# Radial oscillations and gravitational wave echoes of strange stars with
nonvanishing lambda

Jyatsnasree Bora<EMAIL_ADDRESS>Umananda Dev Goswami
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University, Dibrugarh
786004, Assam, India

###### Abstract

We study the effect of the cosmological constant on radial oscillations and
gravitational wave echoes (GWEs) of non-rotating strange stars. To depict
strange star configurations we used two forms of equations of state (EoSs),
viz., the MIT Bag model EoS and the linear EoS. By taking a range of positive
and negative values of cosmological constant, the corresponding mass-radius
relationships for these stars have been calculated. For this purpose, first we
solved the Tolman-Oppenheimer-Volkoff (TOV) equations with a non-zero
cosmological constant and then we solved the pressure and radial perturbation
equations arising due to radial oscillations. The eigenfrequencies of the
fundamental $f$-mode and first 22 pressure $p$-modes are calculated for each
of these EoSs. Again considering the remnant of the GW170817 event as a
strange star, the echo frequencies emitted by such stars in presence of the
cosmological constant are computed. From these numerical calculations, we have
inferred relations between cosmological constant and mode frequency,
structural parameters, GWE frequencies of strange stars. Our results show that
for strange stars, the effective range of cosmological constant is
$10^{-15}\,\mbox{cm}^{-2}\leq\,\Lambda\,\leq 3\times
10^{-13}\,\mbox{cm}^{-2}$.

dense matter – asteroseismology – cosmological constant – gravitational waves

###### pacs:

04.40.Dg,97.10.Sj

##  I Introduction

In 1916 Albert Einstein introduced the term cosmological constant $\Lambda$ as
a modification in his field equation to achieve a static, stationary universe,
which was believed to be the state of the universe at that time. This new term
introduced by him is a dimensionful free parameter carroll . But after the
discovery of the redshift of stars and consequently an expanding universe by
Hubble this idea of cosmological constant was quickly abandoned. Later, it was
reintroduced to overcome the age crisis problem and to construct a universe
satisfying the “perfect cosmological principle” bondi ; hoyle . On the
contrary, current astrophysical and cosmological observational data reveal
that we are in the era of cosmological expansion and the expansion rate of the
universe is increasing rather than decreasing riess ; perlmutter . One of the
strong beliefs is that the present accelerated expansion of the universe is
due to an exotic form of energy known as dark energy. Although it suffers from
the fine-tuning problem, $\Lambda$ has been considered as one of the prime
candidates of dark energy in the sense that it forms the vacuum energy Peebles
. As dark energy pervades throughout the universe, the energy density inside
the compact objects should be affected by it, and hence it is necessary to see
the effect of dark energy content on the behaviour of such objects. So, if we
consider the contribution of cosmological constant $\Lambda$ in compact
objects it will be related to the acceleration of the currently observable
universe or dark energy. This possibility of non-zero $\Lambda$ and it, as a
dominating energy density of the universe, is a fascinating problem in
nowadays research.

One of the most intriguing parts of the issue of compact stars is the study of
unique strange stars. Since the last decade, the subject of strange stars has
attracted much attention from the astrophysicist community. These hypothetical
stars are unique in the sense that the structural behaviour of such objects
rarely matches with that of other compact objects, like white dwarfs and
neutron stars. Due to their unique structural behaviour, the strange stars can
be regarded as excellent natural laboratories to study, test, or perhaps
constrain different modified theories of gravity under extreme conditions that
cannot be reached from the earth-based laboratories. By hypothesis, strange
matters are the true ground state of the hadrons and hence they are stable
matters Witten1984 . So, it could explain the origin of the huge amount of
energy released in superluminous supernovae (100 times brighter than normal
supernovae) ofek ; ouyed . Such strange matters are composed of deconfined
quark matters, mainly of $u$, $d$, $s$ quarks and a small amount of electrons
to maintain charge neutrality aclock ; weber2005 .

Our present study consists of two parts. In the first part, we are seeking
correlations between the cosmological constant $\Lambda$ and radial
oscillation frequencies of strange stars. From long ago, the light variation
in a pulsating star has been used to investigate the physical properties like
mass, radius of a star. This indirect approach to understanding the physical
properties of a star is known as asteroseismology handler2012 . From the birth
to the end of a star, nearly every star undergoes some kind of pulsation. Such
asteroseismic behavioural studies of stars are mainly of two types: radial and
non-radial. For the case of compact objects like white dwarfs, neutron stars,
and strange stars, such radial asteroseismic behaviours are reported earlier
in Chandrasekhar1964a ; Chandrasekhar1964b ; Chanmugam1977 ; datta1992 . Here,
we stick to our study in the case of radial oscillations only of strange
stars.

In the second part of this study, we investigate the effect of $\Lambda$ on
the gravitational wave echoes (GWEs) of strange stars. It has been recently
suggested that some ultra-compact post-merger objects can emit GWEs pani ;
manarelli ; abedi ; urbano . Since strange stars are known to be very compact,
we describe strange stars as ultra-compact objects by using stiffer equations
of state (EoSs). In the recent detection of gravitational waves (GWs) from the
binary neutron star merging event GW170817, the nature of the final massive
remnant formed is still not confirmed. We consider it as a strange star and
evaluate the corresponding echo frequencies emitted by such stars in presence
of the cosmological constant $\Lambda$. The ultra-compact stars are those,
whose compactness, $\mbox{C}=\mbox{M}/\mbox{R}$ exceeds $1/3$ urbano . Again
to be an eligible candidate to emit echo frequencies, the star must possess a
photon sphere. This is the region above the surface of the star at a distance
of $\mbox{R}=3\,\mbox{M}$ pani . However, the compactness should not exceed
Buchdahl’s limit $\mbox{R}=9/4\,\mbox{M}$. Buchdahl’s limit is the upper limit
of compactness of fluid stars which describes the maximum amount of mass that
can exist in a sphere before it must undergo the gravitational collapse urbano
.

In presence of cosmological constant, compact stars are studied earlier in
nayak ; Bordbar ; Largani ; Liu ; arbanil . In the article bohmer , the
equations describing small radial oscillations of relativistic stars in
presence of a cosmological constant $\Lambda$ were reported. They have also
studied the impact of cosmological constant $\Lambda$ on the critical
adiabatic index. Again Hossein et al. Sk  studied the anisotropic compact
stars by considering a variable cosmological constant with a static
spherically symmetric spacetime described by Krori-Barua metric kb . A similar
study was done by Kalam et al. for the anisotropic compact stars with the de
Sitter spacetime kalam . In 2014, O. Zubairi and F. Weber derived the modified
stellar structure equations to account for a finite value of the cosmological
constant in spherically symmetric mass distributions zubairic1 . In zubairic ,
such equations are derived for the compact stellar structures such as quark
star and neutron stars in presence of the cosmological constant $\Lambda$, and
the role of the cosmological constant on stars mass, radius, pressure, and
density profiles along with the gravitational redshifts were discussed. For
compact stars with an equation of state provided by the quark-meson coupling
(QMC) model, such effects of the cosmological constant are discussed in nayak
. Recently, for stable relativistic polytropic objects, the effect of the
cosmological constant is reported in arbanil . A more detailed analysis of the
stability of polytropic spheres in the presence of a cosmological constant can
be found in posada . For the general relativistic case with a vanishing
cosmological constant, the radial oscillation modes and echo frequencies for
strange stars are studied in jb . The present study is the extension of our
previous study jb  in which we had taken into account the dependence of radial
oscillation frequencies and GWE frequencies on different model parameters.
This work will focus on the impact of the cosmological constant on radial
oscillation modes and echo frequencies of strange stars.

