Radial pulsations, moment of inertia and tidal deformability of dark energy stars
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Eur. Phys. J. C (2023) 83:26
https://doi.org/10.1140/epjc/s10052-023-11198-3
Regular Article - Theoretical Physics
Radial pulsations, moment of inertia and tidal deformability
of dark energy stars
Juan M. Z. Pretela
Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150 URCA, Rio de Janeiro, RJ CEP 22290-180, Brazil
Received: 27 October 2022 / Accepted: 4 January 2023
© The Author(s) 2023
Abstract We construct dark energy stars with Chaplygin- vational data [1], but suffers from the well-known coinci-
type equation of state (EoS) in the presence of anisotropic dence problem and the fine-tuning problem [2,3]. The exact
pressure within the framework of Einstein gravity. From the physical nature of dark energy is still a mystery and, con-
classification established by Iyer et al. (Class Quantum Grav sequently, the possibility that dark matter and dark energy
2:219, 1985), we discuss the possible existence of isotropic could be different manifestations of a single substance has
dark energy stars as compact objects. However, there is the been considered [4–7]. In that regard, it was shown that the
possibility of constructing ultra-compact stars for sufficiently inhomogeneous Chaplygin gas offers a simple unified model
large anisotropies. We investigate the stellar stability against of dark matter and dark energy [8]. It was also argued that
radial oscillations, and we also determine the moment of if the Universe is dominated by the Chaplygin gas a cos-
inertia and tidal deformability of these stars. We find that mological constant would be ruled out with high confidence
the usual static criterion for radial stability d M/dρc > 0 still [9].
holds for dark energy stars since the squared frequency of the Using the Planck 2015 CMB anisotropy, type-Ia super-
fundamental pulsation mode vanishes at the critical central novae and observed Hubble parameter data sets, the full
density corresponding to the maximum-mass configuration. parameter space of the modified Chaplygin gas was measured
The dependence of the tidal Love number on the anisotropy by Li et al. [10]. Based on recent observations of high-redshift
parameter α is also examined. We show that the surface grav- quasars, Zheng and colleagues [11] investigated a series of
itational redshift, moment of inertia and dimensionless tidal Chaplygin gas models as candidates for dark matter-energy
deformability undergo significant changes due to anisotropic unification. The application of the Hamilton–Jacobi formal-
pressure, primarily in the high-mass region. Furthermore, in ism for generalized Chaplygin gas models was carried out in
light of the detection of gravitational waves GW190814, we Ref. [12]. Additionally, it is worth mentioning that Odintsov
explore the possibility of describing the secondary compo- et al. [13] considered two different equations of state for dark
nent of such event as a stable dark energy star in the presence energy (i.e., power-law and logarithmic effective corrections
of anisotropy. to the pressure). They showed that the power-law model only
yielded some modest results, achieved under negative values
of bulk viscosity, while the logarithmic scenario provide good
1 Introduction fits in comparison to the CDM model.
Another way to give rise to an accelerated expansion of
Different types of observations (such as Type Ia supernovae, the Universe is by modifying the geometry itself [14,15],
structure formation and CMB anisotropies) indicate that our namely, considering higher curvature corrections to the stan-
Universe is not only expanding, it is accelerating. Within the dard Einstein–Hilbert action. Under this outlook, the cosmic
standard CDM model (which is based on cold dark matter acceleration can be modeled in the scope of a scalar-tensor
and cosmological constant in Einstein gravity), this cosmic gravity theory [16,17]. Moreover, within the context of the
acceleration is due to a smooth component with large nega- so-called f (R) theories [18,19], the quadratic term in the
tive pressure and repulsive gravity, the so-called dark energy. Ricci scalar R leads to an inflationary solution in the early
Such a model gives a good agreement with the recent obser- Universe [20], although such a model does not provide a
late-time accelerated expansion. Nevertheless, the late-time
a e-mail: juanzarate@cbpf.br (corresponding author)
0123456789().: V,-vol 12326 Page 2 of 16 Eur. Phys. J. C (2023) 83:26
acceleration era can be realized by terms containing inverse gravity. Indeed, it has been argued that the deformation near
powers of R [21], though it was shown that this is not com- the maximum neutron-star mass comes from the anisotropic
patible with the solar system experiments [22]. For a com- pressure within these stars, which is caused by the distor-
prehensive study on the evolution of the early and present tion of Fermi surface predicted by the equation of state of
Universe in f (R) modified gravity, we refer the reader to the models [56]. Becerra-Vergara et al. [57] showed that the
the review articles [23–25] and references contained therein. contribution of the fourth order corrections parameter (a4 )
On the other hand, the astrophysical implications due to the of the QCD perturbation on the radial and tangential pres-
f (R) modified gravitational Lagrangian on compact stars sure generate significant effects on the mass-radius relation
have been intensively investigated in the past few years [26– and the stability of quark stars. It has also been shown that
33]. the stellar structure equations in Eddington-inspired Born–
According to the aforementioned works, different dark Infeld theory with isotropic matter can be recast into GR with
energy models have been proposed in order to explain the a modified (apparent) anisotropic matter [58].
mechanisms that lead to the cosmic acceleration. Only about Motivated by the several works already mentioned, we
4% of the Universe is made of familiar atomic matter, 20% aim to discuss the impact of anisotropy on the macroscopic
dark matter, and it turns out that roughly 76% of the Uni- properties of dark energy stars with Chaplygin-like EoS. We
verse is dark energy [34]. Within the context of General will address the following questions: Do these stars belong
Relativity, dark energy is an exotic negative pressure contri- to families of compact or ultra-compact stars? How does
bution that can lead to the observed accelerated expansion. anisotropy affect the compactness and radial stability of dark
In the absence of consensus regarding a theoretical descrip- energy stars satisfying the causality condition? In particular,
tion for the current accelerated expansion of the Universe, by adopting the phenomenological ansatz proposed by Hor-
theorists have proposed using the Chaplygin gas as a use- vat et al. [51], we determine the radius, mass, gravitational
ful phenomenological description [4]. If dark energy is dis- redshift, frequency of the fundamental oscillation mode,
tributed anywhere permeating ordinary matter, then it could moment of inertia and the dimensionless tidal deformability
be present in the interior of a compact star. Therefore, the pur- of anisotropic dark energy stars. The isotropic solutions are
pose of this manuscript is to investigate the possible existence recovered when the anisotropy parameter vanishes, i.e. when
of compact stars with dark energy by assuming a Chaplygin- α = 0.
