage of the binary, the NSs emit large amounts of gravitational waves as the
separation distance decreases. Using the post-Newtonian formalism, we compute
various inspiral properties such as gravitational wave frequency, inspiral
phase, last time in the inspiral stage, strain amplitudes, and post-Newtonian
parameters. With the detection of gravitational waves from the merger of
binary NS in 2017 (GW170817 event), this study offers theoretical insights
into exploring the properties of binary NS, especially if they contain some
amount of DM.

###  6.1 Binary Neutron Star Properties

####  6.1.1 Tidal Love number and deformability

In a BNS system, tidal interactions come into the picture when the two stars
are in close orbit with each other. The mutual gravitational interaction
increases with time because the inspiralling stars emit gravitational
radiation due to their acceleration in the gravitational field. The shape of
the star becomes increasingly tidally deformed. Analogous to electromagnetic
radiation, here also, the gravitational field is generated from the two types
of tidal fields, i.e., (i) electric (even parity or polar) ${\cal E}_{L}$ and
(ii) magnetic (odd parity or axial) ${\cal M}_{L}$ fields, where $L$
represents the space indices. These two fields induce a mass multipole moment
($Q_{L}$) and a current multipole moment ($S_{L}$), which are defined as [303,
304]:

$\displaystyle Q_{L}=\lambda_{l}{\cal E}_{L},$ (6.1) $\displaystyle
S_{L}=\sigma_{l}{\cal M}_{L},$ (6.2)

where $\lambda_{l}$ and $\sigma_{l}$ are the gravitoelectric and
gravitomagnetic tidal deformability of order $l$. The mass and current
multipole moments come into the picture in general relativity compared to
Newtonian dynamics, where only the mass multipole moment plays a role. The
gravitoelectric ($k_{l}$) and gravitomagnetic Love numbers ($j_{l}$) are
related with the deformability parameters $\lambda_{l}$ and $\sigma_{l}$,
respectively, as [304]:

$\displaystyle k_{l}=\frac{(2l-1)!!}{2}\frac{\lambda_{l}}{R^{2l+1}},$ (6.3)
$\displaystyle j_{l}=4(2l-1)!!\frac{\sigma_{l}}{R^{2l+1}}.$ (6.4)

The dimensionless tidal deformability parameters correspond to electric
$\Lambda_{l}$ and magnetic $\Sigma_{l}$ multipole moments, which help express
the tidal deformability of the BNS. These quantities are often used
conveniently in various measurements and related to the Love numbers and
compactness of the star $C$ with the relation [304]:

$\displaystyle\Lambda_{l}=\frac{\lambda_{l}}{M^{2l+1}}=\frac{2}{(2l-1)!!}\frac{k_{l}}{C^{2l+1}}$
(6.5)
$\displaystyle\Sigma_{l}=\frac{\sigma_{l}}{M^{2l+1}}=\frac{1}{4(2l-1)!!}\frac{j_{l}}{C^{2l+1}}.$
(6.6)

As discussed earlier in this paper, the leading order perturbation of the
tidal deformations is the gravitoelectric and gravitomagnetic quadrupole
deformations $\lambda_{2}$ and $\sigma_{2}$. However, the higher-order
deformations are also quite substantial due to the close orbital separation of
the binary system and at the merging condition. All these tidal deformations
can be expressed with the various Love numbers $k_{l}$ and $j_{l}$ as defined
in Eqs. (6.3) and (6.4). To calculate these tidal Love numbers along with the
dimensionless tidal deformability $\Lambda_{l}$ and $\Sigma_{l}$, we adopt the
formalism developed in Ref. [304]. The gravitoelectric perturbations are
evaluated from the differential equation in terms of metric functions ($H$),
which are written as:

$\displaystyle H_{\ell}^{\prime\prime}(r)$
$\displaystyle+H_{\ell}^{\prime}(r)\biggl{[}1-\frac{2m(r)}{r}\biggr{]}^{-1}\Bigg{\\{}\frac{2}{r}-\frac{2m(r)}{r^{2}}-4\pi
r\left[{\cal E}(r)-P(r)\right]\Bigg{\\}}$
$\displaystyle+H_{\ell}(r)\biggl{[}1-\frac{2m(r)}{r}\biggr{]}^{-1}\times\Bigg{\\{}4\pi\bigg{[}5{\cal
E}(r)+9P(r)+\dfrac{d{\cal E}}{dP}\left[{\cal E}(r)+P(r)\right]\bigg{]}$
$\displaystyle-\frac{\ell(\ell+1)}{r^{2}}-4\biggl{[}1-\frac{2m(r)}{r}\biggr{]}^{-1}\biggl{[}\frac{m(r)}{r^{2}}+4\pi\,r\,P(r)\biggr{]}^{2}\Bigg{\\}}=0,\;$
(6.7)

and the gravitomagnetic perturbations are obtained by solving the differential
alignment, which is given as:

$\displaystyle\widetilde{H}_{\ell}^{\prime\prime}(r)$
$\displaystyle-\widetilde{H}_{\ell}^{\prime}(r)\biggl{[}1-\frac{2m(r)}{r}\biggr{]}^{-1}\times
4\pi r\left[P(r)+{\cal E}(r)\right]$
$\displaystyle-\widetilde{H}_{\ell}(r)\biggl{[}1-\frac{2m(r)}{r}\biggr{]}^{-1}\times\left\\{\frac{\ell(\ell+1)}{r^{2}}-\frac{4m(r)}{r^{3}}+8\pi\theta\left[P(r)+{\cal
E}(r)\right]\right\\}=0,$ (6.8) $\displaystyle
y(R)=\frac{R\beta(R)}{H_{\ell}(R)},\ {\rm and}\
\widetilde{y}(R)=\frac{R\widetilde{\beta}(R)}{\widetilde{H}_{\ell}(R)}.$ (6.9)

The gravitoelectric Love numbers $k_{2}$, $k_{3}$ and $k_{4}$ are given as
[303, 260, 301, 118, 304]:

$\displaystyle k_{2}$
$\displaystyle=\frac{8}{5}C^{5}(1-2C)^{2}\big{[}2(y_{2}-1)C-y_{2}+2\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}4(y_{2}+1)C^{4}+2(3y_{2}-2)C^{3}-2(11y_{2}-13)C^{2}+3(5y_{2}-8)C$
$\displaystyle-3(y_{2}-2)\big{]}+3(1-2C)^{2}\big{[}2(y_{2}-1)C-y_{2}+2\big{]}\log(1-2C)\Big{\\}}^{-1}\,,$
(6.10) $\displaystyle k_{3}$
$\displaystyle=\frac{8}{7}C^{7}(1-2C)^{2}\big{[}2(y_{3}-1)C^{2}-3(y_{3}-2)C+y_{3}-3\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}4(y_{3}+1)C^{5}+2(9y_{3}-2)C^{4}-20(7y_{3}-9)C^{3}+5(37y_{3}-72)C^{2}$
$\displaystyle-45(2y_{3}-5)C+15(y_{3}-3)\big{]}+15(1-2C)^{2}\big{[}2(y_{3}-1)C^{2}-3(y_{3}-2)C$
$\displaystyle+y_{3}-3\big{]}\log(1-2C)\Big{\\}}^{-1}\,,$ (6.11)

and

$\displaystyle k_{4}$
$\displaystyle=\frac{32}{147}C^{9}(1-2C)^{2}\big{[}12(y_{4}-1)C^{3}-34(y_{4}-2)C^{2}+28(y_{4}-3)C-7(y_{4}-4)\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}8(y_{4}+1)C^{6}+4(17y_{4}-2)C^{5}-12(83y_{4}-107)C^{4}+40(55y_{4}-116)C^{3}$
$\displaystyle-10(191y_{4}-536)C^{2}+105(7y_{4}-24)C-105(y_{4}-4)\big{]}$
$\displaystyle+15(1-2C)^{2}\times\big{[}12(y_{4}-1)C^{3}-34(y_{4}-2)C^{2}+28(y_{4}-3)C$
$\displaystyle-7(y_{4}-4)\big{]}\log(1-2C)\Big{\\}}^{-1}.\,$ (6.12)