Motivated from the previous works mentioned above, we aim at exploring the
effect of the cosmological constant on strange stars’ structural behaviour,
radial oscillation, and echo frequencies emitted by them and also to find
boundary values of it. Here, we take the strange star as a probe to explore
the consequences of a non-vanishing cosmological constant and allow it to vary
inside the star. Also, the cosmological constant is taken as a matter
distribution source of the star and we show that the compactness of the star
varies with the cosmological constant. In turn, it implies that the
cosmological constant varies inside such dense stars depending on their
compactness. The presence of cosmological constant $\Lambda$ describes well
the strange star configurations. In particular, we study the structural change
in strange stars due to a non-vanishing cosmological constant, focusing on its
effects over the radial oscillations and GWEs of strange stars. Here, we have
solved the Einstein field equation for some finite values of cosmological
constant $\Lambda$ for spherically symmetric mass distribution. By solving the
Tolman-Oppenheimer-Volkoff (TOV) equations we have obtained the mass-radius
relationships of strange stars in presence of a non-vanishing cosmological
constant. Then we have computed the fundamental $f$\- mode and first 22
pressure $p$-mode of radial oscillations, and GWE frequencies emitted by
strange stars in two types of EoSs.

We have organized the rest of the paper as follows. In Sec. II the general
relativistic formulations including TOV equations and perturbation equations
are discussed briefly. In Sec. III the considered EoSs are described along
with a brief note on the cosmological constant. The emission of GWEs from an
ultra-compact object is discussed in Sec. IV. In Sec. V we have discussed the
results that are obtained from this study, which is finally followed by the
concluding section, i.e. Sec. VI. Here we follow the natural unit system by
considering $c=\hbar=1$ and $G=1$ with the metric convention
$(-,\,+,\,+,\,+)$.

##  II General relativistic formulation

###  II.1 Equations for stellar structure

In this subsection, we discuss briefly about the strange stars in general
relativity (GR) in a spacetime with a cosmological constant, i.e. with
$\Lambda\neq 0$. The Einstein’s field equation with the cosmological constant
reads,

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}.$ (1)

We consider the strange star as isotropic, stable, and non-rotating mass
distribution and the perfect fluid inside the strange star is described by the
stress-energy tensor,

$T_{\mu\nu}=(p+\rho)U_{\mu}U_{\nu}+p\,g_{\mu\nu},$ (2)

where $p$ is the fluid pressure, $\rho$ is the fluid energy density and
$U_{\mu}$ are its four-velocities. The line element for a spherically
symmetric mass distribution has the form:

$ds^{2}=-\,e^{\chi(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$
(3)

with the unknown metric functions $\chi(r)$ and $\lambda(r)$ being dependent
on the radial coordinate $r$ only. For the exterior problem the field equation
(1) gives,

$e^{\chi(r)}=e^{-\lambda(r)}=1-\dfrac{2M}{r}-\dfrac{\Lambda r^{2}}{3},$ (4)

where $M$ is the total mass of the star. Similarly for the interior problem we
have,

$e^{\chi(r)}=e^{-\lambda(r)}=1-\dfrac{2m(r)}{r}-\dfrac{\Lambda r^{2}}{3}.$ (5)

Here $m(r)$ is the mass function within the radius $r$. On matching the
exterior and interior solutions at the surface of the star we get, $m(R)=M$,
the total mass of the stellar configuration with $R$ being the radius of the
star.

The addition of cosmological constant $\Lambda$ in Einstein’s field equation
(1) will change the structure of compact objects. Now solving Einstein’s field
equation for the energy-momentum tensor (2) we get the stellar structure
equations, known as the TOV equations tolman ; tov  with the introduction of
cosmological constant zubairic  as

$\dfrac{dm}{dr}=4\pi\rho(r)r^{2},$ (6) $\dfrac{dp}{dr}=-(\rho+p)\dfrac{m+4\pi
pr^{3}-\dfrac{\Lambda r^{3}}{3}}{r^{2}\left(1-\dfrac{2m(r)}{r}-\dfrac{\Lambda
r^{2}}{3}\right)},$ (7) $\dfrac{d\chi}{dr}=-\dfrac{2}{\rho+p}\dfrac{dp}{dr}.$
(8)

To find static equilibrium stellar configurations equations (6) - (8) are to
be integrated along the radial coordinate $r$ with the initial conditions:
$m(r=0)=0$, $p(r=0)=p_{c}$ and $\rho(r=0)=\rho_{c}$. The radius of the star
can be determined by using the fact that the pressure vanishes at the surface
of the star, i.e. $p(r=R)=0$ (see Fig. 2). These equations for stellar
structure can be solved for a given EoS. Solutions of these equations can lead
us to know about the stellar mass, radius, pressure, density profile, and the
gravitational redshift.

###  II.2 Radial stability equations

The theory of infinitesimal, adiabatic, radial oscillations and the radial
stability of relativistic stars was first derived by S. Chandrasekhar in 1964
(Chandrasekhar1964a, ; Chandrasekhar1964b, ). The solutions of these
perturbation equations give information about the eigenfrequencies of radial
oscillations. In the presence of cosmological constant $\Lambda$, the radial
instability equations were investigated earlier in bohmer ; stuchlik . By
introducing two dimensionless parameters $\xi=\Delta r/r$ and $\eta=\Delta
p/p$, where $\Delta r$ is the radial perturbation and $\Delta p$ is the
corresponding Lagrangian perturbations of the pressure, Bhömer and Harko
bohmer  presented the radial and pressure perturbation equations as

$\small{\dfrac{d\xi}{dr}=-\,\dfrac{1}{r}\left(3\xi+\dfrac{\eta}{\gamma}\right)-\dfrac{dp}{dr}\dfrac{\xi}{p+\rho}},$
(9)
$\displaystyle\dfrac{d\eta}{dr}=\xi\left[\omega^{2}e^{\lambda-\chi}(1+\dfrac{\rho}{p})r-\dfrac{4}{p}\dfrac{dp}{dr}+\;\dfrac{r}{p\,(\rho+p)}{\left(\dfrac{dp}{dr}\right)}^{2}-e^{\lambda}\left(8\pi-\dfrac{\Lambda}{p}\right)(\rho+p)\,r\right]$
$\displaystyle+\,\eta\left[\dfrac{1}{(\rho+p)}\dfrac{dp}{dr}-4\pi(\rho+p)\,r\,e^{\lambda}\right],$
(10)

where $\gamma=\dfrac{dp}{d\rho}(1+\rho/p)$ is the relativistic adiabatic
index.

