# Non-radial oscillations in anisotropic dark energy stars

O. P. Jyothilakshmi<EMAIL_ADDRESS>Lakshmi J. Naik
<EMAIL_ADDRESS>V. Sreekanth<EMAIL_ADDRESS>Department of Physics, Amrita School of Physical Sciences, Amrita Vishwa
Vidyapeetham, Coimbatore, India

###### Abstract

We study the non-radial $f$-mode oscillations of both isotropic and
anisotropic dark energy stars by using the modified Chaplygin prescription of
dark energy to model the stellar matter. The anisotropic pressure in the
system is modeled with Bowers-Liang prescription. By solving the stellar
structure equations in presence of anisotropy, we study the global properties
of the dark energy star and compare the mass-radius profiles with data from GW
events and milli-second pulsars. We proceed to determine the prominent non-
radial $l=2$ $f$-mode frequencies of the anisotropic dark energy star by
employing the Cowling approximation and analyse and quantify the spectra by
varying the anisotropic parameter. We report that $f$-mode spectra of dark
energy star have distinctly different behaviour compared to neutron star and
quark star, and this may possibly help in its future identification. Further,
the tidal deformability factors of the anisotropic dark energy stars have also
been analyzed.

††preprint:

##  I Introduction

Dark energy is the fluid component that governs the accelerated expansion of
the universe Riess  _et al._ (1998); Perlmutter  _et al._ (1999); Freedman and
Turner (2003). It is found from various observations that dark energy fills up
almost $70\%$ of the universe Hinshaw  _et al._ (2009). The $\Lambda$CDM model
(also known as the concordance model) Einstein (1917) is one of the most
accepted cosmological models describing dark energy. However, the $\Lambda$CDM
model suffers from problems like the cosmological constant problem Weinberg
(1989); Zeldovich (1967), and the Hubble tension Mörtsell and Dhawan (2018);
Verde  _et al._ (2013); Bolejko (2018). Therefore, many alternative models
have been proposed in the last few decades. A detailed review on various
dynamical models of dark energy such as cosmological constant, quintessence,
K-essence, tachyon field, phantom field, dilatonic field, Chaplygin gas model,
etc. are given in Ref. Copeland  _et al._ (2006).

The fact that interior composition of a compact star is unknown to this date
has motivated a lot of researchers to construct different models of compact
stars. There are several stellar models that include dark energy such as false
vacuum bubbles Coleman and De Luccia (1980), non-singular black holes
Dymnikova (1992), gravastars Mazur and Mottola (2023), and dark energy stars
Chapline (2004). Among these models, we are interested in the dark energy
star, which was first proposed by Chapline Chapline (2004). The author gave an
alternative explanation to astrophysically observed black holes suggesting
that these compact objects may be dark energy stars. The author also proposed
that matter is converted to dark energy when it falls through the event
horizon.

Using the general relativistic prescriptions, the structure of compact stars
can be obtained by describing the interior composition with an equation of
state (EoS). Thus obtained global properties of compact objects are compared
with the astrophysical observations to constrain the EoS. Recently detected
gravitational waves (GWs) from binary neutron star mergers introduced a new
way of restricting the dense matter EoS within neutron stars Abbott  _et al._
(2018). The constraints of EoS also include the maximum mass limit set by
various observational data from pulsars Cromartie  _et al._ (2019); Antoniadis
_et al._ (2013). GW asteroseismology is the study of the interior of compact
objects using GW observations. The stellar oscillations can produce GWs, this
has drawn a great interest recently in the field of GW asteroseismology. The
analysis of stellar oscillations is used to understand the microscopic and
macroscopic properties of compact objects Lindblom and Owen (2002); Jha  _et
al._ (2010); Sotani  _et al._ (2011); Doneva and Yazadjiev (2012); Flores and
Lugones (2014); Jyothilakshmi  _et al._ (2022); Tran  _et al._ (2023). The
non-radial modes of stellar oscillations are an important class among these
and the general relativistic treatment of the same can be found in Ref. Thorne
and Campolattaro (1967); Detweiler and Lindblom (1985).

These stellar oscillations can be classified into various quasi-normal modes
based on the force that restores the system back to equilibrium. Some examples
include fundamental $f$-mode, pressure $p$-mode, gravity $g$-mode, rotational
$r$-mode, and space-time $w$-mode Kokkotas and Schmidt (1999). Among these
modes, $f$-modes are considered to be a promising candidate for GW emissions,
which has frequencies in the sensitivity range of upcoming GW detectors
Andersson (2003).

Another important constraint imposed on EoS is by the tidal distortions caused
from binary neutron star inspiral Hinderer (2008); Hinderer  _et al._ (2010).
The tidal deformability parameter $\Lambda$ describes the gravitational wave
signal emitted during a binary neutron star inspiral Hinderer (2008); Hinderer
_et al._ (2010). It determines a star’s quadrupole deformation caused by its
companion star’s tidal field.