Similarly, the gravitomagnetic Love numbers $j_{2}$, $j_{3}$ and $j_{4}$ are
given as [304]:

$\displaystyle j_{2}$
$\displaystyle=\frac{24}{5}C^{5}\big{[}2(\widetilde{y}_{2}-2)C-\widetilde{y}_{2}+3\big{]}\times\Big{\\{}2C\big{[}2(\widetilde{y}_{2}+1)C^{3}+2\widetilde{y}_{2}C^{2}+3(\widetilde{y}_{2}-1)C$
$\displaystyle-3(\widetilde{y}_{2}-3)\big{]}+3\big{[}2(\widetilde{y}_{2}-2)C-\widetilde{y}_{2}+3\big{]}\log(1-2C)\Big{\\}}^{-1}\,,$
(6.13) $\displaystyle j_{3}$
$\displaystyle=\,\frac{64}{21}C^{7}\big{[}8(\widetilde{y}_{3}-2)C^{2}-10(\widetilde{y}_{3}-3)C+3(\widetilde{y}_{3}-4)\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}4(\widetilde{y}_{3}+1)C^{4}+10\widetilde{y}_{3}C^{3}+30(\widetilde{y}_{3}-1)C^{2}-15(7\widetilde{y}_{3}-18)C+45(\widetilde{y}_{3}-4)\big{]}$
$\displaystyle+15\big{[}8(\widetilde{y}_{3}-2)C^{2}-10(\widetilde{y}_{3}-3)C+3(\widetilde{y}_{3}-4)\big{]}\log(1-2C)\Big{\\}}^{-1}\,,$
(6.14) $\displaystyle j_{4}$
$\displaystyle=\frac{80}{147}C^{9}\big{[}40(\widetilde{y}_{4}-2)C^{3}-90(\widetilde{y}_{4}-3)C^{2}+63(\widetilde{y}_{4}-4)C-14(\widetilde{y}_{4}-5)\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}4(\widetilde{y}_{4}+1)C^{5}+18\widetilde{y}_{4}C^{4}+90(\widetilde{y}_{4}-1)C^{3}-5(137\widetilde{y}_{4}-334)C^{2}$
$\displaystyle+105(7\widetilde{y}_{4}-26)C-210(\widetilde{y}_{4}-5)\big{]}+15\big{[}40(\widetilde{y}_{4}-2)C^{3}-90(\widetilde{y}_{4}-3)C^{2}$
$\displaystyle+63(\widetilde{y}_{4}-4)C-14(\widetilde{y}_{4}-5)\big{]}\log(1-2C)\Big{\\}}^{-1}\,.$
(6.15)

The dimensionless tidal deformability $\Lambda_{l}$ and $\Sigma_{l}$ are
calculated using Eq. (6.5) and (6.6) for different values of $l$. The Love
numbers play a major role in the inspiral phases of the BNS merger. The
dominant contribution comes mainly from the $l=2$ mode compared to higher
order polarities, which will be discussed in sub-section 6.2.2.

####  6.1.2 Shape/Surficial Love Numbers

The surficial Love numbers $h_{l}$ are associated with the surface deformation
of a body. Damour and Nagar gave a relativistic theory of $h_{l}$ of a body in
the influence of tidal forces [303]. It is further extended in Ref. [305] to
understand it in terms of a deformed curvature of the body’s surface. For a
perfect fluid, the surficial Love numbers are related with $k_{l}$ as
$h_{l}=1+2k_{l}$ in Newtonian case [303, 305]. We calculate the $h_{l}$ in the
limit of a perfect fluid using the relation given in [303, 305, 261]:

$\displaystyle h_{\ell}=\Gamma_{1}+2\Gamma_{2}\,k_{\ell},$ (6.16)

where $\Gamma_{1}$ and $\Gamma_{2}$ are defined as

$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{\ell+1}{\ell-1}(1-M/R)F(-\ell,-\ell;-2\ell;2M/R)-\frac{2}{\ell-1}F(-\ell,-\ell-1;-2\ell;2M/R),$
(6.17) $\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\ell}{\ell+2}(1-M/R)F(\ell+1,\ell+1;2\ell+2;2M/R)+\frac{2}{\ell+2}F(\ell+1,\ell;2\ell+2;2M/R),$
(6.18)

The $F(a,b;c;z)$ is the hypergeometric function and $\Gamma_{1}$ and
$\Gamma_{2}$ can be approximated as follow for $C<<1$ [305]:

$\displaystyle\Gamma_{1}$
$\displaystyle=1-(\ell+1)(M/R)+\frac{\ell(\ell+1)(\ell^{2}-2\ell+2)}{(\ell-1)(2\ell-1)}(M/R)^{2}+\cdots,$
(6.19) $\displaystyle\Gamma_{2}$
$\displaystyle=1+\ell(M/R)+\frac{\ell(\ell+1)(\ell^{2}+4\ell+5)}{(\ell+2)(2\ell+3)}(M/R)^{2}+\cdots.$
(6.20)

When compactness goes to zero, the surficial Love number attains the Newtonian
limit $h_{l}=1+2k_{l}$. In the case of a non-rotating black hole, the
surficial Love numbers also play a crucial role, as discussed in Refs. [306,
303, 305]. The surficial Love number is independent of the star’s internal
structure, which is a function of the mass and radius, i.e., the compactness
of the body.

####  6.1.3 Tidal effects on the inspiral phase of BNS

During different phases of BNS systems, including the inspiral, merger, and
post-merger phases, GWs are emitted. The tidal interaction between the two NSs
during the inspiral phase significantly affects the waveforms of the GWs. This
study explores the effects of DM on the inspiral properties of BNS and
calculates the tidal corrections to GWs waveforms for DM-admixed BNS,
including the higher-order tidal properties. The effects of DM on the inspiral
properties mainly depend on (i) the type of DM, (ii) the percentage of DM
present in the star, and (iii) the lifetime of the BNS evolution. We also
calculate the effects of tidal corrections to GWs waveforms for DM admixed BNS
in the last part of this chapter.

To calculate the effects of DM on the inspiral/emitted properties of GWs, we
use the PN formalism [307, 308, 309, 310, 311, 312]. The detailed methods can
be found in our previous study [130]. We solve the energy balance equation
with energy and luminosity up to 3 and 3.5 PN order, respectively. From the
energy balance equation, the PN parameter ($x$) is evaluated, which is again
used for amplitude, and phase calculations as in Ref. [130]. The
gravitoelectric Love numbers $k_{l}$ lead to a correction of order ($2l+1$)PN
to the phase of the gravitational-wave signal [313, 304]:

$\displaystyle\Psi_{\ell}$
$\displaystyle=-\sum_{i=1}^{2}\Biggl{[}\frac{5}{16}\frac{(2\ell-1)!!(4\ell+3)(\ell+1)}{(4\ell-3)(2\ell-3)}\Lambda_{\ell,i}X_{i}^{2\ell-1}x^{2\ell-3/2}+\frac{9}{16}\delta_{\ell
2}\Lambda_{2,i}\frac{X_{i}^{4}}{\eta}x^{5/2}\Biggr{]}$
$\displaystyle+\mathcal{O}(x^{2\ell-1/2})\,,$ (6.21)

with $X_{i}=M_{i}/M$ and $\eta=M_{1}M_{2}/M^{2}$. The
$\delta_{\ell\ell^{\prime}}$ is the Kronecker delta, and $i=1,2$ is used to
distinguish the two stars of the binary system.

###  6.2 Results and Discussions

This section presents our calculated results for the NS’s EOS, mass, radius,
various Love numbers, and deformabilities in Figs. 6.1–6.6. The tidal
deformability is expressed in the gravitoelectric and gravitomagnetic Love
numbers, and the surficial Love numbers describe the surface deformation of
the star. The detailed discussions are outlined in the subsequent subsections.