To study the stability of the stellar object against a small radial
perturbation we have to integrate equations (9) and (II.2) from the centre to
the boundary along with two boundary conditions: one at the centre and other
at the surface of the star. As $r\rightarrow 0$, the coefficient of $1/r$ in
equation (9) must vanish to avoid the singularity in the equation. So at the
centre of the star we get the boundary condition as

$3\gamma\xi+\eta=0.$ (11)

From this condition it can be inferred that $\Delta p=-\,3\gamma p\xi$ at the
centre of the star, i.e. $(\Delta p)_{r\,=\,0}=-{(3\gamma p\xi)}_{r\,=\,0}$.
Again when $r\rightarrow R$, i.e. at the surface of the star, $p\rightarrow
0$. Thus again to avoid singularity this will eventually require that the
coefficient of $1/p$ in the equation (II.2) must vanish at the boundary of the
star. Using equation (7) in equation (II.2), it can be found from this
requirement that

$\displaystyle\xi\left[\omega^{2}R\left(\dfrac{M}{R^{2}}-\dfrac{\Lambda
R}{3}\right)^{-1}\left(1-\dfrac{2M}{R}-\dfrac{\Lambda
R^{2}}{3}\right)^{-1}+R\left(\dfrac{M}{R^{2}}-\dfrac{\Lambda
R}{3}\right)\left(1-\dfrac{2M}{R}-\dfrac{\Lambda R^{2}}{3}\right)^{-1}\right]$
$\displaystyle+\xi\left[\Lambda R\left(\dfrac{M}{R^{2}}-\dfrac{\Lambda
R}{3}\right)^{-1}+4\right]=0.$ (12)

This boundary condition implies that $\Delta p=\,0$ at the stellar surface,
i.e. $(\Delta p)_{r\,=\,R}=0$. These coupled differential equations constitute
the Sturm- Liouville type eigenvalue problem. These equations (9) and (II.2)
together with the boundary conditions (equations (11), (II.2)) determine the
eigenvalues or eigenfrequencies $\omega$ of radial oscillations jb . These
equations are solved by using the shooting method as described in panotopoulus
. If $\omega$ is real, i.e. $\omega^{2}>0$, the configuration is stable and
for $\omega^{2}<0$, the configuration becomes unstable against radial
oscillations. Again for the stable fundamental mode ($f$-mode) i.e. for
$\omega_{0}^{2}>0$, all other higher-order modes will also be stable.

To find the radial oscillation frequencies we first calculated the quantity
$\bar{\omega}=\omega\,t_{0}$, which is dimensionless, and here
$t_{0}=1\,\mbox{ms}$. The oscillation frequencies are then calculated by

$\nu=\dfrac{\overline{\omega}}{2\pi}\;\;\mbox{kHz}.$ (13)

The obtained frequency $\nu$ is now allowed to take some eigenvalues $\nu_{n}$
and for each values of $\nu_{n}$ we have obtained a specific oscillation mode
of the star jb .

##  III Equations of State and the cosmological constant

Before proceeding to discuss the role of the cosmological constant on strange
stars’ oscillation and echo frequencies emitted by them, we wish to add a few
comments on our choice of stellar models. It is well known that the
macroscopic properties of compact objects, such as mass and radius, depend
crucially on the EoSs of ultra-compact matter, which is unfortunately not
confirmed clearly to date. In this study, the chosen EoSs are those that (i)
should be stiff enough to emit GWE, (ii) the model parameters associated with
each EoS lie inside the desired range, and (iii) the mass-radius relations of
which are within the accepted limits. To describe the stellar structure, in
this present work, we are using two EoSs, viz., the MIT Bag model EoS and the
linear EoS as mentioned above.

The MIT Bag model corresponds to a relativistic gas of deconfined quark matter
with energy density. This EoS satisfies all necessary criteria for strange
matter while retaining an elegant simplicity haensel . It was first used by
Witten in 1984 Witten1984 . The form of this equation is

$p=\dfrac{1}{3}(\rho-4\,B).$ (14)

In this EoS, the density at the stellar surface is given by
$\rho(R)=4\,\mbox{B}$, where $p$ is the isotropic pressure, $\rho$ is the
energy density and $B$ is the Bag constant. Instead of taking this usual form
(14) of the MIT Bag model EoS we have chosen the stiffer form of this EoS as

$p=\rho-4\,B$ (15)

The reason behind doing this is that in EoS of the form (14) the compactness
of the stellar structure obtained is not enough to emit GWE frequencies.
Making it a stiffer one of the form (15) allows the star to get enough
compactness to emit GWE frequencies. As reported in aziz , the acceptable
range of Bag constant is
$(133.68\,\mbox{MeV})^{4}<\mbox{B}<(222.54\,\mbox{MeV})^{4}$. So in this work
we choose B as $(190\,\mbox{MeV})^{4}$, which is well inside the desired
range.

Other EoS we have used is the linear EoS of the form:

$p=b\,(\rho-\rho_{s}),$ (16)

where $b$ is the linear constant and $\rho_{s}$ is the surface energy density
rosinska . This EoS was developed by Dey et al. in 1998 dey . We have chosen
the linear constant $b=0.910$ and the corresponding value of the surface
energy density $\rho_{s}$ is taken. While choosing the constant value, we have
kept in mind the conditions for echoing GWs, which restrict that the
compactness should be larger than $1/3$, and also it should respect the
causality condition ($b\leq 1$) jb . Our chosen value of $b$ lies under these
two restrictions.

The dimensionful parameter $\Lambda$ introduced by Einstein in his field
equation has the unit of $\mbox{(length)}^{-2}$. We already have mentioned in
Sec. I about the various studies on compact objects in presence of
cosmological constants nayak ; Bordbar ; Largani ; Liu ; arbanil .
Specifically, we would like to mention here that in Ref. zubairic1 , O.
Zubairi and F. Weber obtained that the properties of compact objects in our
Universe, where $\Lambda$ is very small (predicted via cosmological
observations), do not depend on the cosmological constant. They observed that
to yield an observable effect, $\Lambda$ would have to be of the order of
nuclear density scales $\sim 10^{-14}\,\mbox{cm}^{-2}$. For neutron stars, a
maximum mass of $\sim$ 1.68 $\mbox{M}_{\odot}$ was reported by G.H. Bordbar et
al. Bordbar  using $\Lambda\sim 10^{-18}\,\mbox{cm}^{-2}$. For the case of
white dwarfs, a recent study demands the upper limit on $\Lambda$ as,
$\Lambda<3\times 10^{-18}\,\mbox{cm}^{-2}$ Liu . Largani et al. found that for
neutron stars to have an observable effect of $\Lambda$ in their structure,
the typical values of $\Lambda$ to be $\sim 10^{-12}\,\mbox{cm}^{-2}$Largani .
In the article arbanil , the authors reported the stellar configuration of
polytropic spheres using $\Lambda$ values within the range of
$-10^{-15}\leq\Lambda\leq 10^{-15}\,\mbox{cm}^{-2}$. In bohmer , for a star
composed of matter having a density equivalent to nuclear energy density, the
upper limit of $\Lambda$ was reported to be less than $3\times
10^{-13}\,\mbox{cm}^{-2}$. Thus, motivated by the search of new equilibrium
configurations, corresponding radial oscillations, and also the properties of
GWE frequencies of strange stars, in this study we have considered a
cosmological constant value larger than the one predicted by cosmological
observations planck . We have chosen a set of positive and negative values of
$\Lambda$ to see its impact on radial oscillation modes and echo frequencies
of strange stars. Our chosen values of $\Lambda$ are in multiples of the
$\epsilon$ as $-\,150\,\epsilon$, $-\,100\,\epsilon$, $-\,50\,\epsilon$,
$5\,\epsilon$, $10\,\epsilon$, and $15\,\epsilon$. These values are in the
units of nuclear energy density scale, $\epsilon=140\,\mbox{MeV/\,fm}^{3}$.
The study with positive values of cosmological constant $\Lambda$ (which
corresponds to de Sitter space) gives a more realistic description of strange
stars. These are relevant from the observational point of view and also used
in the context of the dark energy model of the universe. Again the negative
cosmological constant $\Lambda$ corresponding to anti-de Sitter space is
important for anti-de Sitter/conformal field theory correspondence. So this
work is a generalized work that is going to contribute significantly in both
observational regime and conformal field theory regime. Again, it is worthy to
mention that besides the case of the configuration of compact stellar objects,
large values of the cosmological constant have been considered in other types
of astrophysical situations also arbanil . For instance, in the case of
accretion in primordial black holes during the very early universe, the
cosmological constant can take values many orders of magnitude greater than
that predicted by current observational data zstuch . As described by Carroll
in classical GR, there is no preferred choice for the length scale defined by
$\Lambda$ carroll . Indeed, the cosmological constant is a measure of the
energy density of the vacuum or the state of lowest energy. This value cannot
be calculated with any confidence. So, it brings flexibility in choosing
scales of various contributions to the cosmological constant carroll .