type EoS. For such stars to exist in nature, they need to be The organization of this paper is as follows: In Sect. 2
stable under small radial perturbations. we start with a brief overview of relativistic stellar struc-
Adopting a description of dark energy by means of a phan- ture, describing the basic equations for radial pulsations,
tom (ghost) scalar field, Yazadjiev [35] constructed a general moment of inertia and tidal deformability. We then introduce
class of exact interior solutions describing mixed relativis- the Chaplygin-like EoS and discuss its relation to the cosmo-
tic stars containing both ordinary matter and dark energy. logical context in Sect. 3, as well as we present the anisotropy
The energy conditions and gravitational wave echoes of such profile. Section 4 provides a discussion of the numerical
stars were recently analyzed in Ref. [36]. Furthermore, the results for the different physical properties of dark energy
effect of the dynamical scalar field quintessence dark energy stars. Finally, our conclusions are summarized in Sect. 5.
on neutron stars was investigated in [37]. Panotopoulos and
collaborators [38] studied slowly rotating dark energy stars
made of isotropic matter using the Chaplygin EoS. Bhar [39] 2 Stellar structure equations
proposed a model for a dark energy star made of dark and
ordinary matter in the Tolman-Kuchowicz spacetime geom- In order to study the basic features of compact stars with dark
etry. For further stellar models with dark energy we also refer energy, in this section we briefly summarize the stellar struc-
the reader to Refs. [40–48]. ture equations in Einstein gravity. In particular, we focus on
In addition, anisotropy in compact stars may arise due hydrostatic equilibrium structure, radial pulsations, moment
to strong magnetic fields, pion condensation, phase transi- of inertia, and tidal deformability.
tions, mixture of two fluids, bosonic composition, rotation, The theory of gravity to be used in this work is general
etc. Thus, regardless of the specific source of the anisotropy, relativity, where the Einstein field equations are given by
it is more natural to think of anisotropic fluids when studying
compact stars at densities above nuclear saturation density. 1
In that regard, the literature offers some physically motivated G μν ≡ Rμν − gμν R = 8π Tμν , (1)
2
functional relations for the anisotropy, see for example Refs.
[49–55]. However, we must point out that these anisotropic with G μν being the Einstein tensor, Rμν the Ricci ten-
models are based on general assumptions (or ansatzes) that sor, R denotes the scalar curvature, and Tμν is the energy-
do not directly relate to exotic modifications of matter or momentum tensor. Since we are interested in isolated com-
123Eur. Phys. J. C (2023) 83:26 Page 3 of 16 26
pact stars, we consider that the spacetime can be described background and allow us to obtain the metric components
by the spherically symmetric four-dimensional line element and fluid variables.
ds 2 = −e2ψ dt 2 + e2λ dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ). (2) 2.2 Radial oscillations
In addition, we model the compact-star matter by an A rigorous analysis of the radial stability of compact stars
anisotropic perfect fluid, whose energy-momentum tensor requires the calculation of the frequencies of normal vibra-
is given by tion modes. Such frequencies can be found by considering
small deviations from the hydrostatic equilibrium state but
Tμν = (ρ + pt )u μ u ν + pt gμν − σ kμ kν , (3) maintaining the spherical symmetry of the star. In the lin-
ear treatment, where all quadratic (or higher-order) or mixed
where ρ is the energy density, σ ≡ pt − pr the anisotropy terms in the perturbations are discarded, one assumes that
factor, pr the radial pressure, pt the tangential pressure, u μ all perturbations in physical quantities are arbitrarily small.
the four-velocity of the fluid, and k μ is a unit four-vector. The fluid element located at r in the unperturbed config-
These four-vectors must satisfy u μ u μ = −1, kμ k μ = 1 and uration is displaced to radial coordinate r + ξ(t, r ) in the
u μ k μ = 0. Notice that the stellar fluid becomes isotropic perturbed configuration, where ξ is the Lagrangian displace-
when σ = 0. ment. All perturbations have a harmonic time dependence of
the form ∼ eiνt , where ν is the oscillation frequency to be
2.1 TOV equations determined. Consequently, defining ζ ≡ ξ/r , the adiabatic1
radial pulsations of anisotropic compact stars are governed
When the stellar fluid remains in hydrostatic equilibrium, by the following differential equations [55]
neither metric nor thermodynamic quantities depend on the
μ
dζ 1 pr 2σ ζ dψ
time coordinate. This allows us to write u μ = e−ψ δ0 and =− 3ζ + + + ζ, (8)
μ dr r γ pr ρ + pr dr
k μ = e−λ δ1 . Accordingly, the hydrostatic equilibrium of an
d(pr ) dpr
anisotropic compact star is governed by the TOV equations: = ζ ν 2 e2(λ−ψ) (ρ + pr )r − 4
dr dr
dm dψ 2
= 4πr 2 ρ, (4) −8π(ρ + pr )e2λr pr + r (ρ + pr )
dr dr
m
dpr 2m −1 2σ 4 dψ dσ dζ
= −(ρ + pr ) 2 + 4πr pr 1− + , +2σ + +2 + 2σ
dr r r r r dr dr dr
(5) dψ 2
−pr + 4π(ρ + pr )r e2λ + δσ, (9)
dψ 1 dpr 2σ dr r
=− + , (6)
dr ρ + pr dr r (ρ + pr )
where pr is the Lagrangian perturbation of the radial pres-
which are obtained from Eqs. (1)–(3) together with the con- sure and γ = (1 + ρ/ pr )dpr /dρ is the adiabatic index at
μ constant specific entropy.