This is in addition to the constraints previously provided by the
electromagnetic studies of neutron stars, which include their masses, radii,
spin, and gravitational redshift. The tidal properties of neutron stars can be
measured from the GW signals Annala  _et al._ (2018).

In most of the astrophysical studies, the fluid matter within dense compact
stars is assumed to be locally isotropic. However, the high density and strong
gravity suggest that the interior of compact stars could be anisotropic,
meaning the pressure in radial and tangential directions are different. Strong
magnetic field, superfluid cores, phase transition, pion condensation etc. may
give rise to pressure anisotropy. Ruderman in 1972 was the first to propose
the idea of anisotropy in compact stars Ruderman (1972) and later, the
relativistic stellar structure equations for anisotropic stars were obtained
by Bowers and Liang Bowers and Liang (1974).

Apart from this, various other anisotropy models are used in the study of
compact stars Horvat  _et al._ (2011); Herrera and Barreto (2013); Raposo  _et
al._ (2019). See Ref. Kumar and Bharti (2022) for a recent review on
anisotropic compact stars.

Non-radial oscillations were studied in anisotropic neutron stars Doneva and
Yazadjiev (2012) and recently, an extension of this work was done for hadronic
and quark stars Curi  _et al._ (2022). These studies show that presence of
anisotropy has significant effect on non-radial oscillations. Further, the
impact of anisotropy on the tidal deformability was also studied Biswas and
Bose (2019). It will be interesting to study these effects in dark energy
stars. There has been a considerable interest to study the structure and
properties of compact objects with dark energy. In Ref. Lobo (2006), the
author introduced a model of dark energy star and studied its stability.
Anisotropic dark energy stars were considered in Ref. Chan  _et al._ (2009).
Another model of dark energy star which includes neutron gas was constructed
and the stability of the model was also analysed Ghezzi (2011). A mixed dark
energy star model made up of baryonic matter and phantom scalar field has also
been considered Yazadjiev (2011). Further, a stable singularity-free
anisotropic dark energy star was constructed in Ref. Rahaman  _et al._ (2012).

A time-dependent solution of the Einstein field equations describing the
collapse of a spherical system resulting in a dark energy core was obtained in
Ref. Beltracchi and Gondolo (2019). The stability of a dark energy star with a
phantom field was also investigated Sakti and Sulaksono (2021). Recently, a
study on dark energy star within Newtonian treatment was performed with the
generalised Chaplygin EoS by including the effects of anisotropy Abellán  _et
al._ (2023). Global properties of slowly rotating isotropic dark energy stars
with extended Chaplygin EoS were analyzed in Ref. Panotopoulos  _et al._
(2021). Radial oscillations and tidal Love numbers using generalised Chaplygin
EoS prescription was performed for the isotropic Panotopoulos  _et al._ (2020)
and anisotropic Pretel (2023) dark energy stars. Further, it was shown that a
neutron star with dark energy core is dynamically stable under small radial
pulsations and is consistent with observational data Pretel  _et al._ (2024).
Dark energy stars within modified theories of gravity have also drawn interest
recently Das  _et al._ (2024); Bagheri Tudeshki  _et al._ (2024). In this
work, we intend to study the prominent non-radial $f$-mode oscillations in
both isotropic and anisotropic dark energy stars constructed using the
modified Chaplygin EoS. The anisotropy is introduced via the Bowers-Liang
model Bowers and Liang (1974). We also calculate tidal deformability for such
a system.

The paper is organised as follows. In Section II, we discuss the modified
Chaplygin EoS describing dark energy used for the analysis. We present the
stellar structure equations and non-radial oscillations for anisotropic dark
energy star in Section III. The results of our analysis are shown in Section
IV and finally we draw the conclusions of the study and summarise in Section
V.

Notations and conventions: Throughout this paper, we set Newton’s universal
gravitational constant $G$ and velocity of light in free space $c$ as $G=c=1$
and follow the metric convention $g_{\mu\nu}=\textrm{diag}(-1,1,1,1)$.

##  II Chaplygin Gas Model of Dark Energy

The Chaplygin fluid model of dark energy was obtained by Kamenshchik et al. in
2001 Kamenshchik  _et al._ (2001), where the universe is assumed to be filled
with the so called Chaplygin gas obeying the equation of state (EoS)

$p=-\hat{B}/\rho^{\omega},$ (1)

where $p$ is the pressure, $\rho$ is the energy density and $\hat{B}$ is a
positive constant with units of energy density and $\omega$ can take values in
the range $0<\omega\leq 1$.