##  IV Gravitational wave echoes

As mentioned in Sec. I, an important property of ultra-compact objects is that
some of them can emit GWE frequency. It is a very interesting property of such
stars. This happens due to the presence of a photon sphere around the surface
of the star. More precisely, GWEs originate from the GWs that are trapped
between the photon sphere and the surface of the ultra-compact star abedi . As
mentioned in abedi , such type of echoes has two natural frequencies: the
harmonic or resonance frequencies and the black hole ringdown or quasi-normal
mode (QNM) frequencies.

In this study, we have chosen the final remnant of the GW170817 event as a
strange star and calculated the GWEs emitted by such ultra-compact objects.
The GWE frequency can be approximately estimated from the inverse of the time
taken by a massless test particle to travel from the unstable light ring to
the centre of the star. This time is the characteristic echo time and can be
expressed as urbano

$\tau_{echo}\equiv\int_{0}^{3M}\\!\\!\\!\\!e^{\,(\lambda(r)-\chi(r))/2}\;\mathrm{d}r.$
(17)

Using the relation for $e^{\,-\lambda(r)}$ from equation (5) in presence of a
non-zero cosmological constant $\Lambda$, we can obtain the following
expression for characteristic echo time,

$\tau_{echo}=\int_{0}^{3M}\\!\\!\\!\\!\dfrac{1}{\sqrt{e^{\,\chi(r)}\left(1-\dfrac{2m(r)}{r}-\dfrac{\Lambda
r^{2}}{3}\right)}}\;\mathrm{d}r.$ (18)

In this equation, the term $m(r)$ and $\chi(r)$ can be obtained from the
solution of TOV equations with non-vanishing cosmological constant $\Lambda$,
i.e. of equations (6) - (8). Finally, the characteristic echo frequency can be
calculated by using the relation, $\omega_{echo}\approx\pi/\tau_{echo}$.

##  V Numerical results

Figure 1: Mass-radius relationships of strange stars for the MIT Bag model EoS
(left plot) and linear EoS (right plot) with different cosmological constants
showing photon sphere limit, Buchdahl’s limit and black hole limit lines for
negative, zero and positive $\Lambda$ values. Mass of stars is expressed in
terms of solar mass $\mbox{M}_{\odot}$.

As mentioned earlier, in comparison to other compact objects the structural
properties of strange stars are unique. The mass-radius relationship of these
stars follows a definite pattern than that of neutron stars. Strange matter
can exist in lumps with the size of few fermis to the size of $\sim 10$ km
radius strange stars aclock . Using the stiffer MIT Bag model EoS and linear
EoS we have plotted the sequences of mass-radius relationships for such
stellar configurations. In this study, with the MIT bag model EoS, the maximum
mass is obtained as $\approx 3.33\,\mbox{M}_{\odot}$ and the corresponding
maximum radius is $\approx 14.34\,\mbox{km}$ and this corresponds to the
$\Lambda$ value $15\,\epsilon$. The maximum mass obtained for the linear EoS
is $\approx 1.78\,\mbox{M}_{\odot}$ and the maximum radius is $\approx
7.61\,\mbox{km}$ corresponding to the same $\Lambda$ value. The first plot of
Fig. 1 shows the relationship for the MIT Bag model EoS and for the linear EoS
it is shown in the second plot. As shown in these two plots, for much of these
sequences strange stars are following the $\mbox{M}\propto\mbox{R}^{3}$
relation. However, for neutron stars, radii decrease with increasing mass for
much of their ranges. For both of the models, with decreasing $\Lambda$,
stiffer configurations are obtained. The stiffness is observed to be maximum
with $\Lambda=-150\,\epsilon$ and for the MIT Bag model it is higher than the
linear EoS. Linear EoSs are giving smaller stellar configurations than the MIT
Bag model EoSs. In case, if we choose the general form of the MIT Bag model
EoS, then the compactness will not be sufficient to overcome the photon sphere
limit as pointed out earlier. The stiffer form of the MIT Bag model and linear
EoSs are giving compact enough configurations to cross the photon sphere limit
but not Buchdahl’s limit. From Fig. 1, it is clear that the sequences of
stellar configurations that are obtained can emit GWE frequencies with the
considered $\Lambda$ values.

Figure 2: Pressure profiles of strange stars with different $\Lambda$ values
for the MIT Bag model EoS (first panel) and the linear EoS (second panel).
Here the pressure is scaled in terms of central pressure $p_{c}$.

At this point for the clarity about the physical boundary condition that we
have used to calculate the radius of a star as mentioned in section II, we
have plotted Fig. 2, which shows the variation of pressure with radial
distance $r$ (in km) for different strange star configurations as predicted by
the MIT Bag model (left panel) and linear EoS (right panel). These two
pressure profiles indicate that pressure near the centre of the star is
maximum and it decreases with an increase in the radial distance $r$. Hence,
at the surface of the star, the pressure becomes zero as expected.

Figure 3: Variation of radial frequencies $\nu_{n}$ with oscillation modes $n$
for different $\Lambda$ values and corresponding to different masses and radii
which are listed in Tab. 1. The first plot is for the MIT Bag model EoS and
the second plot is for the linear EoS.

As mentioned above, we have calculated the fundamental $f$-mode and 22 lowest
$p$-modes of radial oscillation of strange stars for the considered EoSs. The
variation of oscillation frequencies $\nu_{n}$ (in kHz) is plotted against the
oscillation modes $n$ in Fig. 3. Oscillation frequency increases linearly with
the order of modes. For all values of $\Lambda$, it is increasing linearly
with increasing modes for both of the EoSs. We have observed the maximum
oscillation frequency for $\Lambda=-150\,\epsilon$, and the minimum frequency
is obtained for $\Lambda=15\,\epsilon$ for both of the EoSs. The first panel
of this figure is for the MIT Bag model EoS and the second panel is for the
linear EoS. Linear EoS is giving a frequency above $240\,\mbox{kHz}$, while
for the case of the MIT Bag model the maximum frequency is not exceeding
$150\,\mbox{kHz}$.