servation law ∇μ T1 = 0. The metric function λ(r ) is deter-
mined from the relation e−2λ = 1 − 2m/r , where m(r ) is The above first-order time-independent Eqs. (8) and (9)
the gravitational mass within a sphere of radius r . require boundary conditions set at the center and surface of
By supplying an EoS for the radial pressure in the form the star, similar to a vibrating string fixed at its ends. Since
pr = pr (ρ) and a defined anisotropy relation for σ , the Eq. (8) has a singularity at the origin, the following condition
system of differential Eqs. (4)–(6) is then numerically inte- must be required
grated from the center at r = 0 to the surface of the star
2σ ζ
r = R which correspond to a vanishing pressure. Therefore, pr = − γ pr − 3γ ζ pr as r → 0, (10)
the above equations will be solved under the requirement of ρ + pr
the following boundary conditions
while the Lagrangian perturbation of the radial pressure at
the surface must satisfy
1 2M
ρ(0) = ρc , m(0) = 0, ψ(R) = ln 1 − , (7)
2 R
pr = 0 as r → R. (11)
where ρc is the central energy density, and M ≡ m(R) is the
total mass of the star calculated at its surface. The numeri- 1 In the adiabatic theory, it is assumed that the fluid elements of the star
cal solution of the TOV equations describes the equilibrium neither gain nor lose heat during the oscillation.
12326 Page 4 of 16 Eur. Phys. J. C (2023) 83:26
2.3 Moment of inertia for r → ∞, where J is the total angular momentum of the
star [59,61]. Therefore, comparing this with the asymptotic
Suppose a particle is dropped from rest at a great distance behavior of l (r ), we find that l = 1. As a result, is a
from a rotating star, then it would experience an ever increas- function only of the radial coordinate, and Eq. (15) reduces
ing drag in the direction of rotation as it approaches the star. to
Based on this description, we introduce the angular velocity
acquired by an observer falling freely from infinity, denoted eψ−λ d −(ψ+λ) 4 d
e r = 16π(ρ + pt ), (16)
by ω(r, θ ). Here we will calculate the moment of inertia of an r 4 dr dr
anisotropic dark energy star under the slowly rotating approx-
imation [59]. This means that when we consider rotational which can be integrated to give
corrections only to first order in the angular velocity of the R
star , the line element (2) is replaced by its slowly rotating 4 d
r = 16π (ρ + pt )r 4 eλ−ψ dr. (17)
counterpart, namely dr R 0
ds 2 = − e2ψ(r ) dt 2 + e2λ(r ) dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) In view of Eq. (17), we can obtain the angular momentum
J and hence the moment of inertia I = J/ of a slowly
− 2ω(r, θ )r 2 sin2 θ dtdφ, (12)
rotating anisotropic star:
and following Ref. [59], it is pertinent to define the difference R
≡ − ω as the coordinate angular velocity of the fluid 8π
I = (ρ + pr + σ )eλ−ψ r 4 dr, (18)
element at (r, θ ) seen by the freely falling observer. 3 0
Keep in mind that is the angular velocity of the stellar
fluid as seen by an observer at rest at some spacetime point which reduces to the expression given in Ref. [61] for
(t, r, θ, φ), and hence the four-velocity up to linear terms in isotropic compact stars when σ = 0. For an arbitrary choice
can be written as u μ = (e−ψ , 0, 0, e−ψ ). To this order, of the central value (0), the appropriate boundary condi-
the spherical symmetry is still preserved and it is possible to tions for the differential Eq. (16) come from the requirements
extend the validity of the TOV Eqs. (4)–(6). Nonetheless, the of regularity at the center of the star and asymptotic flatness
03-component of the field equations contributes an additional at infinity, namely
differential equation for angular velocity. By retaining only
d
first-order terms in , such component becomes = 0, lim = . (19)
dr r →∞
r =0
eψ−λ ∂ −(ψ+λ) 4 ∂ 1 ∂ 3 ∂
e r + 2 3 sin θ Once the solution for (r ) is found, we can then deter-
r 4 ∂r ∂r r sin θ ∂θ ∂θ
mine the moment of inertia through the integral (18). It is
= 16π(ρ + pt ). (13) remarkable that the above expression for I is referred to as
the “slowly rotating” approximation because it was obtained
As in the case of isotropic fluids, we follow the same to lowest order in the angular velocity [61]. This means
treatment carried out by Hartle [59,60] and we assume that that the stellar structure equations are still given by the TOV
can be written as Eqs. (4)–(6).
∞
−1 d Pl
(r, θ ) = l (r ) , (14) 2.4 Tidal deformability
sin θ dθ
l=1
It is well known that the tidal properties of neutron stars are
where Pl are Legendre polynomials. Taking this expansion measurable in gravitational waves emitted from the inspiral
into account, Eq. (13) becomes of a binary neutron-star coalescence [62,63]. In that regard,
here we also study the dimensionless tidal deformability of
eψ−λ d −(ψ+λ) 4 dl l(l + 1) − 2
e r − l individual dark energy stars. To do so, we follow the pro-
r 4 dr dr r2 cedure carried out by Hinderer et al. [64] (see also Refs.
= 16π(ρ + pt )l . (15) [65–70] for additional results). The basic idea is as follows:
In a binary system, the deformation of a compact star due
At a distance far away from the star, where e−(ψ+λ) to the tidal effect created by the companion star is charac-
becomes unity, the asymptotic solution of Eq. (15) takes the terized by the tidal deformability parameter λ̄ = −Q i j /Ei j ,
form l (r ) → a1 r −l−2 + a2 r l−1 . If spacetime is to be flat at where Q i j is the induced quadrupole moment tensor and Ei j
large r , then ω → 2J/r 3 (or equivalently, → −2J/r 3 ) is the tidal field tensor [68]. Namely, the latter describes the
123Eur. Phys. J. C (2023) 83:26 Page 5 of 16 26
tidal field from the spacetime curvature sourced by the distant with A ≡ dpt /dpr and vsr being the radial speed of sound.
