Dark Energy Stars with Phantom Field

Home , Dark star (dark matter), Gravastar

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[Phys. . oprrypatmdr nry Although energy. dark phantom portray to 1 keeg ihntesa.S far, So star. the within energy rk htDSsalvoaetestrong the violate shall DES that C aetegaiainlwv echo wave gravitational the late h rsn nvre responsible universe, present the tclysuidyt eae to Related yet. studied atically .I sfudta gravastar that found is It 1. = httecnrbto fthe of contribution the that d ei 16424. nesia noei,and Indonesia, (ICTMP), 3 † p = ωρ sln as long as , 2 ergy model that interacts minimally with the ordinary also determines the compactness of DES, it will be inter- matter. Therein, the interior solution of the Einstein esting to analyze its impact on the gravitational echoes. field equation with a phantom scalar field is derived by Moreover, the authors in Ref. [36] introduce that the in- applying a generating method presented in Ref. [26] from formation of dark energy can be accommodated from the the Schwarzschild interior solution or the constant en- gravitational radiation of binary systems of supermassive ergy density star (CDS), namely dark energy star (DES). black holes. However, the feasibility is still needed further There are several similarities and significant differences investigation (see Ref. [37] and the references therein for between DES and CDS, which we can be seen in Ref. [25]. detailed explanation). The gravitational echoes can be Both stars respect the Buchdahl inequality C 4/9 and detected in the near future with the gravitational waves have asymptotically flatness at infinity. However,≤ DES detectors, so investigating the impact of the phantom possesses a coordinate-dependent energy density and a dark energy on the gravitational echoes may be possible non-vanishing charge known as a dark charge from the to probe dark energy locally in DES or phantom black phantom field’s contribution. This dark charge has a crit- holes. ical role in determining compactness. Due to the phan- In Ref. [38], the authors investigate the gravita- tom field’s presence, the star’s pressures are anisotropic, tional echoes for different exotic compact objects in- where the phantom field’s kinetic term appears on the cluding gravastars given in [28, 29]. Gravastar in Refs. radial pressure. Another exciting feature of DES is the [28, 29] is the simplest model of gravastars constructed vanishing tangential pressure when D M while keep- | | → from CDS which possesses three different regions, i.e in- ing R fixed. terior (0 r < R ), thin shell (R r < R) and exterior ≤ 0 0 ≤ We know that CDS is the simplest exact interior so- (R < r). Gravastar suffers a discontinuity in EoS on the lution of the Einstein field equation supported by an surface; however, it can be resolved by the discontinuity isotropic perfect fluid. CDS is also used to mimic the of the extrinsic curvature by employing Darmois-Israel’s Schwarzschild black hole in the limit R 2m [27] and re- junction conditions [39, 40]. Since CDS is the seed solu- sembles the main features of the simplest→ gravatar model tion of DES, we may expect that the gravastar-like solu- proposed in Refs. [28, 29]. From this fact, we may expect tion can also be constructed from DES. that DES presented in [25] will be able to mimic a black In the present paper, we first investigate the existence hole consisting of a phantom dark energy that we call a of the dark energy properties coming from the phantom phantom black hole. Furthermore, DES can also be used field in DES by employing the energy conditions anal- to study the gravastar-like solution with dark energy rep- ysis. As we know, CDS does not meet the criteria of resented by the phantom field. Due to the constancy of causal conditions, so we also check whether DES satis- the energy density, CDS violates the causal conditions, fies these conditions by analyzing the speed of sound in even though it is stable due to the convective motion, the star. Furthermore, it is also interesting to examine which satisfies ρ′′ = 0 and due to the gravitational crack- the stabilities of DES due to the convective motion and ing since the pressure is isotropic. However, the insta- gravitational cracking because the anisotropy from the bility of the isotropic structure can also occur in com- phantom field’s kinetic term may cause instability within pact objects. The stability of the anisotropic structure of the star. The surface redshift of DES will also be studied neutron stars within the Eddington-inspired Born-Infeld to see the phantom field’s influence compared to that of (EiBI) theory has been investigated in Ref. [30]. The CDS. Since we know that CDS is a good toy model for in- primary analysis of those stabilities is given in [31]. How- vestigating the gravitational echoes for an ultra-compact ever, the stabilities due to those conditions for DES are object, we also delve into the properties of DES as an not systematically investigated, yet [25]. Moreover, the ultra-compact object. We compare our results with CDS claim that the phantom field plays a role as dark energy to see the phantom field’s impact on the gravitational is not thoroughly investigated [25] where the dark energy echoes, especially in the echo time and echo frequency. shall violate SEC [32]. Indeed, stars which do not con- We also compare the effective potential between CDS tain dark energy shall satisfy all energy conditions, for and DES to see how this corresponds to the echo time example, the anisotropic neutron stars [33] and strange and echo frequency. Then we examine the gravastar-like stars in non-conserved energy-momentum theory such as solution of DES. We wish to see whether DES can be con- Rastall gravity [34]. structed to be the ordinary gravastar given in ref. [28] or CDS can be used as an illustrative model for exotic exceptional gravastar. Note that we use the convention compact stars which can produce gravitational echoes G = cl = 1 for the whole paper. We refer cl as the speed [35], even though this represents an unphysical star be- of light. cause of the violation of the causal conditions. If CDS This paper is organized as follows. In Sec. II, we review has compactness in the range of 1/3 C 4/9, it can DES as derived in [25]. In Sec. III, we investigate the have a photon sphere or the light ring≤ where≤ the radia- energy conditions, stabilities, and surface redshift analy- tion can be effectively trapped between the light ring the sis. In Sec. IV, the properties of an ultra-compact DES center of the star. Since DES possesses a similar Buch- are studied, including the calculation of echo time and dahl inequality, it may also have the light ring in the echo frequency and the analysis of the effective poten- ultra-compact region like CDS. Because the dark charge tial. Before the conclusions, we provide the gravastar- 3