Table 1:  Radial oscillation frequencies $\nu_{n}$ in kHz for the MIT Bag model EoS for negative, zero and positive values of $\Lambda$. Modes (Order n) | $\Lambda=-150\,\epsilon$ | $\Lambda=-100\,\epsilon$ | $\Lambda=-50\,\epsilon$ | $\Lambda=0$ | $\Lambda=5\,\epsilon$ | $\Lambda=10\,\epsilon$ | $\Lambda=15\,\epsilon$
---|---|---|---|---|---|---|---
$f$ (0) | 10.34 | 8.76 | 7.04 | 5.08 | 4.85 | 4.59 | 4.31
$p_{1}$ (1) | 16.07 | 14.02 | 11.91 | 9.88 | 9.69 | 9.49 | 9.30
$p_{2}$ (2) | 22.09 | 19.28 | 16.52 | 14.09 | 13.88 | 13.68 | 13.48
$p_{3}$ (3) | 28.31 | 24.67 | 21.14 | 18.15 | 17.91 | 17.68 | 17.47
$p_{4}$ (4) | 34.59 | 30.11 | 25.78 | 22.16 | 21.88 | 21.62 | 21.38
$p_{5}$ (5) | 40.90 | 35.58 | 30.43 | 26.16 | 25.83 | 25.52 | 25.25
$p_{6}$ (6) | 47.21 | 41.06 | 35.10 | 30.14 | 29.76 | 29.41 | 29.10
$p_{7}$ (7) | 53.53 | 46.54 | 39.76 | 34.13 | 33.69 | 33.29 | 32.94
$p_{8}$ (8) | 59.84 | 52.02 | 44.43 | 38.11 | 37.62 | 37.17 | 36.77
$p_{9}$ (9) | 66.16 | 57.05 | 49.10 | 42.09 | 41.55 | 41.05 | 40.60
$p_{10}$ (10) | 72.47 | 62.99 | 53.77 | 46.07 | 45.48 | 44.93 | 44.43
$p_{11}$ (11) | 78.79 | 68.47 | 58.45 | 50.06 | 49.41 | 48.81 | 48.27
$p_{12}$ (12) | 85.10 | 73.95 | 63.12 | 54.04 | 53.34 | 52.69 | 52.10
$p_{13}$ (13) | 91.41 | 79.43 | 67.79 | 58.03 | 57.27 | 56.57 | 55.93
$p_{14}$ (14) | 97.71 | 84.91 | 72.47 | 62.01 | 61.20 | 60.44 | 59.76
$p_{15}$ (15) | 104.02 | 90.37 | 77.14 | 66.00 | 65.13 | 64.33 | 63.60
$p_{16}$ (16) | 110.33 | 95.86 | 81.82 | 69.99 | 69.06 | 68.21 | 67.43
$p_{17}$ (17) | 116.65 | 101.36 | 86.49 | 73.98 | 72.99 | 72.09 | 71.26
$p_{18}$ (18) | 122.96 | 106.77 | 91.15 | 77.96 | 76.93 | 75.97 | 75.10
$p_{19}$ (19) | 129.28 | 112.31 | 95.82 | 81.95 | 80.86 | 79.85 | 78.93
$p_{20}$ (20) | 135.60 | 117.81 | 100.50 | 85.94 | 84.79 | 83.74 | 82.77
$p_{21}$ (21) | 141.91 | 123.33 | 105.19 | 89.93 | 88.73 | 87.62 | 86.61
$p_{22}$ (22) | 148.23 | 128.77 | 109.85 | 93.92 | 92.66 | 91.50 | 90.44

In Tables 1 and 2, we summarize the radial oscillation frequency modes of
strange stars obtained for the MIT Bag model EoS and the linear EoS
respectively for all considered values of $\Lambda$. From these tables, it is
inferred that for all these modes, more positive $\Lambda$ values are giving
smaller oscillation frequencies i.e. with increasing $\Lambda$ values
oscillation frequencies are getting smaller for each oscillation mode.

Table 2:  Radial oscillation frequencies $\nu_{n}$ in kHz for the linear EoS for negative, zero and positive values of $\Lambda$. Modes (Order n) | $\Lambda=-150\,\epsilon$ | $\Lambda=-100\,\epsilon$ | $\Lambda=-50\,\epsilon$ | $\Lambda=0$ | $\Lambda=5\,\epsilon$ | $\Lambda=10\,\epsilon$ | $\Lambda=15\,\epsilon$
---|---|---|---|---|---|---|---
$f$ (0) | 18.67 | 16.44 | 14.03 | 11.44 | 11.18 | 10.91 | 10.64
$p_{1}$ (1) | 28.90 | 25.78 | 22.42 | 18.86 | 18.49 | 18.13 | 17.76
$p_{2}$ (2) | 38.98 | 34.90 | 30.52 | 25.89 | 25.41 | 24.94 | 24.47
$p_{3}$ (3) | 49.06 | 43.97 | 38.52 | 32.79 | 32.21 | 31.63 | 31.05
$p_{4}$ (4) | 59.17 | 53.04 | 46.50 | 39.64 | 38.94 | 38.25 | 37.56
$p_{5}$ (5) | 69.32 | 62.12 | 54.47 | 46.46 | 45.65 | 44.84 | 44.04
$p_{6}$ (6) | 79.51 | 71.22 | 62.44 | 53.27 | 52.35 | 51.42 | 50.50
$p_{7}$ (7) | 89.72 | 80.35 | 70.42 | 60.08 | 59.03 | 58.00 | 56.96
$p_{8}$ (8) | 99.95 | 89.49 | 78.41 | 66.88 | 65.72 | 64.57 | 63.41
$p_{9}$ (9) | 110.21 | 98.64 | 86.41 | 73.69 | 72.41 | 71.14 | 69.87
$p_{10}$ (10) | 120.47 | 107.81 | 94.42 | 80.50 | 79.10 | 77.71 | 76.32
$p_{11}$ (11) | 130.76 | 116.98 | 102.43 | 87.32 | 85.80 | 84.29 | 82.78
$p_{12}$ (12) | 141.05 | 126.17 | 110.45 | 94.13 | 92.50 | 90.86 | 89.24
$p_{13}$ (13) | 151.35 | 135.37 | 118.48 | 100.96 | 99.20 | 97.44 | 95.70
$p_{14}$ (14) | 161.66 | 144.57 | 126.52 | 107.78 | 105.91 | 104.03 | 102.17
$p_{15}$ (15) | 171.97 | 153.78 | 134.56 | 114.61 | 112.61 | 110.62 | 108.63
$p_{16}$ (16) | 182.30 | 162.99 | 142.60 | 121.45 | 119.33 | 117.21 | 115.11
$p_{17}$ (17) | 192.62 | 172.21 | 150.65 | 128.28 | 126.04 | 123.81 | 121.58
$p_{18}$ (18) | 202.95 | 181.43 | 158.70 | 135.12 | 132.76 | 130.40 | 128.06
$p_{19}$ (19) | 213.28 | 190.65 | 166.76 | 141.97 | 139.48 | 137.00 | 134.53
$p_{20}$ (20) | 223.62 | 199.88 | 174.82 | 148.81 | 146.20 | 143.60 | 141.01
$p_{21}$ (21) | 233.96 | 209.11 | 182.88 | 155.66 | 152.93 | 150.21 | 147.50
$p_{22}$ (22) | 244.30 | 218.35 | 190.94 | 162.51 | 159.65 | 156.81 | 153.98

Also, each mode frequency for all $\Lambda$ values is larger for the linear
EoS than that for the MIT Bag model EoS. These are already visualized in Fig.
3.

Figure 4: Variation of difference between consecutive modes of radial
frequencies ($\Delta\nu_{n}$) with radial frequencies $\nu_{n}$ for different
$\Lambda$ values. First panel and second panel are for the MIT Bag model EoS
and linear EoS respectively.