This model provides a possible solution for the unification of dark matter and
dark energy Bilic  _et al._ (2002); Bento  _et al._ (2002); Gorini  _et al._
(2003); Xu  _et al._ (2012). A more generalised version of the Chaplygin EoS
was introduced in Ref. Zhang  _et al._ (2006). Furthermore, a modified
Chaplygin EoS was later constructed by including the effects of viscosity
Pourhassan (2013); Saadat and Pourhassan (2013)

$p=\hat{A}\rho-\frac{\hat{B}}{\rho^{\omega}},$ (2)

where $\hat{A}$ is a dimensionless positive constant. The first term
represents a barotropic and the second term corresponds to the Chaplygin gas.
By setting the value of $\omega=1$, one obtains the modified Chaplygin EoS in
the form Kahya and Pourhassan (2015)

$\displaystyle p=A^{2}\rho-\frac{B^{2}}{\rho}.$ (3)

Since the pressure vanishes at the surface of the star, the energy density
towards the surface takes the value of $\rho_{s}=B/A$. This EoS is used in
various studies on dark energy stars Panotopoulos  _et al._ (2020, 2021);
Pretel (2023); Abellán  _et al._ (2023); Das  _et al._ (2024); Bagheri
Tudeshki  _et al._ (2024); Pretel  _et al._ (2024) and following Refs.
Panotopoulos  _et al._ (2021, 2020); Pretel (2023), we take the values of the
constants as $A=\sqrt{0.4}$ and $B=0.23\times 10^{-3}$/km2.

Next, we proceed to study the non-radial $f$-mode oscillations of anisotropic
dark energy star.

##  III Anisotropy and Non-radial oscillation

In this section, we describe the stellar structure equations, non-radial
oscillations and tidal deformability for an anisotropic stellar system. The
stress-energy tensor $T^{\mu\nu}$ for an anisotropic fluid is given by

$\displaystyle T^{\mu\nu}$ $\displaystyle=\rho
u^{\mu}u^{\nu}+qg^{\mu\nu}+\sigma k^{\mu}k^{\nu};$ (4)

where $\sigma=p-q$ is the anisotropy pressure, $p$ is the radial pressure, $q$
is the tangential pressure, $\rho$ is the energy density, $u^{\mu}$ is the
four-velocity of the fluid, and $k_{\nu}$ is the radial vector. The four-
vectors satisfy the conditions: $u_{\mu}u^{\mu}=-1$, $k_{\mu}k^{\mu}=1$ and
$u^{\mu}k_{\mu}=0$ . We use the Bowers-Liang model for anisotropy Bowers and
Liang (1974),

$\displaystyle\sigma=-\lambda_{BL}\frac{r^{3}}{3}\left(\rho(r)+3p(r)\right)\left(\frac{\rho(r)+p(r)}{r-2m(r)}\right);$
(5)

where free parameter $\lambda_{BL}$ denotes the measure of anisotropy.
Following Ref. Biswas and Bose (2019), we consider $-2<\lambda_{BL}<2$ in this
work. The line element for a spherically symmetric static star in
Schwarzschild coordinates, ($t,\,r,\,\theta,\,\phi$) is given by

$\displaystyle ds^{2}$
$\displaystyle=-e^{2\Phi}dt^{2}+e^{2\lambda}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (6)

Here, $\Phi(r)$ and $\lambda(r)$ are the metric functions. Solving the
Einstein field equations for the static spherically symmetric metric for an
anisotropic fluid results in the stellar structure equations Bowers and Liang
(1974)

$\displaystyle\frac{dp}{dr}$ $\displaystyle=-\frac{(\rho+p)(m+4\pi
r^{3}p)}{r^{2}(1-2m/r)}-\frac{2\sigma}{r},$ (7a) $\displaystyle\frac{dm}{dr}$
$\displaystyle=4\pi r^{2}\rho.$ (7b)

These equations reduce to the well known Tolman-Oppenheimer-Volkoff (TOV)
equations Tolman (1939); Oppenheimer and Volkoff (1939) of the isotropic case,
if the value of anisotropic factor is set to zero $(\sigma=0)$. The above
shown coupled differential equations are integrated from the center to the
surface of the star for a given EoS describing the stellar matter. The initial
conditions towards the center of the star are pressure
$p_{c}=p(0)=p(\rho_{c})$ and mass $m_{c}=m(0)=0$. The pressure $p$ tends to
zero as it approaches the surface of the star ($r=R$) and the mass of the star
is then obtained as $M=m(R)$. For a given EoS, we can obtain stellar
configurations with different masses and radii by varying the value of central
density $\rho_{c}$.