####  6.2.1 Equation of State and Mass-Radius Relations

The IOPB-I unified EOS (IOPB-I-U), represented as DM0, and two DM admixed
EOSs, such as DM3 and DM5 (DM0, DM3, and DM5 represent the DM Fermi momenta
0.00, 0.03 and 0.05 GeV, respectively.), are shown on the left side of Fig.
6.1. At the lower density region, all three EOSs coincide into a single line
representing the crust part, as shown in the figure. This means the crust EOS
is the same for all three cases. With the addition of DM, the EOSs become
softer, which is clearly seen in the figure. The softening of the EOS depends
on the amount of DM contained inside the NS. Therefore, EOS with 0.05 GeV DM
momentum is the softest compared to the other two cases, DM3 and DM0.

Figure 6.1: Left: EOSs for unified IOPB-I (IOPB-I-U) with different DM
percentages. The red line represents the IOPB-I crust calculated in Ref.
[314]. Right: $M-R$ profiles for DM admixed NS. Different massive pulsars
constraints, such as MSP J0740+6620 by Cromartie et al. [17] and PSR
J0348+0432 by Antoniadis et al. [16] are overlaid with different color bars.
Both old NICER [20, 21] constraints for canonical stars and new NICER limits
for both 1.40 and 2.08 $M_{\odot}$ are also depicted [22].

On the right side of Fig. 6.1, we depict the mass $M$ and radius $R$ of the DM
admixed NS with the IOPB-I-U equation of state. The IOPB-I-U predicts a mass
of $M=2.149\ M_{\odot}$ and radius $R=11.748$ km. Both the mass and radius are
well within the known observational constraints, as shown in the figure. With
the addition of DM, the $M$ and $R$ of the NS decreases as the presence of DM
softens the EOSs. We noticed a certain range for DM momentum that produces
mass–radius values compatible with the latest massive pulsar and NICER limit.
Hence, from the observational data, such as advanced LIGO and NICER, one can
fix the percentage of DM inside the NS. Theoretically, by generating many such
EOSs and using Bayesian analysis, one could also fix the DM percentage inside
the NS.

####  6.2.2 Electric Love Number and Tidal Deformability

Figure 6.2: Gravitoelectric tidal Love numbers for IOPB-I-U EOS with DM Fermi
momenta 0.00, 0.03 and 0.05 GeV with $l=2,3,{\rm and}\ 4$.

The gravitoelectric Love numbers with different values of $l$ are calculated
using Eqs. (6.10)–(6.12), and the results are shown in Fig. 6.2. As expected,
the value of $k_{2}$ is more pronounced than the $k_{3}$ and $k_{4}$. The
effects of crust on the $k_{l}$ are more significant than the EOS without
considering the crust, as observed in Ref. [315]. Hence, we take the unified
EOSs to calculate the $k_{l}$. The effects of DM on the gravitoelectric Love
numbers are also shown in the figure. The maximum value of $k_{l}$ increases
with DM percentage and the differences are seen mainly in the lower mass
regions, where the crust plays an important role. Hence, choosing suitable
crust EOSs for Love numbers and tidal deformability calculations is crucial.

With the increase of $l$, the magnitude of $k_{l}$ decreases more profoundly.
However, the impacts of DM on $k_{3}$ and $k_{4}$ are non-negligible. It is
discussed in Ref. [304] that the symmetry energy is less important on the
crust than the stiffness of the EOS. We also found similar results for the DM
admixed NS, as the EOSs softened with adding DM.

The dimensionless tidal deformability $\Lambda_{l}$ was calculated for DM
admixed NS within the IOPB-I-U equation of state, and the results are shown in
Fig. 6.3. All the three EOSs, such as DM0, DM3, and DM5, well satisfy the
$\Lambda_{1.4}$ constraints for $l=2$. On the other hand, we found a shifting
on the $\Lambda_{l}$ with the increased $l$ values, which do not satisfy the
GW170817 limit except for higher DM momenta.

Figure 6.3: Same as Figure 6.2 but for dimensionless electric tidal
deformability. The purple colour error bar represents the constraint on
$\Lambda_{1.4}$ given by LIGO/Virgo [23, 24] from the BNS merger event
GW170817 with, $\Lambda_{1.4}=190_{-70}^{+390}$. The red colour error bar
represents the $\Lambda_{1.4}$ constraints from the GW190814 event under the
assumption of NSBH scenario with, $\Lambda_{1.4}=616_{-158}^{+273}$ [34].

The secondary component of the GW190814 event is very much in debate, “whether
it is a supermassive NS or the lightest black hole?”. Several attempts have
been made to explore this object [41, 129]. Some of the suggestions are as
follows: (i) It is a heavy NS with deconﬁned QCD core [42], (ii) a super-fast
pulsar [180], (iii) a binary black-hole merger [43, 134] and (iv) the DM
admixed NS with EOS is sufficiently stiff [129] etc. If this event is the
merger of the NS and black hole (NSBH), then the tidal deformability of the NS
for the canonical star is $\Lambda_{1.4}=616_{-158}^{+273}$.

Hence, one can impose a constraint on the NS mass, radius, and tidal
deformability from both GW170817 and GW190814 data. We noticed that using
IOPB-I EOS, DM0, and DM3 satisfies the values of $\Lambda_{l}$ given by
GW190814 for the lowest multipole moment. For higher values of $l$, the values
of $\Lambda_{l}$ are shifted and do not satisfy the GW190814 limit.

####  6.2.3 Magnetic Love Number and Tidal Deformability

We calculate the gravitomagnetic Love numbers and tidal deformabilities for
$l=2,3,4$ using Eqs. (6.6) and (6.13)–(6.15), and the results are depicted in
Figs. 6.4 and 6.5. The magnitude of $j_{2}$ is more negative than both $j_{3}$
and $j_{4}$, i.e., with the addition of DM in the EOSs, the $j_{l}$ values
correspond to DM3 and DM5 decrease as compared to the DM0 case. Hence, the
effects of DM on the $j_{l}$ are significant. This also depends on the DM
contained inside the NS. The magnitudes of the gravitomagnetic tidal Love
numbers for both static and irrotational fluids are almost comparable;
however, their values are in opposite signs (as seen from the figures). These
tidal Love numbers also impinge less effect on the waveform of GWs, which
might be detected in the upcoming modern detectors.

Figure 6.4: Magnetic tidal Love numbers for IOPB-I EOS with DM Fermi momenta
0.00, 0.03 and 0.05 GeV with three different $l$ values for static fluid.
Figure 6.5: Same as Fig. 6.4 but for irrotational fluid.

The gravitomagnetic tidal deformability is shown in Fig. 6.6 for DM admixed
NS. The values of $\Sigma_{l}$ increase with DM momenta, contrary to the
$\Lambda_{l}$. However, this effect is less significant on the waveforms of
the emitted GWs since it depends on the value of $j_{l}$ compared to
$\Lambda_{l}$. There are also no significant changes with different values of
$l$ with the DM percentage. We also observed that the changes are seen mainly
in the core part of the NS, as the crust has less contribution to $j_{l}$ and
$\Sigma_{l}$. Hence, the crust part does not affect the gravitomagnetic Love
numbers and tidal deformabilities.

####  6.2.4 Surficial Love Numbers

The surficial Love numbers describe the surface deformation of a body in the
presence of a tidal field. These are expressed in terms of the compactness of
the NS in these calculations. The calculated surficial Love numbers $h_{l}$
with DM admixed NS are shown in Fig. 6.7 for different multipole moments $l$.
The pattern of the variation of $h_{l}$ is almost similar to the $k_{l}$, but
the magnitudes $h_{l}$ are higher compared to $k_{l}$. When the compactness
$C$$\rightarrow$$0$, all the values of $h_{l}$, i.e., $h_{2}$, $h_{3}$ and
$h_{4}$ approach $\sim{1}$, irrespective of the moments (see Fig. 6.7).

Figure 6.6: Same as Fig. 6.5 but for dimensionless magnetic tidal
deformability.