In Fig. 4, the variation of separation between two consecutive modes of radial
frequencies ($\Delta\nu$) with respect to the mode frequencies $\nu$ is shown.
The first panel is for the MIT Bag model and the second panel is for the
linear EoS. For the MIT Bag model with positive $\Lambda$ values, a gradually
decreasing pattern is observed. The difference between the oscillation
frequencies of $f$-mode and $p_{1}$-mode is larger than that of the other
values. However, in the case of more negative values of $\Lambda$, the
opposite behaviour is observed. Unlike other $\Lambda$ values, for
$\Lambda=-150\,\epsilon$ and $-100\,\epsilon$ the difference between
oscillation frequencies of $f$-mode and $p_{1}$-mode is smaller than the rest
of the variations. In the case of linear EoS, nearly a smooth variation for
all the considered $\Lambda$ values is observed. For $\Lambda=-150\,\epsilon$
both EoSs give maximum value of $\Delta\nu$ and for $\Lambda=15\,\epsilon$
they give a minimum value of $\Delta\nu$ in case of modes other than $f$-mode
and $p_{1}$-mode with MIT Bag model EoS. That is under this condition
$\Delta\nu$ decreases with increasing values of $\Lambda$. This figure also
clarifies that for $\Lambda=-150\,\epsilon$, the radial frequencies are
maximum whereas the frequencies are minimum for $\Lambda=15\,\epsilon$ for
both MIT Bag model EoS and linear EoS as also seen from Tables 1 and 2
respectively.

Figure 5: Variation of $f$-mode of radial perturbations $\xi(r)$ with radial
distance $r$ (in km) obtained by using the MIT Bag model EoS (first panel) and
the linear EoS (second panel) for different $\Lambda$ values.

From the Sturm-Liouville dynamic pulsation equations in presence of a
cosmological constant (equations (9)-(II.2)), we have studied the variation of
radial and pressure perturbations with the radial distance $r$. In Fig. 5 and
6 the variation of radial perturbation $\xi(r)$ with radial distance $r$ is
shown for $f$-mode and $p_{22}$-mode respectively. In the first panel of Fig.
5, the variation of radial perturbation $\xi(r)$ with $r$ (in km) is shown for
the MIT Bag model and in the second panel, the variation is shown for the
linear EoS. From these two plots, it is clear that the radial perturbation is
larger only near the centre of the star. It is diminishing near the surface of
the star for both these EoSs. As the EoSs with different $\Lambda$ values are
mimicking strange stars of different sizes, the perturbation along the radial
distances are found different.

Figure 6: Variation of radial perturbations $\xi(r)$ with radial distance $r$
(in km) obtained by using the MIT Bag model EoS (first panel) and the linear
EoS (second panel) for different $\Lambda$ values for the case of pressure
$p_{22}$-mode.

Similarly in Fig. 6, the variation is shown for higher-order $p_{22}$-mode.
For the MIT Bag model, it is shown in the first panel of the figure. As like
the $f$-mode, the perturbation is larger near the centre of the star. Towards
the surface, a distinct damping nature of radial perturbation is noticed. The
different $\Lambda$ values are also showing distinct variations in their
respective perturbations. As shown in the second panel of Fig. 6, for the case
of linear EoS, the variation of $p_{22}$ mode is behaving similarly to that of
the MIT Bag model EoS. The perturbation is decreasing along the surface of the
star. Intermediate pressure perturbations curves can be drawn which will lie
in between $f$ and $p_{22}$ modes.

Figure 7: Variation showing pressure perturbations $\eta(r)$ with radial
distance $r$ (in km) found by using MIT the Bag model EoS (first panel) and
the linear EoS (second panel) for different $\Lambda$ values for fundamental
$f$-mode.

The variation of pressure perturbation is different from the variation of
radial perturbation. As clear from Fig. 7 and 8, the variation is larger near
the centre and surface of each star. Whereas, for the case of radial
perturbation, it is larger near the centre of the star only. In the first
panel of Fig. 7, the change in pressure perturbation with distance from the
centre of the star is showing for the MIT Bag model EoS for fundamental
$f$-mode due to different $\Lambda$ values. The different sizes of each star
for each $\Lambda$ values are also clear from this plot. Similar behaviour is
observed for linear EoS with fundamental $f$-mode. This can be visualised from
the second panel of this Fig. 7.These variations shown in this figure
correspond to different values of mass and radius corresponding to considered
$\Lambda$ values. As an example, for the MIT Bag model EoS i.e. in the left
panel of Fig. 7 and for $\Lambda=15\,\epsilon$, the corresponding mass and
radius are $\approx 3.3\,\mbox{M}_{\odot}$ and $\approx 14.3\,\mbox{km}$
respectively. The other values of mass and radius corresponding to these
variations can be found in Table 3.

Figure 8: Variation showing pressure perturbations $\eta(r)$ with radial
distance $r$ (in km) found by using the MIT Bag model EoS (first panel) and
the linear EoS (second panel) for different $\Lambda$ values for the pressure
$p_{22}$-mode.

This pressure variation for higher-order oscillation mode i.e. for
$p_{22}$-mode is shown in Fig. 8. The first panel corresponds to the MIT Bag
model and the second panel corresponds to the linear EoS. Similar to the
$f$-mode of oscillation this variation is also larger near the centre and near
the surface of the star. For all other modes in between $f$-mode and
$p_{22}$-mode, pressure perturbation will stay in between.

Figure 9: The variation of compactness of strange stars with different
$\Lambda$ values for the MIT Bag model EoS (first panel) and the linear EoS
(second panel).

The value of cosmological constant $\Lambda$ varying with different strange
star configurations can be visualized in Fig. 9. The left panel of this figure
is for the MIT Bag model and the right panel is for the linear EoS. For both
of the cases, with increasing $\Lambda$ value compactness is found to
decrease. The fall is somewhat rapid for the MIT Bag model EoS than that of
the linear EoS. The different values of these plots can be found in Table 1
and in Table 2.

Figure 10: The variation of GWE frequencies with different $\Lambda$ values
for the MIT Bag model EoS (first panel) and the linear EoS (second panel).

After the numerical analysis of the perturbation equations, another important
part of this work is to find the effect of cosmological constant $\Lambda$ on
GWE frequencies. The GWE frequencies are found to diminish with an increase in
$\Lambda$ values, for both of the EoSs. For the MIT Bag model with negative
$\Lambda$ values the variation is straightaway, however, a rapid drop is
observed for positive $\Lambda$ values. The linear EoS is showing almost a
straight variation for both positive and negative values of $\Lambda$. These
variations are shown in Fig. 10, where the first panel is for the MIT Bag
model and the second panel is for the linear EoS.

Table 3:  Mass, radius, compactness of strange stars, characteristic echo time and GWEs from strange stars for the MIT Bag model EoS with the bag constant $B=(190\,\mbox{MeV)}^{4}$. | Value of $\Lambda$ | Radius R | Mass M | Compactness | Echo | GWE
---|---|---|---|---|---|---
(in terms of $\epsilon$) | (in $\,\mbox{cm}^{-2}$) | (in km) | (in $\mbox{M}_{\odot}$) | (M/R) | time (ms) | frequency (kHz)
-150$\,\epsilon$ | -2.66$\times 10^{-12}$ | 13.003 | 3.327 | 0.3784 | 0.048 | 64.90
-100$\,\epsilon$ | -1.77$\times 10^{-12}$ | 13.090 | 3.315 | 0.3745 | 0.055 | 56.74
-50$\,\epsilon$ | -8.88$\times 10^{-13}$ | 13.259 | 3.300 | 0.3681 | 0.065 | 48.44
0 | 0 | 13.766 | 3.295 | 0.3540 | 0.078 | 39.91
5$\,\epsilon$ | 8.88$\times 10^{-14}$ | 13.893 | 3.300 | 0.3512 | 0.080 | 38.86
10$\,\epsilon$ | 1.77$\times 10^{-13}$ | 14.067 | 3.308 | 0.3477 | 0.083 | 37.42
15$\,\epsilon$ | 2.66$\times 10^{-13}$ | 14.340 | 3.326 | 0.3430 | 0.089 | 35.08
Table 4:  Mass, radius, compactness of strange stars, characteristic echo time and GWEs from strange stars for the linear EoS with the linear constant $b=0.910$. | Value of $\Lambda$ | Radius R | Mass M | Compactness | Echo | GWE
---|---|---|---|---|---|---
(in terms of $\epsilon$) | (in $\,\mbox{cm}^{-2}$) | (in km) | (in $\mbox{M}_{\odot}$) | (M/R) | time (ms) | frequency (kHz)
-150$\,\epsilon$ | -2.66$\times 10^{-12}$ | 7.200 | 1.742 | 0.3577 | 0.030 | 103.42
-100$\,\epsilon$ | -1.77$\times 10^{-12}$ | 7.271 | 1.749 | 0.3557 | 0.033 | 94.24
-50$\,\epsilon$ | -8.88$\times 10^{-13}$ | 7.372 | 1.759 | 0.3529 | 0.037 | 84.12
0 | 0 | 7.535 | 1.775 | 0.3484 | 0.043 | 72.91
5$\,\epsilon$ | 8.88$\times 10^{-14}$ | 7.558 | 1.778 | 0.3478 | 0.044 | 71.71
10$\,\epsilon$ | 1.77$\times 10^{-13}$ | 7.584 | 1.780 | 0.3472 | 0.045 | 70.51
15$\,\epsilon$ | 2.66$\times 10^{-13}$ | 7.610 | 1.783 | 0.3464 | 0.045 | 69.40