M m like solution of DES in Sec. V. Finally, in conclusion, we c = , C = , m = M 2 D2, (7) summarize the results of the whole paper. m R − p and M,D,R are the mass, dark charge and the star II. DARK ENERGY STAR SOLUTION radius, respectively. Furthermore, the pressure, energy density, charge density and the phantom field are given by [25] It is calculated by Yazadjiev [25] using his method in [26] that one can obtain an interior solution of the Ein- 1 1 stein field equation containing an ordinary matter and 2C 2 2 2 3C 2λ(c−1) 1 R2 r (1 2C) a scalar phantom field with no phantom potential. The − − − 1 p = 2 e 1 , (8) 4πR  2 2C 2 2  equation of motion of this system is described by 3 (1 2C) 1 2 r − − − R 1   R = κ T g T 2∂ ϕ∂ ϕ, (1) µν µν − 2 µν − µ ν 3M 2λ(c−1) ρ = 3 e + 3(c 1)p, (9) where κ = 8π and Tµν is the energy-momentum tensor 4πR − of the ordinary matter described by 3D 3D Tµν = (ρ + p)uµuν + pgµν . (2) ρ = e2λ(c−1) + p, (10) D 4πR3 m The phantom kinetic term is given on the second term of right hand side of Eq. (1). ρ and p are energy density D and pressure of the isotropic perfect fluid, respectively. ϕ = λ. (11) The equation of motion of the phantom field is given by m κ DES satisfies the following equilibrium equation µϕ = ρ , (3) ∇µ∇ 2 D ∂rp + (ρ + p)∂rU = ρD∂rϕ. (12) where ρD is the charge density of the phantom field ϕ. He introduces ρD as the source of the dark energy. When Above equation explains that the dark energy term pro- considering an astrophysical scale of the manifestation vides effective pressure to balance the gravitational force of the dark energy, the phantom field’s potential can be and hydrodynamic force in the left hand side. DES also neglected. Hence, within the derivation of the Einstein- satisfies the following Buchdahl inequality [25] perfect fluid-phantom field system’s interior and exterior solutions, one can consider the vanishing potential. It is 4 C , (13) also considered that there is no interaction between the ≤ 9 phantom field and the ordinary matter. The ansatz of the metric solution is given by which is similar as CDS. This interior solution will reduce to CDS when the dark charge vanishes. On the ds2 = e2U dt2+e−2U+2λ e−2χdr2 + r2 dθ2 + sin2 θdφ2 , − surface of the star (r = R), the interior solution matches (4) continuously with the exterior solution which is given by where the functions U,λ,χ depend on coordinate r only. c −(c−1) 2 Moreover, the phantom field ϕ and charge density ρD 2m 2m dr ds2 = 1 dt2 + 1 are also dependent on radial coordinate only. For the ext − − r − r 1 2m ordinary matter, it is imposed that p,ρ are dependent − r on r only while u dxµ = eU dt. To produce an exact 2 2 2 2 µ − +r dθ + sin θdφ . (14) interior solution of the Einstein-phantom field system (1), one can employ the mathematical methods as given in This is the asymptotically flat phantom black hole so- Ref. [26]. We need CDS as a seed metric to generate lution. When c = 1, it will reduce to the Schwarzschild a new interior solution with an additional phantom field black hole. Note that this black hole solution corresponds contribution. The resulting interior solution is given as with the phantom field through the charge D where the follows phantom field is given by dr2 ds2 = e2λcdt2 + e−2λ(c−1) + r2dθ2 int 2C 2 D 2m − 1 2 r R ϕ = ln 1 . (15) − 2m − r +r2 sin2 θdφ2 , (5) At spatial infinity, the phantom field will vanish. where In [25], Yazadjiev also considers an extremal condition 1 of DES. Nevertheless, this extremality is distinct from the 3 1 1 2C 2 extremality on the black holes, which indicates the condi- eλ = (1 2C) 2 1 r2 , (6) 2 − − 2 − R2 tion where the horizons coincide with being one horizon. 4