The value of $h_{l}$ increases with the percentage of DM contained in the NS.
Thus, the surface of the NS is more distorted for the DM5 case than in the
other two DM3 and DM0 conditions. A comparative analysis among the multipole
moments tells that the surface deformation is mainly due to the quadrupole
deformation of the star. However, the contributions of $h_{l}$ for $l=3$ and 4
are non-negligible. In all the cases of $l$, the maximum distortion of the
star takes place at around $C\sim{0.04}$.

Figure 6.7: Shape/surficial Love numbers for IOPB-I EOS with DM Fermi momenta
0.00, 0.03, and 0.05 GeV with three different $l$. Figure 6.8: Tidal
corrections as a function of gravitational frequency for IOPB-I EOS with DM
Fermi momenta 0.00, 0.03, and 0.05 GeV with three different $l$.

####  6.2.5 Tidal Effects on the GW Signal

The tidal deformations have significant effects on the gravitational wave in
the later stage of the inspiral phase. In this calculation, we study the
effects of DM on tidal corrections on the gravitational wave. The tidal
corrections are calculated using a PN approximation. The correction is only
related to NS observable through the tidal deformability parameter, which is
model-dependent. In our previous study [130], we calculated the inspiral
properties with a DM equation of state and observed that the tidal
deformability of the star decreased with increasing DM percentage, which
slowed down the merger dynamics. The effects of DM on the GW amplitude (such
as $h_{+}$, $h_{\times}$ and $h_{22}$), inspiral phase ($\Phi$), etc., are
calculated. Here, we calculate the tidal corrections ($\Psi_{l}$) and depict
these in Fig. 6.8 for different values of $l$ as a function of frequency
($f$). Similar to all previous cases, we observe that the leading order
contribution comes from the $l=2$ case. The magnitude of the tidal
contribution was found to be greater in the higher frequency region. Hence,
the corrections also accelerate the merger of the BNS for a higher massive
star than the less massive one.

The mass of the NS decreases with the addition of DM to the BNS. From our
calculated results (see Fig. 6.8), it is clear that the magnitude of
$\Psi_{l}$ decreases with the increasing percentage of DM. We can connect this
behavior with the decreasing nature of $\Lambda_{l}$, as $\Psi_{l}$ is related
to $\Lambda_{l}$ (see Eq. (6.21)). Thus, the merging time of the binary system
increases with the addition of DM in the NSs. Similar to other quantities, the
contribution to waveform $\Psi_{l}$ has less significance than the higher-
order deformation of the binary system. Thus, most of the GWs signal comes
from the quadrupole deformations.

###  6.3 Inspiral properties

####  6.3.1 Post-Newtonian expanded Taylor-T4

The PN formalism is a slow-motion, weakly stressed, and weak field
approximation to the general theory of relativity and is valid for any mass
ratio [307, 316]. In the PN method, the equations of motion are obtained by
systematically expanding the metric. Einstein’s field equations are expressed
in powers of the dimensionless parameter
$\epsilon=\sqrt{GM_{t}/c^{2}d}\sim\frac{v}{c}$, where $M_{t}$ is the total
mass of the system, $d$ is the distance between two NS, and $v$ is the
characteristic velocity of particles. The metric is then solved iteratively in
powers of $\epsilon$, and the equations of motion are evaluated from the
metric using the geodesic equation. An expansion containing terms up to
$\epsilon^{n}$ or equivalently $(1/c)^{n}$ is denoted an $\frac{n}{2}$PN
expansion [307, 308]. This calculation takes the energy and luminosity up to
3PN and 3.5PN order, respectively, for the in-spiraling binary in quasi-
circular orbits.

The BNS emit GWs in their inspiral phases. Because of the emission of the GWs
for a classical system, the BNS has to rotate in quasi-circular orbits. The
flux (luminosity) emitted by the system balances the rate of change of energy
with time in that orbit, and the energy balance equation is given as [308,
317, 318]:

${\cal{L}}=-\frac{dE}{dt}=-\frac{dE/dx}{dt/dx},$ (6.22)

where $E$ is the energy of the system, $t$ is the time taken and $x$ is the PN
parameter which is defined as [308, 317, 318]

$x=\Big{(}M_{t}\frac{d\Phi}{dt}\Big{)}^{2/3}=\Big{(}M_{t}\Omega\Big{)}^{2/3},$
(6.23)

where $M_{t}$ is the total mass of BNS systems, $\Phi$ is the orbital phase,
$\Omega=2\pi\omega$ is the orbital angular velocity ($\omega=f/2$ is the
orbital angular frequency), and $f$ is the frequency of the emitted GW. The
energy of the system calculated up to 3PN order in terms of $x$ is given as
[307, 317, 318]:

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{1}{2}M_{t}\nu
x\Bigg{\\{}1+\Bigg{(}-\frac{3}{4}-\frac{\nu}{12}\Bigg{)}x+\Bigg{(}-\frac{27}{8}+\frac{19\nu}{8}-\frac{\nu^{2}}{24}\Bigg{)}x^{2}$
(6.24)
$\displaystyle+\Bigg{[}-\frac{675}{64}+\Bigg{(}\frac{34445}{576}-\frac{205\pi^{2}}{96}\Bigg{)}\nu-\frac{155\nu^{2}}{96}-\frac{35\nu^{3}}{5184}\Bigg{]}x^{3}\Bigg{\\}},$

where $\nu=m_{1}m_{2}/M_{t}^{2}$ is the symmetric mass ratio, $m_{1}$ and
$m_{2}$ are the individual masses of the binary. The luminosity ${\cal{L}}$ is
calculated by the time derivative of Eq. (6.24) using PN expansions up to 3.5
order is given as [307, 317, 318]

$\displaystyle{\cal{L}}=$
$\displaystyle\frac{32}{5}\nu^{2}x^{5}\Bigg{\\{}1+\Bigg{(}-\frac{1247}{336}-\frac{35\nu}{12}\Bigg{)}x+4\pi
x^{3/2}+\Bigg{(}-\frac{44711}{9072}+\frac{9271\nu}{504}+\frac{65\nu^{2}}{18}\Bigg{)}x^{2}$
$\displaystyle+$
$\displaystyle\Bigg{(}-\frac{8191}{672}-\frac{583\nu}{24}\Bigg{)}\pi
x^{5/2}+\Bigg{[}\frac{6643739519}{69854400}+\frac{16\pi^{2}}{3}-\frac{1712\gamma_{E}}{015}-\frac{856}{105}\ln(16x)$
$\displaystyle+$
$\displaystyle\Bigg{(}-\frac{134543}{7776}+\frac{41\pi^{2}}{48}\Bigg{)}\nu-\frac{94403\nu^{2}}{3024}-\frac{775\nu^{3}}{324}\Bigg{]}x^{3}+\Bigg{(}-\frac{16285}{504}+\frac{214745\nu}{1728}$
$\displaystyle+$ $\displaystyle\frac{193385\nu^{2}}{3024}\Bigg{)}\pi
x^{7/2}\Bigg{\\}},$ (6.25)

where $\gamma_{E}\approx 0.5772$ is the Euler’s constant. The phase evolution
of the binary system can be calculated by solving the following equations:

$\displaystyle\frac{dx}{dt}=\frac{dE/dt}{dE/dx}=-\frac{{\cal{L}}}{dE/dx},$
(6.26) $\displaystyle\frac{d\Phi}{dt}=\frac{x^{3/2}}{M_{t}}.$ (6.27)

There are various methods to integrate this system of equations labeled as
TaylorT1-TaylorT4 [308, 319, 318]. In the TaylorT1 method, the Eqs. (6.24) and
(6.25) are inserted in Eq. (6.26), and then the integration is performed by
using the initial conditions $x_{0}=(M_{t}\Omega_{0})^{2/3}$ and $\Phi_{0}$.
In the TaylorT2 method, the equations are written by starting with a
parametric solution of energy balance equations. Then, each expression
integrand is re-expanded as a single PN parameter x and truncated at the
appropriate order [308, 318]

$\displaystyle t(x)=t_{0}+\int_{x}^{x_{0}}dx\frac{(dE/dx)}{{\cal{L}}},$ (6.28)
$\displaystyle\Phi(x)=\Phi_{0}+\int_{x}^{x_{0}}dx\frac{x^{3/2}}{M_{t}}\frac{(dE/dx)}{{\cal{L}}}.$
(6.29)

In the present calculations, we take the TaylorT4 approximation. In this
method, the right-hand side of the Eq. (6.28) is re-expanded as a single
series and truncated at 3.5PN order before doing the integration. After
calculating $x$, $t$, and $\Phi$, one can know the properties related to the
in-spiraling binaries for the point-particle system.