The properties of stellar structure such as mass, radius, and compactness
along with the results for the GWE frequencies and characteristic echo times
that are obtained by using the MIT Bag model EoS are compiled in Table 3. From
this table, the change in these properties of strange stars for the MIT Bag
model with different $\Lambda$ values can be seen clearly. With increasing
$\Lambda$ values the size of stars is increasing. The compactness of the stars
is decreasing with more positive values. Thus the upper limit on cosmological
constant $\Lambda$ value is depicting a star with compactness such that it is
just able to echo the falling GWs. For our considered maximum value of
$\Lambda$ (15$\,\epsilon$), the compactness is found to be 0.3430, which is
greater than 1/3. The characteristic echo time obtained for this EoS increases
with the increasing $\Lambda$ value, and hence the reverse effect is noticed
for the GWE frequencies. For this EoS, all echo frequencies obtained are in
the range of tens of kilohertz.

The sizes of stars obtained by using the linear EoSs are smaller than that
obtained for the MIT Bag model EoSs. Eventually, the masses are also small in
comparison to that for the MIT Bag model and hence fulfilling the criteria for
echoing GWs. These results are shown in Table 4. For all the chosen $\Lambda$
values, the stars are found to be with enough compactness to emit GWE
frequencies. The characteristic echo times obtained are smaller than that of
the MIT Bag model EoS and hence in turn the larger echo frequencies are
obtained for this EoS.

Table 5:  Properties of strange stars for the MIT Bag model and linear EoS with different values of $\Lambda$. | Value of $\Lambda$ |  |  | MIT Bag model |  |  |  | Linear EoS |
---|---|---|---|---|---|---|---|---|---
(in $\,\mbox{cm}^{-2}$) | (in terms of $\epsilon$) | Radius | Mass | Compactness | Echo freq- | Radius | Mass | Compactness | Echo freq-
|  | (in km) | (in $\mbox{M}_{\odot}$) | (M/R) | uency (kHz) | (in km) | (in $\mbox{M}_{\odot}$) | (M/R) | uency (kHz)
$1\times 10^{-18}$ | 5.6$\times 10^{-5}\,\epsilon$ | 13.766 | 3.295 | 0.354 | 39.91 | 7.535 | 1.775 | 0.348 | 72.91
$1\times 10^{-16}$ | 5.6$\times 10^{-3}\,\epsilon$ | 13.766 | 3.295 | 0.354 | 39.91 | 7.535 | 1.775 | 0.348 | 72.91
$1\times 10^{-15}$ | 5.6$\times 10^{-2}\,\epsilon$ | 13.767 | 3.296 | 0.354 | 39.90 | 7.535 | 1.775 | 0.348 | 72.89
$1\times 10^{-14}$ | 5.6$\times 10^{-1}\,\epsilon$ | 13.778 | 3.296 | 0.354 | 39.80 | 7.538 | 1.775 | 0.348 | 72.76
$5\times 10^{-14}$ | 2.8$\,\epsilon$ | 13.834 | 3.298 | 0.353 | 39.34 | 7.548 | 1.776 | 0.348 | 72.23
$1\times 10^{-13}$ | 5.6$\,\epsilon$ | 13.912 | 3.301 | 0.351 | 38.66 | 7.558 | 1.778 | 0.348 | 71.71
$2\times 10^{-13}$ | 11.2$\,\epsilon$ | 14.122 | 3.312 | 0.347 | 36.99 | 7.590 | 1.781 | 0.347 | 70.20
$3\times 10^{-13}$ | 16.8$\,\epsilon$ | 14.492 | 3.338 | 0.341 | 35.26 | 7.621 | 1.784 | 0.346 | 68.28

##  VI Conclusions

In this study, we look into the role of the cosmological constant on two
interesting aspects of strange stars. In one part we have tried to understand
its impact over the radial oscillations of strange stars and in another, we
have investigated its effect on GWE frequencies emitted by a strange star,
like the star formed in the binary merging event GW170817. This study is made
in the general relativistic framework. To study its role over radial
oscillations we solved the TOV equations for two EoSs, viz., the MIT bag model
EoS and the linear EoS. The solution of TOV equations leads us to know the
structure of strange stars in presence of cosmological constant $\Lambda$,
which is found to be different from the structure obtained by using the
vanishing $\Lambda$ jb . Again as discussed in Sec. V, different $\Lambda$
values give us different strange star configurations. Moreover, the pulsation
equations developed by Chandrasekhar are modified for spacetime with a
cosmological constant by introducing two dimensionless parameters $\xi$ and
$\eta$. The solution of these pulsation equations gave us eigenfrequencies of
oscillations and hence we have calculated the $f$-mode and first 22 $p$-modes
of oscillations. We found that the large value of $\Lambda$ decreases the
radial oscillation frequencies of stars for both of the EoSs. Further, to see
the role of $\Lambda$ on GWEs first we calculated the characteristic echo time
of GWs falling on strange stars.