companion. By matching the internal solution with the external solu-
The tidal parameter is related to the tidal Love number k2 tion of the perturbed variable H at the surface of the star
through the relation2 r = R, we obtain the tidal Love number [72]
2 8
λ̄ = k2 R 5 , (20) k2 = (1 − 2C)2 C 5 [2C(y R − 1) − y R + 2]
3 5
× 2C[4(y R + 1)C 4 + (6y R − 4)C 3
but it is common in the literature to define the dimension-
less tidal deformability = λ̄/M 5 , so in our results we + (26 − 22y R )C 2 + 3(5y R − 8)C − 3y R + 6
will focus on . The calculation of λ̄ requires considering −1
linear quadrupolar perturbations (due to the external tidal + 3(1 − 2C)2 [2C(y R − 1) − y R + 2] log(1 − 2C) ,
field) to the equilibrium configuration. Thus, the spacetime
(27)
metric is given by gμν = gμν 0 + h , where g 0 describes
μν μν
the equilibrium configuration and h μν is a linearized met-
where C ≡ M/R is the compactness of the star, and y R ≡
ric perturbation. For static and even-parity perturbations in
y(R) is obtained by integrating Eq. (23) from the origin up
the Regge-Wheeler gauge [71], the perturbed metric can be
to the stellar surface.
written as [64]
h μν = diag −e2ψ(r ) H0 , e2λ(r ) H2 , r 2 K , r 2 sin2 θ K Y2m (θ, φ),
3 Equation of state and anisotropy model
(21)
As it is well known, a possible alternative to the Phantom
where H0 , H2 and K are functions of the radial coordinate, and Quintessence fields is the Chaplygin gas, where the EoS
and Ylm are the spherical harmonics for l = 2. assumes the form pr = −B/ρ, with B being a positive con-
Since the perturbed energy-momentum tensor is given by stant (given in m−4 units). In fact, it was argued that such gas
δTμν = diag(−δρ, δpr , δpt , δpt ), the linearized field equa- could provide a solution to unify the effects of dark matter in
tions imply that: the early times and dark energy in late times [4,11]. Although
⎧
⎪ the literature provides a more generalized version for such
⎨ H0 = −H2 ≡ H from δG 22 − δG 33 = 0,
EoS in the context of the Friedmann-Lemaître-Robertson-
K = 2H ψ + H from δG 21 = 0,
⎪
⎩ H −2λ
Walker Universe [5–7,73–77], here we will use the simplest
δpt = 8πr e (λ + ψ )Y2m from δG 22 = 8π δpt . form plus a linear term corresponding to a barotropic fluid,
namely
In addition, from δG 00 − δG 11 = −8π(δρ + δpt ), we can
obtain the following differential equation [72]
B
pr = Aρ − , (28)
ρ
H + P H + QH = 0, (22)
where A is a positive dimensionless constant. Our model is
or alternatively, characterized by two free parameters A and B. Nevertheless,
we must emphasize here that Li et al. [10] considered an
r y = −y 2 + (1 − r P)y − r 2 Q, (23) equation of state with three degrees of freedom, namely p =
Aρ − B/ρ α , where α is an extra parameter. They carried
where we have defined out a statistical treatment of astronomical data in order to
constrain the parameter space. In the light of the Markov
H
y≡r , (24) chain Monte Carlo method, they found that at 2σ level, α =
H −0.0156+0.0982+0.2346 +0.0018+0.0030
−0.1380−0.2180 and A = 0.0009−0.0017−0.0030 from
2 2m
P≡ + e2λ + 4πr ( pr − ρ) , (25) CMB+JLA+CC data sets. In other words, the constants α
r r2 and A are very close to zero and hence the nature of unified
ρ + pr 6e2λ dark matter-energy model is very similar to the cosmological
Q ≡ 4π e 2λ
4ρ + 8 pr + (1 + v 2
sr ) − − 4ψ 2 ,
Avsr
2 r2 standard CDM model.
(26) On the other hand, at astrophysics level, compact stars
obeying the EoS (28) have been investigated by several
2 It should be noted that the tidal deformability parameter is being authors, see for example Refs. [38,41,43–45]. In this work
denoted by λ̄ in order not to be confused with the metric component λ. we will adopt values of A and B for which appreciable
12326 Page 6 of 16 Eur. Phys. J. C (2023) 83:26
changes in the mass-radius diagram can be visualized in 4 Numerical results
order to compare our theoretical results with observational
measurements of massive pulsars. 4.1 Equilibrium configurations
In order to describe physically realistic compact stars, the
causality condition must be respected throughout the interior So far we do not know exactly whether the millisecond
region of the star. In other words, the speed of sound (defined pulsars (observed in compact binaries from optical spectro-
√
by vs ≡ dp/dρ) cannot be greater than the speed of light. scopic and photometric measurements) are hadronic, quark
Thus, in view of Eq. (28), we have or hybrid stars. In fact, it has been theorized that cold quark
matter might exist at the core of heavy neutron stars [84].
dpr B Despite the precise measurements of masses [85–87] and
vsr
2
≡ = A + 2, (29)
dρ ρ radii [88–90], such constraints are still unable to distin-
guish the theoretical predictions coming from the different
and since the radial pressure vanishes at the surface of the star,
models for strange stars and (hybrid) neutron stars. This
then B = Aρ 2 . Thereby, the causality condition vsr 2 (R) =
means that the dense matter EoS within compact stars still
2 A < 1 implies that A < 0.5.
remains poorly understood. Furthermore, a realistic compact
Besides, it is more realistic to consider stellar models
star possesses high magnetic fields and rotation properties,
where there exists a tangential pressure as well as a radial
which significantly alter its internal structure. For compari-
one, since anisotropies arise at high densities, i.e. above
son reasons, it is therefore common to use the observational
the nuclear saturation density as considered in this work.
mass-radius measurements (in view of the detection of grav-
Although the literature offers different functional relations
itational waves and electromagnetic signals) on the mass-
to model anisotropic pressures at very high densities inside
radius diagrams for any type of EoS even being of different
compact stars [49–54], here we adopt the simplest model,
microscopic compositions. In that perspective, our theoret-
which was proposed by Horvat and collaborators [51]
ical results will be compared with observational measure-
ments.