The solutions obtained by solving Eqs. (7a) and (7b) are used to study the
non-radial oscillations of anisotropic dark energy stars. Here, we employ the
well known Cowling approximation which assumes an unperturbed space-time while
studying the fluid pertubations Cowling (1941). It was found that the results
obtained using this approximation differ from the general relativistic
solutions only by less than $20\%$ Sotani and Takiwaki (2020). The oscillation
equations in the Cowling approximation are obtained by considering the
perturbation of energy-momentum tensor conservation Flores and Lugones (2014);
Doneva and Yazadjiev (2012); Sotani  _et al._ (2011).

The system of differential equations in terms of the fluid perturbations
$V(r)$ and $W(r)$ representing the anisotropic non-radial oscillations are
given by Doneva and Yazadjiev (2012)

$\displaystyle\frac{dW}{dr}$ $\displaystyle=$
$\displaystyle\frac{d\rho}{dp}\Big{[}\omega^{2}\frac{\rho+p-\sigma}{\rho+p}\left(1-\frac{\partial\sigma}{\partial
p}\right)^{-1}e^{\lambda-2\Phi}r^{2}V$
$\displaystyle+\frac{d\Phi}{dr}W\Big{]}-l(l+1)e^{\lambda}V$
$\displaystyle+\frac{\sigma}{\rho+p}\left[\frac{2}{r}\left(1+\frac{d\rho}{dp}\right)W+l(l+1)e^{\lambda}V\right],$
$\displaystyle\frac{dV}{dr}$ $\displaystyle=$ $\displaystyle
2V\frac{d\Phi}{dr}-\left(1-\frac{\partial\sigma}{\partial
p}\right)\frac{\rho+p}{\rho+p-\sigma}\frac{e^{\lambda}}{r^{2}}W$
$\displaystyle+$
$\displaystyle\Big{[}\frac{\sigma^{\,\prime}}{\rho+p-\sigma}+\left(\frac{d\rho}{dp}+1\right)\frac{\sigma}{\rho+p-\sigma}\left(\frac{d\Phi}{dr}+\frac{2}{r}\right)$
$\displaystyle-$ $\displaystyle\frac{2}{r}\frac{\partial\sigma}{\partial
p}-\left(1-\frac{\partial\sigma}{\partial
p}\right)^{-1}\left(\frac{\partial^{2}\sigma}{\partial
p^{2}}p^{\prime}+\frac{\partial^{2}\sigma}{\partial
p\partial\mu}\mu^{\prime}\right)\Big{]}V.$

Here, $\mu=2m/r=1-e^{-2\lambda}$. These differential equations are solved from
the center to the surface of the star, which obey the following conditions as
$r\rightarrow 0$:

$\displaystyle W=Ar^{l+1},\;\;\;V=-\frac{A}{l}r^{l};$ (9)

where $A$ is an arbitrary constant. The boundary condition towards the surface
of the star is Doneva and Yazadjiev (2012)

$\displaystyle\omega^{2}\left(1-\frac{\sigma}{\rho+p}\right)\left(1-\frac{\partial\sigma}{\partial
p}\right)^{-1}e^{-2\Phi}V$
$\displaystyle+\left(\frac{d\Phi}{dr}+\frac{2}{r}\frac{\sigma}{\rho+p}\right)e^{-\lambda}\frac{W}{r^{2}}=0.$
(10)

The coupled differential equations Eqs. (8) and (8) are integrated using the
boundary conditions given by Eqs. (9) and (III). Initially, we assume a value
for $\omega^{2}$ and after each integration, the value of $\omega^{2}$ is
modified until the surface boundary condition is satisfied.

Figure 1: Mass-radius curve of static anisotropic dark energy stars with
$\lambda_{BL}$ ranging from $-2$ to $+2$. $\lambda_{BL}=0$ corresponds to the
isotropic case. The mass-radius curves of isotropic neutron star (NS), with
Sly4 EoS and quark star (QS), with modified Bag model EoS are also shown. The
horizontal bands are observational constraints from GW 190814 events Abbott
_et al._ (2020) and various pulsars (PSR J2215+5135
($M=2.27_{-0.15}^{+0.17}M_{\odot}$) Linares  _et al._ (2018), PSR J0348+0432
($M=2.01_{-0.04}^{+0.04}M_{\odot}$) Antoniadis  _et al._ (2013) and PSR
J0740+6620 ($2.14_{-0.09}^{+0.10}M_{\odot}$ ($68.3\%$ credible) ) Cromartie
_et al._ (2019)). We indicate the observational measurements from GW 170814
event Abbott  _et al._ (2018). The Buchdahl limit ($2M/R\leq 0.88$) Buchdahl
(1959) is also shown here.

To obtain the tidal perturbations in presence of anisotropy, we follow the
prescription developed in Ref. Biswas and Bose (2019) based on Refs. Hinderer
(2008); Hinderer  _et al._ (2010).