In the case of BNS, the tidal interaction comes into the picture, which has a
significant role in the inspiral properties. Therefore, we have to include the
extra part of tidal interactions with this point-particle approximation. The
system’s motion is tidally interacting, which influences its internal
structures. Also, the tidal interactions affect the evolution of the GW phase
by a parameter $\lambda$. Hence, the BNS system’s tidal contributions must be
added on the right-hand side of Eq. (6.26), which is modified as [310, 311,
312]:

$\frac{dx}{dt}=\frac{64\nu}{5M_{t}}x^{5}\\{F_{3.5}^{\mathrm{Taylor}}(x)+F^{\mathrm{Tidal}}(x)\\},$
(6.30)

where $F_{3.5}^{\mathrm{Taylor}}$is PN-expanded expression for point-mass
contribution using Taylor-T4 approximation, given by [309, 308, 310, 311, 312]

$\displaystyle F_{3.5}^{\mathrm{Taylor}}(x)=$ $\displaystyle
1-\Bigg{(}\frac{743}{336}+\frac{11}{4}\nu\Bigg{)}x+4\pi
x^{3/2}+\Bigg{(}\frac{34103}{18144}+\frac{13661}{2016}\nu+\frac{59}{18}\nu^{2}\Bigg{)}x^{2}$
$\displaystyle-$
$\displaystyle\Bigg{(}\frac{4159}{672}+\frac{189}{8}\nu\Bigg{)}\pi
x^{5/2}+\Bigg{[}\frac{16447322263}{139708800}-\frac{1712}{105}\gamma_{E}-\frac{56198689}{217728}\nu$
$\displaystyle+$
$\displaystyle\frac{541}{896}\nu^{2}-\frac{5605}{2592}\nu^{3}+\frac{\pi^{2}}{48}(256+451\nu)-\frac{856}{105}\ln(16x)\Bigg{]}x^{3}$
$\displaystyle+$
$\displaystyle\Bigg{(}-\frac{4415}{4032}+\frac{358675}{6048}\nu+\frac{91495}{1512}\nu^{2}\Bigg{)}\pi
x^{7/2},$ (6.31)

With the addition of the tidal part to the PN formalism, we adopt the method
developed by Vines et al. [320] up to 1PN accuracy, which is expressed as

$\displaystyle F^{\mathrm{Tidal}}(x)=$
$\displaystyle\frac{32\chi_{1}\lambda_{2}}{5M_{t}^{6}}\Big{[}12(1+11\chi_{1})x^{10}+\Big{(}\frac{4421}{28}-\frac{12263}{28}\chi_{2}+\frac{1893}{2}\chi_{2}^{2}-661\chi_{2}^{3}\Big{)}x^{11}$
$\displaystyle+$ $\displaystyle(1\leftrightarrow 2)\Big{]},$ (6.32)

where $\lambda_{2}$ is the tidal deformability of the second star and
$\chi_{1}$ is the mass fraction of the first star ($\chi_{1}=M_{1}/M_{t}$).
The numerical values of $k_{2}$ and $R$ are given in Table 6.1 for different
parameter sets with three DM fractions. The numerical values are different
from others because they are model dependent. Baiotti et al. [310] re-
expressed the Eq. (6.32) in terms of tidal Love number $k_{2}$ and compactness
parameter $C$ as

$F^{\mathrm{Tidal}}(x)=\sum_{I=1,2}F_{LO}(\chi_{I})x^{5}\big{(}1+F_{1}(\chi_{I})x\big{)},$
(6.33)

with

$F_{LO}(\chi_{I})=4\hat{k_{2}^{I}}\Bigg{(}\frac{12-11\chi_{I}}{\chi_{I}}\Bigg{)},$
(6.34)

and

$F_{1}(\chi_{I})=\frac{4421-12263\chi_{I}+26502\chi_{I}^{2}-18508\chi_{I}^{3}}{336(12-11\chi_{I})}.$
(6.35)

The expression for $\hat{k_{2}^{I}}$ is defined as:

$\hat{k_{2}^{I}}=k_{2}^{I}\Bigg{(}\frac{\chi_{I}}{C_{I}}\Bigg{)}^{5}\ \
I=1,2.$ (6.36)

For equal-mass binary, $\chi_{1}=\chi_{2}=\chi=1/2$ and $C=C_{1}=C_{2}$, the
Eq. (6.32) can be written as [311]

$F^{\mathrm{Tidal}}(x)=\frac{52}{5M_{t}}\frac{k_{2}}{C^{5}}x^{10}\Big{(}1+\frac{5203}{4368}x\Big{)}.$
(6.37)

Although the tidal interactions affect the NS-NS inspiral only at 5PN order,
its coefficient is 104 order magnitude for a NS with a radius of 10–15 km,
$k_{2}\sim 0.1$, and $C\sim 0.14–0.20$. Hence, it plays a vital role in the
late inspiral stage.

####  6.3.2 Polarization waveforms

From the observational point of view, the GW detector detects the radiation in
the direction of the source. Therefore, the polarization waveforms are
measured in the line-of-sight of the source, which can be expressed in
spherical coordinates ($R,\hat{\theta},\hat{\phi}$) as [317, 321]

$\displaystyle
h_{+}(\mathrm{t})=\frac{4}{r}{\cal{M}}^{5/3}\omega^{2/3}\Bigg{(}\frac{1+\cos^{2}\theta}{2}\Bigg{)}\cos(2\omega
t_{\rm ret}+2\phi),$ $\displaystyle\mathrm{and}$ $\displaystyle
h_{\times}(\mathrm{t})=\frac{4}{r}{\cal{M}}^{5/3}\omega^{2/3}\cos\theta\sin(2\omega
t_{\rm ret}+2\phi),$

where the chirp mass ${\cal{M}}$ is defined as ${\cal{M}}=\nu^{3/5}M$. The
amplitudes of the GW polarization with a fixed $\omega$ depends on the binary
masses through ${\cal{M}}$, which can also be derived as
${\cal{M}}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}$. It is measured by
LIGO with a good precision ${\cal{M}}=1.188_{-0.002}^{+0.004}M_{\odot}$ [23].
If we see the orbit edge-on, $\theta=\pi/2$, then $h_{\times}$ vanishes, and
the GW is linearly polarized. If we put $\theta=0$, then $h_{+}$ and
$h_{\times}$ have the same amplitude, but the only difference is the phase,
which is circularly polarized.