2m
σ =α pr = α 1 − e−2λ pr , (30) We begin our discussion of dark energy stars by consid-
r
ering the isotropic case (i.e., when σ = 0 in the TOV equa-
where α is a dimensionless parameter that controls the tions). We numerically integrate Eqs. (4)–(6) from the center
amount of anisotropy within the stellar fluid. This parameter up to the surface of the star through the boundary conditions
can assume positive or negative values of the order of unity, (7). As usual, the radius R is determined when the pressure
see Refs. [26,32,51,52,55,78–82]. Notice that the isotropic vanishes, and the total mass M is calculated at the surface.
solutions are recovered when the value of α vanishes. Specifi- The felt panel of Fig. 1 exhibits the mass-radius relations of
cally, the anisotropy ansatz (30) has two important character- dark energy stars for different values of parameters A and B
istics: (i) the fluid becomes isotropic at the center generating in the EoS (28). Remark that we have adopted values of A less
regular solutions and (ii) the effect of anisotropy vanishes in than 0.5 in order to respect the causality condition. One can
the hydrostatic equilibrium equation in the Newtonian limit. observe that small values of A (see black curve) do not pro-
Unlike this profile, the effect of anisotropy does not vanish vide compact stars that fit current observational data. How-
in the hydrostatic equilibrium equation in the non-relativistic ever, higher values of maximum mass can be obtained for
regime for the Bowers-Liang model [49], which could be an larger values of A, see for example red and green curves. For a
unphysical trait as argued in Ref. [79]. For a broader dis- fixed value of A, the maximum mass decreases as the param-
cussion on the different ways of generating static spherically eter B increases. We perceive that the secondary component
symmetric anisotropic fluid solutions, we refer the reader to resulting from the gravitational-wave signal GW190814 [91]
the recent review article [83]. can be consistently described as a compact star with Chap-
Since the Eulerian perturbation for the metric potential λ lygin EoS (28) for A = 0.4 and B ∈ [4, 5]μ. Furthermore,
can be written as δλ = −4πr (ρ + pr )e2λ ξ [55], then δσ the magenta curve fits very well with all observational data,
takes the form but its maximum-mass value is above 3M .
Another interesting feature of these stars is their compact-
δσ = α (1 − e−2λ )δpr − 8π pr (ρ + pr )r 2 ζ , (31) ness, defined by C ≡ M/R. According to the classification
adopted by Iyer et al. [92], the configurations shown in the
where it should be noted that the relation between the Eule- mass-radius diagram correspond to compact stars, see the
rian and Lagrangian perturbations for radial pressure is given right plot of Fig. 1. Besides, we can appreciate that the com-
by pr = δpr + r ζ pr . The above expression will be substi- pactness of dark energy stars is of the order of the compact-
tuted in Eq. (9) when we discuss later the radial pulsations ness of hadronic-matter stars, as is the case of the SLy EoS
in the stellar interior for at least some values of α. [93], despite the fact that the maximum mass in the magenta
123Eur. Phys. J. C (2023) 83:26 Page 7 of 16 26
configuration sequence can exceed 3M . Nonetheless, as we lead to an increase in compactness, mainly in the high-
will see later, the introduction of anisotropy can turn such central-density branch. Remarkably, for sufficiently large
stars into ultra-compact objects. Of course, this will depend values of α (see purple curve), it is possible to obtain
on the amount of anisotropy in the stellar interior. anisotropic dark energy stars as ultra-compact objects.
In order to include anisotropic pressures and investigate The gravitational redshift, conventionally defined as the
their effects on the internal structure of dark energy stars, we fractional change between observed and emitted wave-
will adopt two specific models with the following parameters lengths compared to emitted wavelength, in the case of a
Schwarzschild star is given by [61]
Model I: A = 0.3, B = 6.0μ ,
Model II: A = 0.4, B = 5.2μ , λ(R) 2M −1/2
z sur = e −1= 1− − 1. (32)
R
which are models favored by observational measurements
according to the left panel of Fig. 1. Moreover, model II In the right plot of Fig. 4, the surface gravitational redshift is
precisely corresponds to the first model considered by Pan- plotted as a function of the total mass for both models I and
otopoulos et al. [38]. II. This plot indicates that the gravitational redshift of light
Similar to the isotropic case, we numerically solve the emitted at the surface of a dark energy star is substantially
hydrostatic background Eqs. (4)-(6) with boundary con- affected by the anisotropy in the high-mass region, while
ditions (7), but taking into account the anisotropy profile the changes are negligible for sufficiently low masses. For
(30). For instance, for the model I and a central density a fixed value of central density, Table 2 shows that positive
ρc = 2.0 × 1018 kg/m3 , Fig. 2 illustrates the mass density, (negative) anisotropy increases (decreases) the value of the
pressure and squared speed of sound as functions of the radial redshift.
coordinate for different values of the free parameter α. We
can see that the internal structure of a dark energy star is 4.2 Oscillation spectrum
affected by the presence of anisotropy. In effect, the radius
of the star increases (decreases) for more positive (negative) A necessary condition (the well-known M(ρc ) method) for
values of α. In addition, we remark that the speed of sound, stellar stability is that stable stars must lie in the region where
both radial and tangential, respect the causality condition. d M/dρc > 0. According to the right plot of Fig. 3, the full
This has also been verified for other values of central density blue and orange circles on each curve indicate the onset of
considered in the construction of Fig. 1. instability for each family of equilibrium solutions. However,
Varying the central density, we obtain the mass-radius a sufficient condition is to calculate the frequencies of the
diagrams and mass-central density relations for models I radial vibration modes for each central density [61]. Here we
and II, as shown in Fig. 3. We observe that the substan- will analyze if both methods are compatible in the case of
tial changes introduced by anisotropy in dark energy stars dark energy stars including anisotropic pressure.
occur in the high-mass branch (close to the maximum-mass Once the equilibrium Eqs. (4)–(6) are integrated from the
point), while the effects are irrelevant at low central densi- center to the surface of the star, we then proceed to solve
ties. The maximum-mass values increase as the parameter the radial pulsation Eqs. (8) and (9) with the correspond-
α increases (see also the data in Table 1). Note that model ing boundary conditions (10) and (11) using the shooting
I without anisotropic pressures is not capable of generating method. Namely, we integrate from the origin (where we
maximum masses above 2M . Nevertheless, the inclusion of consider the normalized eigenfunctions ζ (0) = 1) up to
anisotropies (see the blue curve for α = 0.4) allows a signif- the stellar surface for a set of trial values ν 2 satisfying the
icant increase in the maximum mass and hence a more favor- condition (10). In this way, the appropriate eigenfrequen-
able description of the compact objects observed in nature. cies correspond to the values for which the boundary con-
On the other hand, model II with anisotropies (see orange dition (11) is fulfilled. For instance, for a central density
curves) fits better with the observational measurements. In ρc = 1.5 × 1018 kg/m3 , α = 0.4 and parameters given by
particular, in view of the lower mass of the compact object model I, Fig. 5 displays the radial behavior of the perturba-
from the coalescence GW190814 [91], two curves are par- tion variables for the first five squared eigenfrequencies νn2 ,
ticularly outstanding. In other words, such object can be well where n indicates the number of nodes inside the star. This
described as an anisotropic dark energy star when α = 0.2 frequency spectrum forms an infinite discrete sequence, i.e.