The tidal deformability $\Lambda$ of a static, spherically symmetric star
placed in a static external quadrupolar tidal field $\mathcal{E}_{ij}$ is
defined in terms of induced quadrupole moment $Q_{ij}$ as
$\Lambda=-Q_{ij}/\mathcal{E}_{ij}$ and can be expressed in terms of stellar
radius $R$, mass $M$ and tidal Love number $k_{2}$ as $\Lambda=2k_{2}R^{5}/3$
Hinderer (2008); Hinderer  _et al._ (2010). In order to study the tidal
perturbations, we use the set of coupled differential equations in terms of
the fluid perturbation function $H$ given by Biswas and Bose (2019)

$\displaystyle\frac{dH}{dr}$ $\displaystyle=$ $\displaystyle\beta,$ (11)
$\displaystyle\frac{d\beta}{dr}$ $\displaystyle=$
$\displaystyle\frac{2rH}{r-2m}\left\\{-2\pi\left[4\rho+8p+\frac{\rho+p}{\mathcal{A}c_{s}^{2}}(1+c_{s}^{2})\right]\phantom{\frac{3}{r^{2}}}\right.$
(12) $\displaystyle+$
$\displaystyle\left.\frac{3}{r^{2}}+\frac{2r}{r-2m}\left(\frac{m}{r^{2}}+4\pi
rp\right)^{2}\right\\}$ $\displaystyle+$
$\displaystyle\frac{2\beta}{r-2m}\left\\{-1+\frac{m}{r}+2\pi
r^{2}(\rho-p)\right\\}.$

Here, $\mathcal{A}=dq/dp$ and $c_{s}^{2}=dp/d\rho$ is the speed of sound
squared. By setting $\mathcal{A}=1$, the above equations reduce to the
isotropic case Hinderer (2008); Hinderer  _et al._ (2010). The integration is
performed outwards starting from the stellar center ($r\to 0$) with the values
$H(r)=a_{0}r^{2}$ and $\beta(r)=2a_{0}r$, where $a_{0}$ is an arbitrary
constant. The interior and exterior solutions are matched at the surface of
the star and the solution obtained can be used to calculate the $l=2$ tidal
Love number $k_{2}$ Hinderer (2008); Hinderer  _et al._ (2010)

$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{8C^{5}}{5}(1-2C)^{2}[2+2C(y-1)-y]$
$\displaystyle\times\bigg{\\{}2C[6-3y+3C(5y-8)]$
$\displaystyle+4C^{3}[13-11y+C(3y-2)+2C^{2}(1+y)]$
$\displaystyle+3(1-2C)^{2}[2-y+2C(y-1)]\ln(1-2C)\bigg{\\}}^{-1};$

where $y=R\beta(R)/H(R)$ and compactness of the star $C=M/R$, which gives a
measure of the strength of star’s gravity.

##  IV Results and discussions

Now, we proceed to obtain the stellar configurations, non-radial $f$-mode
oscillations and tidal deformability factors of both isotropic and anisotropic
dark energy stars that obey the modified Chaplygin EoS discussed in Sec. II.
We begin our analysis by numerically solving the static stellar structure
equations: Eqs. (7a) and (7b) from the center to the surface of the star with
the appropriate boundary conditions given in Sec. III. The $f$-mode
frequencies are calculated by numerically solving the coupled differential
equations: Eqs. (8) and (8) with the boundary conditions given by Eqs. (9) and
(III). Here, we use Ridder’s method to obtain the eigen or $f$-mode frequency
with sufficient precision. Finally, we obtain the tidal deformability
$\Lambda$ by solving the second-order differential equation Eq. (12) from the
center to the surface of the star. We also compare our results with that
obtained for isotropic neutron and quark stars. For neutron star (NS) matter,
we consider the Sly4 EoS Douchin and Haensel (2001) ; whereas for the
(strange) quark star (QS), the modified Bag model Fraga  _et al._ (2001) is
used.