The polarization waveforms for an equal mass binary NS (EMBNS) systems along
the z-axis (optimally oriented observer) calculated up to 2.5PN are given by
[322, 323, 324, 308]

$\displaystyle h_{+}^{(z)}=$ $\displaystyle\frac{M_{t}}{2D}x\Bigg{(}\cos
2\Phi\bigg{\\{}-2+\frac{17}{4}x-4\pi x^{3/2}+\frac{15917}{2880}x^{2}+9\pi
x^{5/2}\bigg{\\}}$ $\displaystyle+$ $\displaystyle\sin
2\Phi\Big{\\{}\frac{59}{5}x^{5/2}\Big{\\}}\Bigg{)},$ (6.39) $\displaystyle
h_{\times}^{(z)}=$ $\displaystyle\frac{M_{t}}{2D}x\Bigg{(}\sin
2\Phi\bigg{\\{}-2+\frac{17}{4}x-4\pi x^{3/2}+\frac{15917}{2880}x^{2}+9\pi
x^{5/2}\bigg{\\}}$ $\displaystyle+$ $\displaystyle\cos
2\Phi\Big{\\{}-\frac{59}{5}x^{5/2}\Big{\\}}\Bigg{)}.$ (6.40)

The polarization waveforms are shown in Fig. 6.11 for IOPB-I parameter sets.
The dominant (2,2) mode up to 3PN order is written as follow [309, 325, 318]

$\displaystyle h_{22}=$
$\displaystyle-8\nu\sqrt{\frac{\pi}{5}}\frac{M_{t}}{D}e^{-2i\Phi}x\Bigg{\\{}1+\Bigg{(}-\frac{107}{42}+\frac{55\nu}{42}\Bigg{)}x+2\pi
x^{3/2}$ $\displaystyle+$
$\displaystyle\Bigg{(}-\frac{2173}{1512}-\frac{1069\nu}{216}+\frac{2047\nu^{2}}{1512}\Bigg{)}x^{2}+\Bigg{[}\frac{-107\pi}{21}+\Bigg{(}\frac{34\pi}{21}-24i\Bigg{)}\nu\Bigg{]}x^{5/2}$
$\displaystyle+$
$\displaystyle\Bigg{[}\frac{27027409}{646800}-\frac{856\gamma_{E}}{105}+\frac{2\pi^{2}}{3}+\frac{428i\pi}{105}-\frac{428}{105}\ln(16x)$
$\displaystyle+$
$\displaystyle\Bigg{(}\frac{41\pi^{2}}{96}-\frac{278185}{33264}\nu\Bigg{)}-\frac{20261\nu^{2}}{2772}+\frac{11463\nu^{3}}{99792}\Bigg{]}x^{3}\Bigg{\\}},$
(6.41)

where $D$ is the distance between source and observer, taken as 100 Mpc in our
calculations.

####  6.3.3 Tidal deformability

The tidal deformability of the NS decreases with DM’s addition, as shown in
Fig. 6.9. The same trend was also obtained for $\Lambda$. This is because,
with the addition of DM, the EOS becomes softer, and the mass and radius
change accordingly. Therefore, the softer EOS gives a lower $\lambda$ than the
stiffer one. The vertical double-headed line represents the constraints from
GW170817 with $\Lambda_{1.4}=190_{-120}^{+390}$ [24]. The values of
$\Lambda_{1.4}$ are 473.22, and 169.644 for DM momentum 0.03 and 0.05 GeV,
respectively. This implies that the value of $\Lambda_{1.4}$ with these DM
momenta satisfies the constraints given by GW170817.

Figure 6.9: Left: The mass variation of tidal deformability of the NS is shown for IOPB-I set with two different DM momentum. Right: The dimensionless tidal deformability vs. masses of the NS with the addition of DM are shown. The vertical line with bar represents the GW170817 constraints on $\Lambda_{1.4}$ [23, 24]. Table 6.1: The maximum mass of the spherical NS $M_{\rm max}$ obtained with different EOS, the radius $R_{1.35}$, the tidal deformability $\lambda_{1.35}$, the dimensionless tidal deformability $\Lambda_{1.35}$, the tidal love number $k_{2,1.35}$, the compactness $C_{1.35}$ for NS mass 1.35 $M_{\odot}$. The total mass of the BNS system is $M_{t}=2.7M_{\odot}$, and $M_{t}\Omega_{0}=M_{t}\pi f_{0}=0.0155$ are taken in this calculation with the minimum gravitational wave frequency $f_{0}$ $\approx$ 371 Hz. $\lambda_{1.35}^{*}$ is in $10^{36}$ g cm2 sec-2 unit. EOSs |  $M_{\rm max}$ ($M_{\odot})$ |  $R_{1.35}$ (km) | $\lambda_{1.35}^{*}$ | $\Lambda_{1.35}$ | $k_{2,1.35}$ | $C_{1.35}$ | $M_{t}\Omega_{0}$
---|---|---|---|---|---|---|---
NL3 | 2.775 | 14.568 | 7.415 | 1575.239 | 0.1130 | 0.137 | 0.0155
G3 | 1.979 | 12.476 | 2.754 | 585.182 | 0.0911 | 0.159 | 0.0155
IOPB-I | 2.149 | 13.168 | 4.026 | 855.304 | 0.1017 | 0.151 | 0.0155
IOPB-I+DM3 | 2.051 | 12.047 | 2.717 | 577.237 | 0.1071 | 0.165 | 0.0155
IOPB-I+DM5 | 1.788 | 10.204 | 0.990 | 210.387 | 0.0896 | 0.195 | 0.0155

####  6.3.4 Gravitational waves properties

In this sub-sec., we describe the inspiral properties of the BNS using the PN
formalism applied to a quasi-circular orbit. For simplicity, we restrict our
calculations only to EMBNS. Again, we take IOPB-I as a representative EOS for
the DM admixed NS since it predicts the mass of the NS as 2.149 $M_{\odot}$
which is consistent with PSR J0740+6620 [17]. Here, we calculate the
polarization waveforms and the GW properties, such as amplitude, frequency,
and orbital phase, for the EMBNS. The observables are calculated at retarded
time defined by [311]:

$\displaystyle t_{\rm ret}=t-r_{*},$ (6.42)

where $r_{*}$ is the tortoise coordinate defined as

$\displaystyle r_{*}=r_{A}+2M_{t}\ln\Big{(}\frac{r_{A}}{2M_{t}}-1\Big{)},$
(6.43)

where $r_{=}\sqrt{A/4\pi}$ and A is the proper surface area of the sphere. In
a simplified manner, $r_{A}$ is written as [311]

$\displaystyle r_{A}=r\Bigg{(}1+\frac{M_{t}}{2r}\Bigg{)}^{2},$ (6.44)

where $r\approx 200M_{t}$ is the finite spherical-coordinate radius.

Figure 6.10: (2,2) mode waveforms are shown with and without tidal
contributions for the IOPB-I parameter sets as a representative case. Figure
6.11: Plus and cross-polarization waveforms are shown for the IOPB-I parameter
sets. Figure 6.12: (2,2) mode waveform, frequency, phase, and PN-parameters
are shown for three parameter sets of the EMBNS in the retarded time interval
calculated for $D=100$ Mpc.

Fig. 6.12 shows the waveforms, frequencies, phases, and $x$ for NL3, G3, and
IOPB-I sets. The stiff EOS, like NL3, predicts higher mass and tidal
deformability than the softest EOS, like G3. Therefore, the massive NS is
easily deformed by the presence of tidal fields created by its companion star.
Hence, in the inspiral phase, a less massive BNS system sustains a longer time
than a more massive BNS system. This is because the tidal interactions
accelerate the orbital evolution in the late inspiral phases due to increased
interactions between two NSs. Some significant effects have been seen in Fig.
6.10 with and without tidal interactions.

To have a quantitative and comparative estimation for the inspiral properties
of the BNS merger, we enumerate the numerical value of $t_{\rm ret}$, $\Phi$,
$f$ and $x$ in Table 6.2 using NL3, IOPB-I, and G3. The last orbital time is
0.0583 sec. for NL3, which is less than from other two sets and confirms the
model-dependent prediction. There is significant influence seen for $t_{\rm
ret}$, $\Phi$ and $f$, except the value of $x$. The frequency and phase are
also increased for the BNS of soft EOS. The pictorial representations of $f$,
$\Phi$, and $x$ are compared in Fig. 6.12 with and without DM contains.