and α = 0.4. Moreover, model II with negative anisotropies ν02 < ν12 < ν22 < · · · , where the eigenvalue corresponding to
(such as α = −0.4) favors the description of the massive n = 0 is the lowest one (or equivalently, the longest period of
pulsar J2215+5135 [94]. all the allowed vibration modes) and it is known as the fun-
The left panel of Fig. 4 describes the behavior of compact- damental mode. Such mode has no nodes, whereas the first
ness as a function of central density. Positive anisotropies overtone (n = 1) has one node, the second overtone (n = 2)
12326 Page 8 of 16 Eur. Phys. J. C (2023) 83:26
Fig. 1 Left panel: Mass-radius diagrams for dark energy stars with and the cyan area is the mass-radius constraint from the GW170817
Chaplygin-like EoS (28) and isotropic pressure (σ = 0) for several val- event. Moreover, the NICER measurements for PSR J0030+0451 are
ues of the positive parameters A and B. Here the constant B is given displayed by black dots with their respective error bars [95,96]. Right
in μ = 10−20 m−4 units. The gray horizontal stripe at 2.0M stands panel: Variation of the compactness with total gravitational mass,
for the two massive NS pulsars J1614-2230 [85] and J0348+0432 [86]. where the gray and orange stripes represent compact and ultra-compact
Yellow and blue regions represent the observational measurements of objects, respectively, according to the classification given in Ref. [92].
the masses of the highly massive NS pulsars J0740+6620 [87] and For comparison reasons, we have included the results corresponding to
J2215+5135 [94], respectively. The filled pink band stands for the lower the SLy EoS [93] by blue curves in both plots
mass of the compact object detected by the GW190814 event [91],
Table 1 Maximum-mass
Model α ρc [1018 kg/m3 ] R [km] M [M ]
configurations with
Chaplygin-like EoS (28) for I −0.4 2.424 9.812 1.786
model I and II. The energy
density values correspond to the −0.2 2.364 9.902 1.852
critical central density where the 0 2.295 9.994 1.919
function M(ρc ) is a maximum 0.2 2.219 10.086 1.988
on the right plot of Fig. 3
0.4 2.135 10.180 2.059
II −0.4 1.777 11.630 2.320
−0.2 1.721 11.738 2.402
0 1.661 11.845 2.486
0.2 1.594 11.955 2.570
0.4 1.523 12.065 2.565
has two, and so on. Stable stars are described by their oscil- the fundamental mode is exactly zero at the critical-central-
latory behavior so that νn2 > 0 (i.e., νn is purely real). On the density value corresponding to the maximum-mass configu-
other hand, if any of these is negative for a particular star, the ration as shown in the right plot of Fig. 3, see the full blue
frequency is purely imaginary and hence the star is unstable. and orange circles for both models. Furthermore, according
Since each higher-order mode has a squared eigenfre- to the right plot of Fig. 6, the maximum-mass values (that
quency that is larger than in the case of the preceding mode, is, when d M/dρc = 0) can be used as turning points from
it is enough to calculate the frequency of the fundamental stability to dynamical instability. Therefore, we can conclude
pulsation mode for the equilibrium sequences presented in that the usual criterion to guarantee stability d M/dρc > 0
Fig. 3. With this in mind, in Fig. 6 we plot the squared fre- is still valid for the case of anisotropic dark energy stars. In
quency of the fundamental oscillation mode as a function of other words, the conventional M(ρc ) method is compatible
the central density (left panel) and gravitational mass (right with the calculation of the eigenfrequencies of the normal
panel). According to the left plot, the squared frequency of vibration modes.
123Eur. Phys. J. C (2023) 83:26 Page 9 of 16 26
Table 2 Radius, mass, redshift, fundamental mode frequency ( f 0 = parameter α. Remarkably, with the exception of the fundamental mode
ν0 /2π ), moment of inertia and dimensionless tidal deformability of frequency and tidal deformability, these properties undergo a significant
dark energy stars with central energy density ρc = 1.5 × 1018 kg/m3 increase as α increases
as predicted by models I and II for several values of the anisotropy
Model α R [km] M [M ] z sur f 0 [kHz] I [1038 kg · m2 ]
I −0.4 10.062 1.713 0.418 2.414 1.695 13.278
−0.2 10.163 1.781 0.440 2.312 1.820 10.709
0 10.263 1.852 0.463 2.201 1.957 8.598
0.2 10.361 1.926 0.489 2.081 2.105 6.868
0.4 10.456 2.003 0.518 1.950 2.265 5.454
II −0.4 11.767 2.310 0.543 1.131 3.298 4.889
−0.2 11.859 2.395 0.574 0.998 3.531 3.823
0 11.944 2.481 0.609 0.840 3.778 2.978
0.2 12.019 2.569 0.647 0.637 4.037 2.309
0.4 12.083 2.656 0.688 0.315 4.303 1.782
Fig. 2 Radial behavior of the mass density (left panel), pressures (mid- curves represent the isotropic solutions. Note that both the radial and
dle panel) and squared speed of sound (right panel) inside an anisotropic tangential speed of sound obey the causality condition. Furthermore,
dark energy star with central density ρc = 2.0 × 1018 kg/m3 and several one can observe that the increase in α leads to larger radii, and the
values of the parameter α. All plots correspond to model I and the black anisotropy is more pronounced in the intermediate regions
If the anisotropic dark energy star has a central density stable branches in the mass-radius diagram for hybrid stars
higher than one corresponding to the maximum-mass con- [100].