In Fig. 1, we plot the mass-radius profiles of the anisotropic dark energy
stars for different values of anisotropic strength. Following Ref. Biswas and
Bose (2019), we consider the range: $-2<\lambda_{BL}<2$ in this work. Here,
$\lambda_{BL}=0$ corresponds to the isotropic stellar profile. As
$\lambda_{BL}$ varies positively (negatively) from the isotropic value, both
mass and radius of the star increase (decrease). This trend is in agreement
with Ref. Pretel (2023), although a different model of anisotropy was employed
in their analysis of dark energy stars. The maximum mass values corresponding
to $\lambda_{BL}=-2,\,-1,\,0,\,1,$ and $2$ are obtained as
$2.06,\,2.25,\,2.47,\,2.74,$ and $3.07M_{\odot}$ respectively; the
corresponding central density values are $(2.24,\,1.85,\,1.69,\,1.43,$ and
$1.22)\times 10^{15}$ g cm-3. Thus, we see that anisotropy has significant
impact on the maximum mass and radius of the stellar profiles. We note that
the effect of anisotropy on the mass and radius values are more prominent at
higher central densities. Further, one can see that neutron star profiles are
different: results in lower value of maximum mass and corresponding radius and
higher radius for lower central densities. However, we see that the isotropic
quark star profile is qualitatively similar to that of the dark energy star;
the maximum mass and radius values are lesser than that of the dark star. The
maximum mass (corresponding radius) of NS and QS are $2.06M_{\odot}$ ($10.02$
km) and $2.15\,M_{\odot}$ ($11.71$ km) respectively.

We now compare the dark energy star results shown in Fig. 1 with some of the
observational measurements given by GW events and milli-second pulsars. We
have shown the observational data from GW 170817 Abbott  _et al._ (2018) and
GW 190814 Abbott  _et al._ (2020) events. We have also indicated the
observational limits from PSR J2215+5135 ($M=2.27_{-0.15}^{+0.17}M_{\odot}$)
Linares  _et al._ (2018), PSR J0348+0432 ($M=2.01_{-0.04}^{+0.04}M_{\odot}$)
Antoniadis  _et al._ (2013) and PSR J0740+6620
($2.14_{-0.09}^{+0.10}M_{\odot}$ ($68.3\%$ credible)) Cromartie  _et al._
(2019). We find that the maximum masses of anisotropic dark energy stars with
$\lambda_{BL}$ values $-1$ and $-2$ are consistent with the PSR J0740+6620,
PSR J0348+0432 and PSR J2215+5135. We also note that the mass-radius values of
anisotropic dark energy stars fall within the limits imposed by GW 170817;
whereas, the values of mass and radius obey the limit imposed by GW 190814
only for $\lambda_{BL}=-1$ and $-2$. Furthermore, we have shown the Buchdahl
limit Buchdahl (1959) in Fig. 1 and it can be seen that the anisotropic dark
energy star profiles considered in this work fall well within this limit.

Figure 2: The scaled (a) radial pressure $p/p_{c}$ and (b) anisotropic
pressure $\sigma/p_{c}$ as a function of radial co-ordinate
$r=\sqrt{x^{2}+y^{2}}$ in $z=0$ plane of the anisotropic dark energy star with
central pressure $p_{c}=6.75\times 10^{35}$ g cm-1s-2.

Figure 3: $f$-mode frequency as a function of (a) mass and (b) compactness,
$C=M/R$ of the anisotropic dark energy star for $l=2$ mode. The results for
neutron star (NS) and quark star (QS) are also depicted.

Figure 4: The (a) tidal Love number $k_{2}$ against mass of the anisotropic
dark energy star and (b) $f$-mode frequency as a function of tidal
deformability $\Lambda$.

Next, we present the variation of scaled radial pressure $p/p_{c}$ and
anisotropic pressure $\sigma/p_{c}$ of dark energy stars as a function of
radial co-ordinate $r=\sqrt{x^{2}+y^{2}}$ with $z=0$ for different values of
$\lambda_{BL}$ in Fig. 2. The stellar profiles are obtained by keeping a fixed
value of central pressure $p_{c}=6.75\times 10^{35}$ gcm-1s-2, with
corresponding central density being $2.0\times 10^{15}$ g cm-3. The radius of
the star increases marginally with an increase in the value of $\lambda_{BL}$.
For $\lambda_{BL}=-2,-1,0,1,2$, the radius values are obtained as $11.04$,
$11.29$, $11.59$, $11.84$, and $12.04$ km respectively. The radial pressure
for any given value of $r$ is found to increase (decrease) with respect to the
isotropic case $i.e.,\,\lambda_{BL}=0$, with increase (decrease) in the value
of $\lambda_{BL}$, as shown in Fig. 2. The variation of radial pressure from
the isotropic case is maximum towards the mid of the radii of the profiles.
Now, coming to the anisotropic pressure profiles, we find that it changes
drastically as the value of $\lambda_{BL}$ varies. From Fig. 2, it can be seen
that the anisotropic pressure increases from $0$ ($\lambda_{BL}=0$), reaches a
maximum value and then slowly decreases with $r$ for positive values of
$\lambda_{BL}$ and vice versa for negative values. Further, the anisotropic
pressure profiles show an overall decrement with increase in the value of
$\lambda_{BL}$ from $-2$ to $2$. It can be seen that, $\sigma/p_{c}$ becomes
more and more negative with increasing values of $\lambda_{BL}$ in the
positive direction; vice-versa with negative increment of $\lambda_{BL}$. We
also note that, for positive values of $\lambda_{BL}$, the deviation from the
isotropic case is higher compared to negative values of $\lambda_{BL}$. We
note that qualitatively same trend is observed in the case of neutron stars
with a different anisotropy prescription in Ref. Doneva and Yazadjiev (2012).