The extremality on DES refers to D M when keep- ing R ﬁxed. In this limit, the space-time| | → metric of DES 3 reduces to

M r2 M r2

− 3− 3− ] ds2 = e R R2 dt2 + e R R2 dr2 + r2dθ2 2 − -2 +r2 sin2 θ2dφ2 . (16) [10 1 This corresponds to the following conditions

2 0 3M − M 3− r ρ = e R R2 , ρ = ρ, p =0. (17) 0 1 2 3 4 4πR3 D ± r The phantom ﬁeld is then given by 0 2

M r ]

ϕ = 3 . (18) -1 ∓2R − R2 -1

[10

Note that D M corresponds with ϕ = U and ρD = ρ r

→ p while D M corresponds with ϕ = U and ρD = ρ. -2 The extremal→− DES (16) matches continuously− with− the following exterior phantom black hole, D M, | | → -3 2 2M 2 2M 2 2 2 2 2 2 0 1 2 3 4 ds = e r dt + e r dr + r dθ + sin θ dφ , r − (19) and the phantom ﬁeld now is 3 C = 4/9 M C = 3.75/9 ϕ = . (20) ] 2 C = 3.25/9

∓ r -2 C = 3/9

In this work, we do not intend to discuss furthermore [10 DES’s extremal condition. One can see the elaborate t p 1 explanation of this extremal DES in Ref. [25].