The study clearly shows that at the present cosmological constant value around
$\Lambda=(4.24\,\pm\,0.11)\times 10^{-66}\,\mbox{eV}^{2}$ ($\equiv\;\sim
10^{-56}$ cm-2), the maximum mass and radius, radial oscillations, and echo
frequencies of strange stars are identical with the GR values and can’t be
distinguished both theoretically and experimentally. Hence, the observations
of such parameters of a strange star can’t be used to differentiate between GR
and the $\Lambda$CDM model for the present value of $\Lambda$. However, bounds
on $\Lambda$ from the stellar stability and interior structure nayak ; Bordbar
; Liu ; arbanil  are comparatively weak, which gives us a wide range of
possible $\Lambda$ values. We have shown that within this boundary, the
cosmological constant $\Lambda$ with a large magnitude can effectively put
significant impacts on the mass, radius, radial oscillations, and echo
frequencies of a strange star. Our results showed that for strange stars, the
effective upper limit of $\Lambda$ is $\sim 3\times 10^{-13}\,\mbox{cm}^{-2}$
and the lower limit is $\sim 10^{-15}\,\mbox{cm}^{-2}$. The maximum mass,
radius, radial oscillation frequencies, and echo frequencies of strange star
configurations have changed significantly within these boundary values. These
impacts again depend on the EoS of the star. These results are presented in
Table 5. According to the results of this table the cosmological constant has
no significant effect on the properties of a strange star when its value is
less than $10^{-15}\,\mbox{cm}^{-2}$. This table generalizes our results,
whereas in the graphs only large values of $\Lambda$ are taken to show the
changes distinctly and also to visualize the variation pattern or dependency
of the same with $\Lambda$ efficiently. Again some Ref.s Bordbar ; Liu ;
arbanil ; wei  suggest a possibility of varying $\Lambda$ with the
cosmological evolution. Similarly, in some modified gravity theories like
$f(R)$ gravity in Palatini formalism, the field equations can be written in GR
form with an effective $\Lambda$ term pano . This indicates the fact that the
cosmological evolution obtained by using a feasible $f(R)$ gravity model can
mimic the $\Lambda$CDM model in the present and early universe effectively.
However, depending on the functional form and uniqueness of such $f(R)$
models, the effective $\Lambda$ term varies differently dj ; santos .
Therefore, consideration of different values of $\Lambda$ in a wide range can
be helpful to see the possible impacts of it on the strange star structures in
theories with varying dark energy or $f(R)$ gravity in Palatini formalism,
where the effective $\Lambda$ term is not a constant. However, we agree that a
functional variation of $f(R)$ gravity models can have other impacts on the
strange star properties also and we leave it as the future scope of our study.
An important outcome of the study is that a $\Lambda$ value below $1\times
10^{-15}\,\mbox{cm}^{-2}$ does not have significant observable impacts on the
maximum mass, maximum radius, radial oscillations, and echo frequencies of a
strange star with the MIT Bag model and linear EoSs. The boundary values of
$\Lambda$ are found to be the same for these two EoSs. As these both EoSs are
stiff EoSs, with almost similar compactness so the boundary values on
$\Lambda$ are similar for them.

Another important aspect of the study is that the study gives a theoretical
idea of the structure of a strange star in terms of a few properties like
maximum mass, maximum radius, radial oscillations, and echo frequencies in
both de Sitter and anti-de Sitter regime. The study with positive values of
cosmological constant $\Lambda$ (which corresponds to de Sitter space) gives a
more realistic description of strange stars. These are relevant from the
observational point of view and also used in the context of the dark energy
model of the universe. On the other hand, the negative cosmological constant
$\Lambda$ corresponding to anti-de Sitter space is important for anti-de
Sitter/conformal field theory correspondence as mentioned earlier.

To know the enigmatic stellar interior of compact stars using asteroseismology
several ground-based missions are proposed in addition to few early space
missions in this regard. Some of such important missions are MOST most , CoRoT
corot , BRITE brite , Kepler/K2 kepler , PLATO plato , and TESS tess
missions. The unplanned pioneer mission in the field of observational
asteroseismology was NASA’s WIRE mission wire . The Canadian MOST mission was
the first space telescope dedicated to asteroseismology and was proposed to
monitor relatively bright stars. The French-led CoRoT mission was a great
success for massive star variability studies. The BRITE-constellation of
nanosatellites is another important photometry mission available for
asteroseismology and it is ranked amongst the highest givers for providing
excellent asteroseismic returns. The most famous of all space telescopes
providing time series photometry is the Kepler mission launched in 2009. It is
targeted to study solar like stars all over the HR diagram. The next
generation of stellar missions such as PLATO and other current and future
Earth based radio telescopes, such as Arecibo qui  or SKA ska  have the
potential to put important constraints in the stellar properties, or to find
such compact stars. Furthermore, the recently launched NASA’s mission NICER
nicer , designed primarily to observe thermal X-rays emitted by several
millisecond pulsars, is another ray of hope to detect such stars.

Again, the current GW detectors have comparatively low sensitivity (detectable
frequency range is $\sim$ 20 Hz - 4 kHz martynov ; abbot ), so unfortunately
the ongoing detectors does not allow the detection of such compact stellar
oscillation frequencies. The next generation of ground-based GW detectors,
e.g., the Einstein Telescope (ET) et  and the Cosmic Explorer (CE) ce  are
expected to have a sensitivity much higher than an order of magnitude in
comparison to the Advanced LIGO. These detectors are supposed to detect such
emission and provide information on the oscillation of compact stars chiren .
If in near future such frequencies are detected, they could also provide the
simultaneous measurements of compact star masses, tidal Love numbers benni ,
frequency, damping time, amplitude of the modes. Furthermore, the expectations
for the detection of the compact star’s oscillations will also be increased by
the launch of the eXTP extp  and other X-ray missions. When the detection of
the radial oscillations of neutron stars and other compact stars will become
possible, then it can be used to constrain the EoSs of compact stars with high
precision and hence the corresponding stellar properties.

From the emission of GWs, the most important modes to be detected are the
fundamental $f$-modes and the first few pressure $p$-modes, in addition to the
$r$-modes and $w$-modes kokkotas . The $f$-mode of oscillation is the mode
through which most of the gravitational radiation of the star is radiated away
kokkotas97 . The $p$-mode is associated with the acoustic sound speed inside
the star. These $p$\- or the low order radial acoustic modes can be observed
through the associated GW perturbations. One of the obvious sources of
excitation of these radial (or non-radial) acoustic modes is the supernova
explosion gpano . The pulsating compact objects formed from such events emit
GWs. Strong star-quakes can also arise with their associated pulsar glitches
that can excite global stellar pulsations. Again, the compact stars that
undergo a phase transition, e.g. when a neutron star collapses into a strange
star and the coalescence of two compact stars may form a pulsating remnant
compact star lusky . These mechanisms of excitation generally give $f$-modes
as well as radial $p$-modes. Moreover, due to the similar excitation mechanism
the $f$-modes and $p$-mode can be determined using similar strategy glam .
Besides these excitation mechanisms, such modes can also be excited by tidal
forces in a close eccentric binary system chirenti  and a resonant excitation
in binaries hind . Accretion in Low-Mass X-ray Binaries (LMXBs) can also
excite these modes ander . Star-quakes caused by cracks in the crust, magnetic
reconfiguration or any other dynamical instabilities are also associated with
the excitation of stellar oscillation modes franco ; tsang .

It is also worth mentioning that in actual practice, the large amplitude of
oscillation modes only occurs in catastrophic situations, such as in core-
collapse supernovae. Such events are highly non-spherical and hence in such
cases, the non-radial modes could become more applicable. So a possible
extension of our present work would be to study the non-radial oscillations of
strange stars and the corresponding GW signals associated with them. In abedi
, the authors have claimed a tentative detection of echoes from the remnant of
GW170817 event at a frequency $\simeq 72\,\mbox{Hz}$. Although the consistent
mass ($2.6-2.7\,\mbox{M}_{\odot}$) of this remnant almost lies within the
range of masses of strange stars predicted by our EoSs with different
$\Lambda$ values, the GWE frequencies found from our calculations are much
higher (in kHz range) than this tentative detection value. The basic reason
for the difference between this result and our results is that instead of
considering the remnant of GW170817 as an exotic compact object (ECO), we have
considered the star of this mass range as a strange star. Further, while
considering the EoSs, we have neglected the possible temperature effect on
EoS. Again we choose strange stars as non-rotating ones and solved the TOV
equations for the static stellar model while investigating these properties.
So, the study of rotating, anisotropic strange stars or, other possible ECO is
a topic that will be considered for future work.

## Acknowledgments

JB would like to thank Dibrugarh University, India for the financial support
through the grant ‘DURF (Extension)-2020-21’ while carrying out this work.

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