For $f$-mode analysis, we employ the Cowling approximation discussed in Sec.
III. In Figs. 3 and 3, we plot the $l=2$ $f$-mode frequencies as a function of
dark energy stellar mass and compactness $C=M/R$ respectively. We observe
that, the $f$-mode oscillation frequencies of isotropic dark energy star
($\lambda_{BL}=0$) lie within a smaller range compared to that of the
isotropic NS considered. Also, the oscillation frequency corresponding to the
maximum mass, $f_{max}$ of the dark energy star is observed at a lower value
compared to that of the NS. Now, considering the anisotropic dark energy star,
the $f$-mode frequencies corresponding to different values of $\lambda_{BL}$
can be observed distinctly from the isotropic case. We find that the frequency
range of $f$-modes expands for more positive values of $\lambda_{BL}$. From
Fig. 3, the $f$-modes of dark energy star lie in the range of $2.38-2.61$ kHz
and $2.86-2.02$ kHz for $\lambda_{BL}=-2$ and $+2$ respectively, with the mass
ranging from $1.00M_{\odot}$ to the maximum mass of the star obtained for each
case. The deviation of the $f$-mode spectra from the isotropic case is more
prominent for larger absolute values of $\lambda_{BL}$. This is because, the
variation of anisotropic pressure of the star is observed to be large for
higher values of $|\lambda_{BL}|$. Further, we note that the $f$-mode
frequencies show only a small deviation from isotropic case for negative
$\lambda_{BL}$ values. For example, the values of $f$ corresponding to
$1.00M_{\odot}$ are $2.39$, and $2.38$ kHz respectively for $\lambda_{BL}=-1,$
and $-2$; while the frequencies are obtained as $2.43,\,2.55$, and $2.86$ kHz
for $\lambda_{BL}=0,\,1,$ and $2$ respectively with the same mass. Also, the
$f_{max}$ values are found to decrease with increase in the value of
$\lambda_{BL}$. The values of $f_{max}$ for $\lambda_{BL}=-2,\,-1,\,0,\,1,$
and $2$ are $2.61,\,2.48,\,2.36,\,2.20,$ and $2.02$ kHz respectively.

More importantly, we observe that the $f$-mode frequency spectrum of dark
energy stars is distinctively different from that of the neutron stars (NS)
and quark stars (QS). We note that, the $f$-mode frequency corresponding to
positive $\lambda_{BL}$ start from larger values compared to the isotropic
case and decrease, then approach $f_{max}$; while for negative $\lambda_{BL}$
values, the curves start from smaller frequency range, cross the isotropic
curve and approach $f_{max}$. Moreover, we observe that the $f$-modes of dark
energy star begin in the frequency range $\approx 2.4-3.1$ kHz, unlike the
case of NS and QS for which the curve begins at lower frequencies. For NS, the
values of $f$ increase with the stellar mass and $f_{max}$ occur at largest
value in the $f$-mode frequency range obtained for the star. On the other
hand, the $f$-mode values of dark energy star for positive $\lambda_{BL}$
decrease with mass and $f_{max}$ values are obtained at lower frequency range.
Additionally, it can be noted that the $f$-mode curve of dark energy star
appears similar to that of QS when $\lambda_{BL}$ becomes more negative; the
curve remains almost constant when mass is increased, then rises suddenly when
approaching the maximum mass.

The $f$-mode frequency as a function of compactness $C=M/R$ is given in Fig.
3. The values of compactness corresponding to maximum mass $C_{max}$ for dark
energy stars are $0.28,\,0.29,\,0.31,\,0.33,$ and $0.35$ with
$\lambda_{BL}=-2,\,-1,\,0,\,1,$ and $2$ respectively. The values of $C_{max}$
for NS and QS are $0.30$ and $0.27$ respectively. We note that, the
compactness of isotropic dark energy stars is of the order of NS. However, a
positive deviation from isotropy results in ultra-compactness, $C>1/3$ Iyer
_et al._ (1985).