0 III. ENERGY CONDITIONS, STABILITIES 0 1 2 3 4 AND REDSHIFT r

2 2 2 A. Energy conditions FIG. 1. Comparison ofρ ˆ (ρm ), pˆr (prm ), andp ˆt (ptm ) in different compactness. We set M = 5 and R = 9. Note that rˆ = r/m. In the original paper [25], the author claims that the interior solution in Eq. (5) represents a compact object containing an ordinary matter and dark energy with the (SEC). In mathematical form, these energy conditions dark charge D while ρD is the dark source. As we know are given as follows [41] in cosmology, the phantom field is one of the models to NEC : ρ + p 0, ρ + p 0, (21) describe dark energy that will produce an accelerating r ≥ t ≥ expansion of the universe with the EoS parameter ω < WEC : ρ + pr 0, ρ 0, ρ + pt 0, (22) 1 [2]. However, to justify an object whether contains a ≥ ≥ ≥ DEC : ρ pr , ρ pt , (23) dark− energy or not, we need some conditions that really ≥ | | ≥ | | SEC : ρ + p 0, ρ + p +2p 0, (24) describe the properties of dark energy. r ≥ r t ≥ Regarding the properties of dark energy, one can check where pr and pt are the pressure in radial and tangen- the energy condition of the stars. The energy conditions tial direction, respectively. WEC stipulates that the en- are crucial to verify the existence of realistic matter dis- ergy density is required to be non-negative measured by tribution in any astrophysical objects. The energy con- the corresponding observer. DEC stipulates that the ditions are divided into four, i.e. Null Energy Condi- pressures do not exceed the energy density, so that the tion (NEC), Weak Energy Condition (WEC), Dominant sound speed in the fluid is always less than the speed of Energy Condition (DEC) and Strong Energy Condition light. However, SEC may not be satisfied that denotes 5

6 0

] -10

-2 4

[10 -20

D 2 -30

0 -40 0 1 2 3 4 0 1 2 3 4 r r 6 C = 4/9 1.2 C = 4/9 C = 3.75/9 C = 3.75/9 ] 4 C = 3.25/9 C = 3.25/9 -1 C = 3/9 C = 3/9 t 0.8