Table 6.2: The orbital phase ($\Phi$), gravitational wave frequency ($f$), and post-Newtonian parameter ($x$) are given in the last inspiral orbits of the BNS system. EOSs |  $t_{\rm ret}$ (sec.) |  $\Phi$ (radians) |  $f$ (Hz) |  $x$
---|---|---|---|---
NL3 | 0.0583 | 102.505 | 2356.47 | 0.213
G3 | 0.0597 | 106.383 | 2377.13 | 0.214
IOPB-I | 0.0593 | 105.178 | 2343.66 | 0.212
IOPB-I+DM3 | 0.0597 | 106.429 | 2387.09 | 0.215
IOPB-I+DM5 | 0.0603 | 108.224 | 2365.90 | 0.214
Figure 6.13: Same as Fig. 6.12, but for DM admixed NS with IOPB-I parameter
set as a representative case.

With the addition of DM to the NS EOS, the calculated macroscopic properties
($M$, $R$, and $\lambda$) decreases. As a result, the tidal interaction
between the two NSs becomes weaker compared to the normal BNS combination. For
example, $M=2.149M_{\odot}$ without DM and it is 2.051 $M_{\odot}$ with
$k_{f}^{\rm DM}=0.03$ GeV with IOPB-I set. Similarly, the radius of the star
is $R_{1.35}=13.168$ km without DM, and it is 12.047 km when DM is included
(see Table 6.1). The calculated values for $f$, $\Phi$, and $x$ are tabulated
in Table 6.2 for IOPB-I parameter set as a representative case with DM momenta
$k_{f}^{\rm DM}=0.03$ and $0.05$ GeV. The time period $t_{\rm ret}$ is found
to be 0.0603 sec. with IOPB-I+DM5, which is larger than the other two values
obtained with IOPB-I and IOPB-I+DM3. These results infer that the BNS system
with a higher percentage of DM sustains more time in the inspiral phase (see
Fig. 6.13). The other quantities, such as $f$ and $\Phi$ also affected by DM,
as shown in Fig. 6.13.

In general, with the addition of DM, the EOS becomes softer, which affects the
macroscopic properties of the BNS, i.e., the mass of the NS decreases with DM.
However, much ambiguity prevails in the post-merger remnants of BNS. This has
been settled partially after the analysis of GW170817 and GW190425 events.
Some possible pathways can be found in Ref. [326]. The likely theoretical
probabilities for a post-merger NS remnant depend on (i) the remnant mass
($M_{R}$) and (ii) the EOS of the NS, which dictates TOV mass ($M_{\rm TOV}$);

  1. 1.

if $M_{R}\geq 1.5\ M_{\rm TOV}$, the system promptly collapses into a black
hole.

  2. 2.

for $1.2\ M_{\rm TOV}\leq M\leq 1.5\ M_{\rm TOV}$, a hypermassive neutron star
survives the collision but collapses to form a black hole on dynamical
timescales.

  3. 3.

in the case, $M_{\rm TOV}<M_{R}\leq 1.2\ M_{\rm TOV}$, a supermassive NS
survived the collision and will collapse to form a black hole on secular
timescales.

  4. 4.

if $M_{R}\leq M_{\rm TOV}$, a stable NS is formed after the merger.

We refer the readers to see Fig. 1 of this Ref. [326] for a clear picture. The
post-merger remnant of DM admixed BNS merger depends on the percentage of DM
inside the NS. In Ref. [327], it is reported that for a contribution of DM
$\sim 5-10$% of the NS mass, a different peak in the power spectral density of
the GW signal is found. Also, a self-interacting DM of about $\leq 10\%$ leads
to the early formation of a black hole [328], which can be a subsequent seed
for a supermassive black hole. Therefore, to understand the nature of the
post-merger remnant of DM admixed BNS, one has to know (i) the exact nature of
DM particles and (ii) the percentage of DM present inside the NS.

### Conclusion

In this chapter, we have calculated the DM admixed binary NS properties with
RMF and E-RMF equation of states. The shape of the BNS is deformed due to the
mutual interactions between them. The rate of deformations can be measured by
various Love numbers and their corresponding deformabilities. Here, we
calculated those Love numbers and deformabilities for both gravitoelectric and
gravitomagnetic cases for DM admixed NS. It has been observed that the
magnitude of different Love numbers increases with DM percentage for a fixed
value of $l$. However, their corresponding tidal deformabilities decreased
with increasing the value of DM percentage.

In the late inspiral stage, the distance between binary stars decreases by the
emission of GWs. We have calculated the properties of the BNS mainly in the
late stage of inspiral with the post-Newtonian expansion for the equal mass
binary cases. The inspiral properties, such as its phase, frequency, and post-
Newtonian parameter, as well as various waveforms, are calculated. The
properties of waveforms in the inspiral phase are compared with and without
tidal interactions. We found that if the tidal interactions are switched off,
the BNS sustain more time in their inspiral phases. Also, the polarization
waveform $h_{+}$, $h_{\times}$, and the strain amplitude of $h_{22}$ modes
have been calculated for different EOSs in the retarded time interval at a
distance of 100 Mpc between the source and the observer.

We also calculated the last inspiral time of BNS. It is observed that the
binary system favors a longer time for softer EOS (G3) than the stiffer one
(NL3). The other quantities (frequencies and phase) are almost similar to all
the considered sets in the calculations, but the inspiral times in the last
orbit are found to be force-dependent. We get that with the addition of DM,
the binary system becomes less deformed and sustains more time in its inspiral
phases. The DM also affects the other properties compared to the ordinary NS,
and its effects on the inspiral properties of the BNS were significant.
Finally, we suggest that one has to take DM inside the compact objects while
modeling the inspiral waveforms for the BNS systems. One can also use more
accurate methods, such as numerical relativity techniques, to study the DM
effects on the inspiral-merger-ringdown and post-merger properties.

##  Chapter 7 Summary and Conclusion

This thesis explored several areas of nuclear astrophysics, such as nuclear
structures, NM, NSs, and gravitational waves. We briefly introduced the NS,
its birth, and its evolution in the Universe. The detection mechanism and the
internal structure of the NS are also discussed. To study the properties of
NS, one has to solve the TOV equations, where the star EOS is used as an
input. The RMF and E-RMF formalisms are chosen to get the NS equation of
states. In the first part of the thesis, we validated the formalism by
applying the model to finite nuclei and infinite nuclear matter. The last part
of the introduction explained the methodology, problem statement, and
motivation for the thesis.

After a brief introduction, we tried to present the formalism of the
relativistic mean-field models in Chapter-2. In the RMF theory, the nuclear
interactions are carried out by exchanging different mesons. We used the
effective field theory motivated RMF model because it successfully predicts
the properties of finite nuclei and nuclear matter. In recent years, it has
also well explained the NS properties such as $\sim 2\,M_{\odot}$ constraint.
Therefore, in the thesis, we have used RMF and E-RMF to calculate the
properties of the NS. In the E-RMF formalism, we include the self and cross-
couplings of the mesons such as $\sigma$, $\omega$, $\rho$, $\delta$ up to
$4^{\rm th}$ order. Hence, the system’s energy is the addition of nucleons and
mesons. The finite nuclei properties, such as binding energy per nucleon, skin
thickness, charge radius, density distributions, single particle energy
levels, and two neutron separation energy for different successful forces,
along with the recently reported BigApple, are calculated. It has been
observed that almost all predicted properties of the finite nuclei are well
consistent with the experimental data available for known magic and doubly
magic spherical nuclei. The properties of nuclear matter have been calculated
with various RMF parameter sets to check their consistency with the
experimental/empirical data. Lastly, we extended the calculations to
understand the EOS for the beta-equilibrated NS and also checked the speed of
sound for consistency throughout the star for different parameter sets.