Further, we study the tidal properties and plot the relation between $f$-mode
and tidal deformability $\Lambda$ of anisotropic dark energy stars. In Fig. 4,
we plot the tidal Love number $k_{2}$ as a function of the stellar mass. The
value of $k_{2}$ increases until it reaches a maximum and then decreases as
the mass is increased. A similar trend was observed for $k_{2}$ of anisotropic
dark energy stars using a different model of anisotropy in Ref. Pretel (2023).
The value of anisotropy parameter $\lambda_{BL}$ has a significant effect on
the behaviour of tidal Love number $k_{2}$ as well. We find that the tidal
Love numbers show a significant deviation from isotropy for positive values of
$\lambda_{BL}$. Whereas, the deviation observed is comparatively lower for
negative values of $\lambda_{BL}$. The values of the tidal Love number
corresponding to the maximum mass ($k_{2}^{max}$) for
$\lambda_{BL}=-2,\,-1,\,0,\,1,$ and $2$ are
$0.0147,\,0.0125,\,0.0116,\,0.0098,$ and $0.0079$ respectively. Next, we plot
the $f$-mode frequencies as a function of tidal deformability $\Lambda$ in
Fig. 4. The values of tidal deformability corresponding to the maximum mass
($\Lambda_{max}$) for $\lambda_{BL}=-2,\,-1,\,0,\,1,$ and $2$ are
$5.96,\,3.95,\,2.69,\,1.69,$ and $0.97$ respectively. The value of
$\Lambda_{max}$ is found to decrease with an increase in $\lambda_{BL}$. The
values of tidal deformability for $1.4M_{\odot}$ ($\Lambda_{1.4}$) for
$\lambda_{BL}=-2,\,-1,\,0,\,1,$ and $2$ are $49,\,53,\,59,\,60,$ and $76$
respectively. We find that values of $\Lambda_{1.4}$ are in good agreement
with the limit given by GW 170817 ($\Lambda_{1.4}\leq 800$ Abbott  _et al._
(2017)). However, the values of $f$-mode frequency obtained in our analysis
are higher and does not fall within the limit provided in Ref. Wen  _et al._
(2019). We hope that future detection of binary NS mergers may provide further
constraints on $f$-mode frequencies.

##  V SUMMARY AND CONCLUSIONS

We have studied the prominent non-radial $f$-mode oscillations for the
isotropic and anisotropic dark energy star configurations. By using the
modified Chaplygin fluid prescription for the constituent dark energy matter,
we solved the respective coupled differential equations in presence of
anisotropy for stellar structure, $f$-mode and tidal deformability.
Anisotropic pressure of the dark energy fluid is introduced with the Bowers-
Liang model, where the free parameter $\lambda_{BL}$ determines the strength
of it. We find that the internal stellar structure is highly sensitive to the
strength of anisotropy and the maximum mass and radius of the star are found
to vary significantly with $\lambda_{BL}$. As $\lambda_{BL}$ increases from
$-2$ to $+2$, the maximum mass and corresponding radius of the star are found
to increase. Further, we compared the stellar profiles obtained with
observational data from GW events and milli-second pulsars and good agreement
was found; especially for configurations with $\lambda_{BL}<0$. Also, the
mass-radius profiles of the dark energy star show a different trend from that
of the neutron star; while, the curves are qualitatively more similar to that
of a quark star.

The oscillation frequency of $l=2$ $f$-mode of dark energy star was found
under the Cowling approximation. We found that the $f-$mode frequencies span
the range $\sim 2-3.1$ kHz. The range of $f$-mode spectrum obtained for the
isotropic case was found to be smaller compared to that of neutron and quark
stars. Further, the effect of anisotropy is reflected in the $f$-mode
solutions, where the frequencies show a significant deviation from the
isotropic case. This deviation is observed to be high for larger absolute
value of the parameter $\lambda_{BL}$, which is expected, since the
anisotropic pressure variation is found to be appreciably high for large
$|\lambda_{BL}|$ values. Further, the value of $f$ corresponding to the
maximum mass of the star, $f_{max}$ is found to decrease with increase in the
value of $\lambda_{BL}$. Also, the $f$-mode frequencies were obtained as a
function of compactness and tidal deformability. It was found that dark energy
stars become ultra compact objects for positive values of $\lambda_{BL}$. It
was noted that the tidal deformability corresponding to the maximum mass
$\Lambda_{max}$ decreases when $\lambda_{BL}$ is increased. Further, the tidal
deformability of anisotropic dark energy stars was found to be in good
agreement with the limits imposed by GW 170814 event.

Interestingly, we found that the $f$-mode frequency spectrum of the isotropic
and anisotropic dark energy stars is indicatively different from that obtained
for neutron stars (NS) and quark stars (QS). Further, unlike the case of NS
and QS, the $f$-mode spectra of dark energy stars begin within a higher
frequency range at $M\gtrsim 1.00\,M_{\odot}$. We believe that this distinct
behaviour of $f$-modes in dark energy stars may give directions in its
identification.

## Acknowledgements

L. J. N. acknowledges the Department of Science and Technology, Govt. of India
for the INSPIRE Fellowship.

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