[10 2 0.4

0 0.0 0 1 2 3 4 0 1 2 3 4 r r

2 2 FIG. 2. Comparison ofρ ˆD (ρDm ) and ∆ˆ (∆m ) in different FIG. 3. EoS parameter for radial (ωr) and tangential (ωt) compactness (M = 5, R = 9). direction in different compactness (M = 5, R = 9). the strong repulsion of gravity. We can investigate the for C = 4/9 while the tangential pressure is monotoni- energy conditions for this DES of which if this star con- cally decreasing for all C. Similarly with the tangential tains dark energy, it shall violate SEC (24). For DES, pressure, the charge density is also decreasing. note that the radial and tangential pressures are given Then we can see that the anisotropy is higher near by the surface than the center for the compactness below 4/9. Since the anisotropy is positive, it is clear that the p = p 2grr(∂ ϕ)2, p = p, (25) anisotropic force is repulsive rather than attractive. The r − r t phantom field’s contribution is not dominant near the respectively. Furthermore, as we can see, DES is an star’s center for C below the Buchdahl limit. anistropic star where the anisotropy factor is given as For the present universe where the dark energy sup- follows ports its accelerated expansion, EoS is given as ω = p/ρ where the parameter ω needs to be negative in the range 2D2r2 rr 2 2λ(c−2) 1 ω 1/3. However, in the current observation, ∆=2g (∂rϕ) = 6 e . (26) R −the EoS≤ parameter≤ − can still be lower than 1. Itisacon- − This anisotropy contributes only to the radial pressure. dition for a phantom scalar field where the kinetic energy We can see the comparison of ρ, pr , and pt on Fig. 1 term has a negative sign. For DES, the EoS parameter is and ρD and ∆ on Fig. 2 for several values of C when we dependent on radial coordinate, not constant. Because set M = 5 and R = 9. Note that this star’s pressures are of the anisotropic pressures, it is convenient to write the not linear with the energy density as the common model equations of state as follows for dark energy in our universe. For C = 4/9, the pres- p p ω = r , ω = t . (27) sures and energy density are singular at the center of the r ρ t ρ star. The enormous pressure in the center could cause a change on the topology as noted in [15] in which this We denote ωr as radial EoS parameter and ωt as tangen- may produce a tunnel and transform DES into a worm- tial EoS parameter of DES. We show its profile in Fig. 3. hole [42, 43] as we can see that the phantom energy may From those panels, we can see that the condition ωr < 1 aid the traversable wormholes [44, 45]. For C = 3.25/9 is not satisfied near the center of DES for C smaller than− and C = 3/9, the center’s energy density is lower than Buchdahl limit. Moreover, ωt is always positive. on the surface. The radial pressure is minimal at the We have mentioned beforehand that one property of center and becomes more negative on the surface, except the existence of dark energy in an astrophysical object 6 is it should violate the SEC [32]. Hence, we need to motion is given by investigate its energy conditions. As we can see from Fig. 4, DEC is not satisfied. Furthermore, the condition d2ρ ρ′′ = 0. (29) ρ 0 is satisfied. Regarding SEC, it is fully satisfied for dr2 ≤ C≥=4/9 for dark energy condition. For compactness less than C = 4/9, it is obviously seen that near the center Hence, when the fluid of DES satisfies this circumstance, of the star, SEC is not fully violated. Hence, we may the star is stable due to the convective motion. This cri- say that DES still contains dark energy but it is not fully terion for DES is shown in Fig. 6 (top panel). It can distributed in the interior of the star for C below the be seen that ρ′′ is almost approaching zero, except for Buchdahl limit. C = 4/9. Hence, for C below the Buchdahl limit, the fluid is almost stable due to the convective motion. We know that ρ′′ = 0 because the energy density is constant B. Causal conditions for CDS. So, as a toy model, CDS can be stable due to the convective motion. However, the phantom scalar field’s presence does not significantly impact the star be- According to the initial observer, the relativistic prin- cause it omits the stability due to the convective motion ciple of causality says that the cause has to precede its compared to CDS. effect. So, the cause and its consequence are separated Besides the convective motion, as we have mentioned by a time-like interval. If a time-like interval separates before, we also consider the stability due to the grav- two events, it means that a signal could be sent between itational cracking. The anisotropic matter distribution them at less than the speed of light. If there is a signal could cause the gravitational cracking instability to the that moves faster than the speed of light, it will violate stars [31] in which this instability does not occur on CDS. the causality. So, special relativity will not allow an ob- The cracking instability determines the tidal acceleration ject to have a speed faster than light. For an anisotropic profiles produced by the perturbations of the energy den- structure, the causal conditions are given as follows sity and the anisotropic pressures identifying the sign of the total force in the system. The condition to be satis- dpr dpt 0 v2 = 1, 0 v2 = 1. (28) fied by a fluid distribution to avoid gravitational cracking ≤ r dρ ≤ ≤ t dρ ≤ is given by The plots of causal conditions of DES are shown in Fig. 1 v2 v2 0. (30) 5 where the top panel is for the radial speed, and the − ≤ t − r ≤ bottom panel is for the tangential speed. It is seen that the causal conditions are violated. For DES, the plot of the gravitational cracking is shown We know that CDS also does not satisfy the causal in Fig. 6 (bottom panel). From that figure, we may conditions because of the incompressible energy density. conclude that DES is not stable due to the gravitational However, CDS can be a representative toy model for com- cracking. pact objects that respect the Buchdahl limit and have a chance to become the ultra-compact objects but still violate the causal conditions [35]. Because the Buchdahl D. Redshift limit’s derivation does not use the causal conditions, it is plausible that a compact object may not satisfy one of The redshift of DES and its exterior solution can be those conditions. We find that DES does not satisfy the computed using causal conditions but still respects the Buchdahl limit. Hence, we may say that DES may also be a good and 1 representative toy model to study an ultra-compact ob- z = 1/2 1. (31) ( gtt) − ject containing ordinary matter and dark energy. − The plot of the redshift for different value of compactness is given in Fig. 7 where the top panel is for CDS and DES, C. Stabilities due to convective motion and and the bottom panel is for the exterior solutions. Those gravitational cracking figures show that the redshift is monotonically decreasing to the increasing of the radius of the star. The values of We also consider the stabilities of DES due to the con- surface redshift zs of DES and CDS for each compactness vective motion, and gravitational cracking [31]. The sta- are given in Table I. bility due to the convective motion implies that when a From Table I, it is seen that the surface redshift for fluid element is displaced downward, the fluid element DES is larger than CDS for the same compactness. will sink, and the star will be unstable. If the fluid ele- Hence, the phantom field’s presence prolongs the wave- ment’s density is less than its surroundings, it will float length that is measured by the observer at infinity. This back, so that the star will be stable to the convective finding is an excellent sign to observe the existence of motion. The criterion of stability due to the convective such dark star. 7