figuration (indicated by full blue and orange circles in Figs. 3
and 6), the star will become unstable against radial pertur- 4.3 Moment of inertia
bations and collapse to form a black hole. For further details
on the dissipative gravitational collapse of compact stellar To calculate the moment of inertia of anisotropic dark energy
objects we also refer the reader to Refs. [55,97–99]. Nonethe- stars, we first need to solve the differential equation for the
less, we must point out that there are EoS models that allow rotational drag (16) with boundary conditions (19). In partic-
a compact star to migrate to another branch of stable solu- ular, for model I and central density ρc = 1.5 × 1018 kg/m3 ,
tions instead of forming a black hole when it is subjected to a Fig. 7 illustrates the angular velocity everywhere for sev-
perturbation. As a matter of fact, the first-order phase transi- eral values of α. As can be observed in the right plot, the
tion between nuclear and quark matter can generate multiple dragging angular velocity outside the star has the behavior
ω(r ) ∼ r −3 , so that at infinity (where spacetime is flat) the
12326 Page 10 of 16 Eur. Phys. J. C (2023) 83:26 Fig. 3 Mass-radius diagram (left panel) and mass-central density rela- Note that the maximum-mass values for model II correspond to lower tion (right panel) for anisotropic dark energy stars as predicted by model central densities than those for model I, however, model II allows larger I (blue curves) and II (orange curves) with anisotropy profile (30) for masses (see also Table 1). The critical central density corresponding to several values of α. The colored bands in the left plot represent the the maximum point on the M(ρc ) curve is modified by the presence of same as in Fig. 1. Moreover, the full blue and orange circles on the right anisotropy for both models plot indicate the maximum-mass points for model I and II, respectively. Fig. 4 Left panel: Variation of the compactness with central density also that dark energy stars would correspond to ultra-compact objects for several anisotropic dark energy star sequences. The gray and light- if α > 0.4 for model II, see for instance the purple curve for α = 0.7. green stripes represent compact and ultra-compact objects, respectively, Right panel: Surface gravitational redshift as a function of the total according to the classification established by Iyer et al. [92]. Positive mass. In the high-redshift region it can be observed that positive (nega- anisotropy results in increased compactness for sufficiently high cen- tive) anisotropy increases (decreases) the value of z sur . Meanwhile, the tral densities, while the opposite occurs for negative anisotropy. Note effect of anisotropy is irrelevant for sufficiently low redshifts distant local inertial frames do not rotate around the star, given in Eq. (18). For the above central density, we present namely, ω(r ) → 0 for r → ∞. Moreover, anisotropy sig- the moment of inertia of some dark energy configurations nificantly affects the angular velocity of the local inertial for both models in Table 2, where it can be noticed that I frames in the interior region of the star. More specifically, increases as the value of α increases. the dragging angular velocity increases (decreases) for pos- We can now calculate the moment of inertia for a whole itive (negative) values of the anisotropy parameter α. We sequence of dark energy stars by varying the central density can then determine the moment of inertia using the integral ρc . The left panel of Fig. 8 displays the moment of inertia as a 123
Eur. Phys. J. C (2023) 83:26 Page 11 of 16 26
Fig. 5 Numerical solution of the radial pulsation Eqs. (8) and (9) nth vibration mode contains n nodes in the internal structure of the
in the case of an anisotropic dark energy star with central density star. Note that the eigenfunctions ζn (r ) have been normalized assuming
ρc = 1.5 × 1018 kg/m3 , α = 0.4 and EoS parameters given by model I. ζ = 1 at r = 0, and the Lagrangian perturbation of the radial pressure
The radius, mass and the fundamental mode frequency for such config- pr,n (r ) obeys the boundary condition (11) at the stellar surface. Since
uration are found in Table 2. The lines with different colors and styles f 0 is real, this configuration corresponds to a stable anisotropic dark
indicate different overtones so that the solution corresponding to the energy star
Fig. 6 Left panel: Squared frequency of the fundamental pulsation on the right plot of Fig. 3. Right plot: Squared frequency of the funda-
mode as a function of central mass density for anisotropic dark energy mental mode versus gravitational mass, where it can be observed that
stars predicted by Einstein gravity. The full blue and orange circles the maximum-mass values determine the boundary between stable and
indicate the central density values where ν02 = 0, whose values pre- unstable stars
cisely correspond to the maximum-mass points on the M(ρc ) curves
function of the gravitational mass for both models. Remark- moment of inertia induced by the anisotropic pressure, we
ably, model II provides larger values for the moment of inertia can define the following relative difference
than model I. Indeed, the maximum value Imax depends quite
sensitively on the free parameters A and B in the EoS (28). Imax,ani − Imax,iso
I = , (33)
In addition, the main effect of anisotropy on the moment of Imax,iso
inertia for slow rotation occurs in the high-mass region, while
its influence is irrelevant for sufficiently low masses. In order where Imax,iso and Imax,ani are the maximum values of the
to better quantify the changes in the maximum values of the moment of inertia for isotropic and anisotropic configura-
tions, respectively. In the right plot of Fig. 8 we present
12326 Page 12 of 16 Eur. Phys. J. C (2023) 83:26
the dependence I against the anisotropy parameter α. The agree sufficiently with the observational data, e.g. the mass-
impact of anisotropy is getting stronger as |α| grows, reach- radius constraint from the GW170817 event. For isotropic
ing variations (with respect to the isotropic case) of up to configurations, we have shown that various sets of values
∼ 20% for α = 0.5. We can also note that such relative {A, B} can be chosen since they obey the causality condi-
variations are almost independent of the model adopted. tion and consistently describe compact stars observed in the
Universe. Furthermore, we saw that the secondary compo-
4.4 Tidal properties nent resulting from the gravitational-wave signal GW190814
[91] can be described as a dark energy star using A = 0.4
We will now investigate how the anisotropy parameter α and B ∈ [4, 5]μ.