In Chapter-3, in addition to nucleons, we have added extra candidates known as
DM to enumerate their effects on NM quantities. For that, we choose a simple
DM model, where the DM particles interact with nucleons by exchanging standard
model Higgs. The DM density is hypothesized by choosing one-third of the total
baryon density. We have chosen Neutralino as a DM candidate, a weakly
interacting massive particle with a mass of 200 GeV. The coupling constants
between DM-nucleons and nucleons-Higgs are considered by following the earlier
calculations [127] and constrained with the help of experimental data.
Therefore, the energy density and pressure of the NM system are the
combinations of nucleons, mesons, and DM. To estimate the effects of DM on NM
properties such as binding energy, incompressibility, effective mass, symmetry
energy, and its different coefficients with different fractions of DM and
neutron-proton asymmetry, the RMF and E-RMF models are used on top of the DM
EOS. The EOS becomes softer with the increasing percentage of DM. The energy
density also increases with $k_{f}^{\rm DM}$ without adding much to the
pressure. We observed that the DM has marginal effects on the NM properties,
except the EOSs, the binding energy, and $Q_{\rm sym}$. This is because it has
less effect on the system’s pressure due to the contribution of the Higgs
field in the order of $10^{-6}-10^{-8}$ MeV. One can also choose other DM
models by varying their mass and percentage to explore the DM admixed NM
properties.

The isolated static/rotating DM admixed NS properties such as mass, radius,
tidal deformability, and moment of inertia have been calculated with the EOSs
obtained from the RMF and E-RMF models in Chapter-4. It has been observed that
DM has significant effects on both static and rotating NS. The mass and radius
decrease by increasing the DM percentage. However, if we compare the static
and rotating cases, the change is $\sim 20\%$ for different macroscopic
properties. We have also calculated the curvatures for both inside and outside
the NS with DM content. The change of curvature with and without DM is found
to be approximately 33% for the canonical star. This value increases with the
maximum mass of the NS. The surface curvature and the compactness are also
estimated, and it is obtained that their magnitude increased with DM
percentage. For a stable system, the binding energy is always negative (here,
we used the notation negative for a stable system). However, with the addition
of DM inside the NS, the binding energy moves towards positive values, making
the NS unstable. From this, we concluded that a tiny amount of DM can
accumulate inside the NS. A higher amount of DM warms the NS and accelerates
the Urca process, which enhances the cooling of NS and causes instability.
GW19814 event opened a debatable issue regarding the type of secondary
component “whether it is the lightest black hole BH or heaviest NS?”. We have
suggested that the secondary component might be a DM admixed NS if the
underlying nuclear EOS is sufficiently stiff.

In Chapter-5, we have calculated the $f$-mode oscillations of the DM admixed
hyperon stars within the relativistic Cowling approximations. Inside the DM
admixed hyperon star, they interact with DM by exchanging Higgs. Also, the
coupling between DM-Higgs is assumed to be the same as nucleons. We have
calculated the EOSs for various combinations of hyperons-mesons couplings with
different group theory methods. Therefore, total energy density and pressure
are the combinations of nucleons, hyperons, mesons, leptons, and DM. With
those EOSs, we calculated the $f$-mode frequency of the hyperon star with DM
content. The values of these frequencies for softer EOSs are larger (lower
mass NSs) and vice-versa for stiffer EOS. The angular frequencies scaled by
mass and radius are calculated as a function of compactness. It has been found
that both quantities follow a linear relationship. We found the correlations
between $f_{1.4}–\Lambda_{1.4}$ and $f_{\rm max}–\Lambda_{\rm max}$ are almost
consistent with the earlier prediction [298]. Our results correspond to
$f_{1.4}$ is in the range $1.78-2.22$ kHz for the considered EOSs. We hope the
discovery of more BNS merger events will open up tight constraints on the
$f$-mode frequencies in the future.

Chapter-6 is dedicated to binary NS in the presence of DM. It is one of the
most exciting aspects after the discovery of the gravitational waves from the
merger of binary NSs. In this chapter, we have explored the DM admixed binary
NS properties with RMF and E-RMF EOSs. Various Love numbers and corresponding
tidal deformabilities are essential quantities to explain the degree of
deformation for a BNS. Therefore, we have calculated those Love numbers and
their corresponding deformabilities for both gravitoelectric and
gravitomagnetic cases. It has been observed that the magnitude of different
Love numbers increases with DM percentage for a fixed value of $l$. However,
their corresponding tidal deformabilities decrease with increasing DM
percentage. In the late inspiral stage, the BNS system rotates quite first by
emitting a huge amount of GWs. We have calculated those late inspiral
properties, such as the frequency of emitted GWs, waveforms, phase, etc., with
the post-Newtonian expansion for the equal mass binary cases. The waveforms
emitted in the inspiral phase are compared with and without tidal
interactions. We found that if the tidal interactions are switched off, the
BNS sustain more time in their inspiral phases. The polarization waveform
$h_{+}$, $h_{\times}$, and the strain amplitude of $h_{22}$ modes are
calculated in the retarded time interval at a distance of 100 Mpc between the
source and the observer. We observed that the binary system favors a longer
time for softer EoS (G3) than the stiffer NL3. The other quantities
(frequencies and phase) are almost similar for assumed EOSs. However, the
inspiral times in the last orbit are found to be model dependent. For a DM
admixed BNS, the binary system becomes less deformed and sustains more time in
its inspiral phases. The DM also affects the other properties as compared to
the ordinary NS, and the DM effects on the inspiral properties of the BNS were
significant. As a result, we advise that when modeling the inspiral waveforms
for the BNS systems, one can take DM inside the compact objects. Numerical
relativity tools can be used for an accurate description to explore the DM
effects on the inspiral-merger-ringdown and post-merger properties.

#### Future Prospective

##### 1\. Different types of DM model:-

An effective Lagrangian density with different types of DM particles will be
constructed. This may satisfy the relic abundance, and DM detection limit
observed from various experimental data. In our present studies, we are taking
only fermionic DM; however, there is a possibility that the DM may be bosonic
also [329, 330]. In that case, one has to change the framework to calculate
the properties of static and rotating NS and compare with and without DM cases
[331]. In addition to this, the study of symmetric/asymmetric self-
annihilating DM is also interesting.

##### 2\. Effects of DM on hyperon puzzle:-

Hyperons may appear in the inner core of NSs at densities of about
$2-3\rho_{0}$. Although the presence of hyperons in NSs seems energetically
unavoidable, their substantial softening of the EOS leads to a reduction of
maximum mass, which is not compatible with observation. This is known as the
“Hyperon Puzzle”. The DM particles can be accreted not only inside the NS but
also hyperon stars. However, the exact coupling between hyperons with nucleons
and mesons is still a debatable issue. To overcome these difficulties, several
works have been dedicated to explaining both the hyperon puzzle and coupling
strength between different species [50, 332, 333]. From the previous works
[50, 332, 333], we may extend to DM admixed hyperon star. We can put
constraints on the hyperon-DM coupling strength from the different
observational and experimental data. That may provide a narrow window for the
existence of DM and can explain the scattering cross-section estimations from
terrestrial DM detectors.

##### 3\. Binary NS merger and post-merger properties:-

The numerical relativity simulation already opens the door to explore the
second most enigmatic merger between two NSs. We propose to understand more
details in this direction using the DM equation of states. So far, our
analysis reveals that the DM admixed BNS sustains more time in the inspiral
phase than the BNS without DM admixture. We find it interesting to emphasize
the merger and post-merger properties for DM admixed NS with the help of
numerical relativity techniques. In future work, we propose to write our
numerical relativity packages or Einstein toolkit thorns to enumerate the DM
effects on the merger and post-merger systems.

##### 4\. Emitted gravitational waves:-

The GWs emitted from the inspiral-merger-ringdown phases are the primary
source for the GW detectors. The detected GWs properties are compared with
different theoretical templates to estimate the parameters of the GW’s origin,
distance, and mass of the sources. Therefore, making some templates using the
present DM admixed NS equation of states with different analytical techniques
is a fascinating subject. Using such templates, one can estimate the GWs
properties and the source details more accurately. We can use the
observational data of LIGO/Virgo, which will provide direct constraints on our
theoretical model and fix the amount of the accumulated DM inside the star.

#### Expected Outcomes

It has been proposed that the merger of DM admixed binary NS provides an extra
peak in the post-merger signal. From this post-merger data, we may constrain
the percentage of DM inside the NS. Also, we can improve the various existing
templates for the inspiral-merger-ringdown phases for BNS, which may better
estimate the GWs source and its origin in the presence of DM.

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