10 2 C = 4/9 C = 3.75/9 5 1

] ]

-1 -1 0 0 1 0.7

EC [10

EC [10 -5 -1 0 0.0 0.5 1.0 0.0 0.7 -10 -2 0.0 0.8 1.6 2.4 0.0 0.8 1.6 2.4 r r

10 10

+pt |p | +p 5 t 5 r |pr| +pr+2pt

] ]

-2

-2 0 0 2

EC [10 EC [10 1 1 -5 -5 0 0 0 1 C = 3.25/9 0 1 C = 3/9 -10 -10 0.0 1.1 2.2 3.3 0.0 0.9 1.8 2.7 3.6 r r

FIG. 4. Comparison of the energy conditions (EC) in diﬀerent compactness (M = 5, R = 9).

(13). In investigating gravitational echoes, it is impor- TABLE I. Surface redshift zs for DES and CDS. tant to calculate the echo frequency. CDS is a good toy zs zs model to study the echo frequency as it has been calcu- C CDS DES lated in [35, 46] where its frequency is highest compared 4/9 1.0142 1.3995 to other ﬂuid stars with the same mass and radius. The 3.75/9 0.6412 0.9359 echo frequency can be roughly estimated from the inverse 3.25/9 0.2658 0.4371 3/9 0.1581 0.2771 of the time τ for a massless test particle traveling from the unstable light ring or photon sphere to the center of the star. The echo frequency is fecho = π/τecho while the echo time is given by [35, 46, 47] IV. ECHOES FROM ULTRA-COMPACT DARK 3m 1/2 ENERGY STAR g (r) τ = rr dr. (32) echo − g (r) Z0 tt A. Echo time and echo frequency For CDS, we can ﬁnd [46]

−1 4 −1 4 τ cot C 9 + tan 3/ C 9 The production of gravitational waves’ echoes requires echo = − − q q a photon sphere, which implies that the object’s com- m C2 4 9 pactness must be greater than 1/3. For the compactness C − greater than 1/3, we note this object as an ultra-compact 1 q 1 2 ln 2 +3 . (33) object, but it still satisﬁes the Buchdahl inequality Eq. − C − − C 8

2 10

1 5

]

-3 0 [10 0

'' [10

v -1 -5

-2 -10 0 1 2 3 4 0 1 2 3 4 r r 10 2 C = 4/9

C = 3.75/9 ] 5 C = 3.25/9 2 1 C = 3/9

[10

0 r 0

- v C = 4/9

2 -5 t -1 C = 3.75/9 v C = 3.25/9 C = 3/9 -10 -2 0 1 2 3 4 0 1 2 3 4 r r

FIG. 5. Causal conditions in diﬀerent compactness (M = 5, FIG. 6. Stability conditions due to convective motion (top) R = 9). and due to gravitaional cracking (bottom) in diﬀerent compactness (M = 5, R = 9).