affects the tidal properties of dark energy stars. Given a spe- Based on these results, we have established two models
cific value of α, this requires solving the differential equation with different values A and B in order to explore the effects of
(23) for a range of central densities. The left panel of Fig. 9 anisotropy in the interior region of a dark energy star. In par-
is the result of calculating the tidal Love number (27) for a ticular, the maximum-mass values increase as the parameter
sequence of stellar configurations by considering different α increases. We noticed that model I without anisotropic pres-
values of α, where the isotropic case corresponds to α = 0. sures is not capable of generating maximum masses above
Similar to the trends in strange quark stars, as reported in 2M . However, the inclusion of anisotropies (α = 0.4)
Ref. [70], the Love number of dark energy stars grows until allows a significant increase in the maximum mass and thus a
it reaches a maximum value and then decreases as compact- more favorable description of the compact objects observed
ness increases. Note also that the maximum value of k2 is in nature. On the other hand, model II with anisotropies fits
sensitive to the value of α, indicating that the Love num- better with the observational measurements, although such a
ber decreases as the parameter α increases for both models. model can lead to the formation of ultra-compact objects for
Although model II provides larger maximum masses (as well sufficiently large values of α. We also calculated the surface
as redshift and moment of inertia) than model I, we see that gravitational redshift for such stars, and our results indicated
the behavior is different for the maximum values in the tidal that z sur is substantially affected by the anisotropy in the high-
Love number. mass branch, while the changes are irrelevant for sufficiently
Ultimately, in the right plot of Fig. 9, the dimension- low masses.
less tidal deformability = λ̄/M 5 is plotted as a func- A star exists in the Universe only if it is dynamically sta-
tion of mass, where it can be observed that smaller masses ble, so our second task was to investigate whether the dark
yield higher deformabilities. In each model, the presence of energy stars are stable or unstable with respect to an adia-
anisotropy has a negligible effect on for small masses, batic radial perturbation. Our results showed that the stan-
while slightly more significant changes take place only in dard criterion for radial stability d M/dρc > 0 still holds for
the high-mass region. dark energy stars since the squared frequency of the funda-
mental pulsation mode (ν02 ) vanishes at the critical central
density corresponding to the maximum-mass configuration.
5 Conclusions and outlook This has been examined in detail for both isotropic (α = 0)
and anisotropic (α = 0) stellar configurations.
In this work, we have focused on the equilibrium structure In the slowly rotating approximation, where only first-
of dark energy stars by using a Chaplygin-like equation of order terms in the angular velocity are kept, we have also
state under the presence of both isotropic and anisotropic determined the moment of inertia of anisotropic dark energy
pressures within the context of standard GR. Our goal was stars. For this purpose, we first had to calculate the frame-
to construct stable compact stars whose characteristics could dragging angular velocity for each central density. The pres-
be compared with the observational data on the mass-radius ence of anisotropic pressure results in a substantial increase
diagram. In this perspective, the global properties of a com- (decrease) of the angular velocity ω for more positive (neg-
pact star such as radius, mass, redshift, moment of inertia, ative) values of α. We found that the significant impact of
oscillation spectrum and tidal deformability have been cal- the anisotropy on the moment of inertia occurs mainly in the
culated. To describe the anisotropic pressure within the dark high-mass branch for both models. Furthermore, the maxi-
energy fluid we have adopted the anisotropy profile proposed mum value of the moment of inertia can undergo variations
by Horvat et al. [51], where a free parameter α measures the of up to ∼ 20% for α = 0.5 as compared with the isotropic
degree of anisotropy. case.
We have discussed the possibility of observing stable dark We have analyzed the effect of anisotropic pressure on
energy stars made of a negative pressure fluid “−B/ρ” plus a the tidal properties of such stars. In particular, our outcomes
barotropic component “Aρ”. By way of comparison, the EoS revealed that the tidal Love number is sensitive to moderate
parameters A and B have been chosen in such a way that they variations of the parameter α, indicating that the maximum
123Eur. Phys. J. C (2023) 83:26 Page 13 of 16 26
Fig. 7 Left panel: Numerical solution of the differential Eq. (16) for ω(r )/ = 1 − (r )/. It can be observed that the outer solution
a dark energy star described by model I and central density ρc = behaves asymptotically at large distances from the surface of the star
1.5 × 1018 kg/m3 in the presence of anisotropy for several values of (this is, ω → 0 for r → ∞). Furthermore, appreciable changes in
the free parameter α. The solid and dashed lines represent the inte- the angular velocity due to anisotropy can be noticeable, mainly in the
rior and exterior solutions, respectively. Right panel: Ratio of frame- interior region of the star
dragging angular velocity to the angular velocity of the star, namely
Fig. 8 Left panel: Moment of inertia versus mass for anisotropic dark Right panel: Relative deviation (33) as a function of the anisotropy
energy stars, where a higher mass results in larger moment on inertia for parameter. The maximum value of the moment of inertia can undergo
both models. It is observed that the substantial impact of anisotropy on variations with respect to its isotropic counterpart of up to ∼ 20% for
the moment of inertia occurs predominantly in the high-mass branch. α = 0.5
value of k2 can increase as α decreases. In addition, the great- sars. Future research includes the adoption of widespread ver-
est effect of anisotropy on the dimensionless tidal deforma- sions of Chaplygin gas that best fit key cosmological param-
bility takes place only in the high-mass region. Based on the eters. In future studies we will thereby take further steps in
foregoing results, the present work thereby serves to develop that direction, focusing on the different types of generalized
a comprehensive perspective on the relativistic structure of Chaplygin gas models as discussed in Ref. [11]. In addi-
dark energy stars in the presence of anisotropy. tion, as carried out in the case of boson stars [101], it would
Summarizing, we have explored the possible existence be interesting to employ a Fisher matrix analysis in order
of stable dark energy stars whose masses and radii are not to distinguish dark energy stars from black holes and neu-
in disagreement with the current observational data. The tron stars from tidal interactions in inspiraling binary sys-
Chaplygin-like EoS predicts maximum-mass values consis- tems. It is also worth mentioning that Romano [102] has
tent with observational measurements of highly massive pul- recently discussed the effects of dark energy on the propa-
12326 Page 14 of 16 Eur. Phys. J. C (2023) 83:26
Fig. 9 Left panel: Tidal Love number plotted as a function of the com- stantially modified by the anisotropy parameter α for both models, while
pactness C ≡ M/R. Right panel: Dimensionless tidal deformability its greatest effect on tidal deformability occurs only in the high-mass
versus gravitational mass predicted by each model, where larger masses region
yield smaller deformabilities. Note also that the Love number is sub-
gation of gravitational waves. In that regard, we expect that References
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