It is worth noting that m = M for CDS. It is easy to check that when C 4/9, τ . where → echo → ∞ DES with zero dark charge will become CDS. Hence, 1 F = F c, 1+ c;2+ c; , it is interesting to examine the dark charge’s impact on a 2 1 2C the echo frequency of DES. So, we will compare our re- (2c + 1)c sult with the result of CDS. It is worth noting that the F = F c, 1+ c;2+ c; . b 2 1 2 unstable light ring of DES is different from CDS as we derive in Appendix A. For DES, the echo time for a mass- and 2F1(a,b; c; d) is a hypergeometric function. It is ob- less test particle from the photon sphere to the center of viously seen that τecho is dependent on C and c unlike the star can be calculated as CDS that is dependent on C only. The integration of the 1−2c echo time for exterior part yields a complex function. 3 1 2Cr2 √1 2C 1 2 Unlike the echo time of CDS, due to the phantom R 2 − − 2 − R τ = dr field’s presence as the source of dark energy, there is echo q 2 0 2Cr Z 1 R2 an imaginary contribution to the echo time. However, − the imaginary contributions coming from the integration M(2c+1) q 2m −c + 1 dr, (34) from the surface to the photon sphere cancel each other. − r We show the phantom field’s impact by comparing it ZR with CDS shown in the above panels of Fig. 8. We where the photon sphere is located on r = M(2c + 1) show the echo time and frequency as a function of C with (Appendix A). The integration of the interior solution c =1.2626 and c =1.0002. Both echo times go to infin- above cannot be solved analytically. Nonetheless, we can ity when approaching the Buchdahl limit. When c > 1, solve the exterior part analytically with the hyperbolic the echo time is bigger than CDS while the frequency function. By taking r = mrˆ and integrating the exterior is inversely proportional. It denotes that the presence part, we can write the time as of a phantom field as the dark energy model delays the 1−2c R 3 1 3 2 gravitational echoes. The larger c (larger D) delays the τecho √1 2C √1 2C rˆ = 2 − − 2 − drˆ echoes more. When c 1, the echo time of DES ap- 3 2 m 0 √1 2C rˆ ≈ Z − proaches CDS’ echo time. In the bottom panels of Fig. ( 2C)−c ( 2)−c [c(2c + 1)]c+1 8, we show the echo time and echo frequency as a func- − F + − F , (35) − C(1 + c) a 1+ c b tion of c for DES when C = 0.42 and C = 0.44. From 9

20 . For DES, we know that P (r) pr(r). Hence, we can DES find the effective potential as → C = 4/9 C = 3.75/9 l(l + 1) 1 s2 15 C = 3.25/9 V (r)= e2U(r) + − s,l r2 re−2U(r)+2λ(r)−2χ(r) C = 3/9 CDS ′ ′ z 10 C = 4/9 [χ (r) λ (r)] 4π[pr(r) ρ(r)] . (37) × − − − C = 3.75/9 C = 3.25/9 λ(r)−χ(r)−2U(r) C = 3/9 The tortoise coordinate is now dr∗ = e dr. 5 Using Eq. (37), we can analyze the effective potential due to the presence of the phantom field. 0 To analyze the effective potential, we use the ordinary 0 1 2 3 4 dimensionless radial coordinater ˆ and dimensionless tortoise coordinater ˆ∗. Firstly, the comparison of the effec- r tive potential for CDS and DES is explicitly given in the 4 top panel of Fig. 9 for C = 0.44 and c = 1.2626. Note that the discontinuity on the effective potential denotes 3 the surface of the stars. Besides, for Schwarzschild black hole and the phantom black hole, the effective potentials are shown in the bottom panel of Fig. 9. It is seen that the effective potentials of the stars form a potential well z 2 Phantom BH between the center of the star to the unstable circular c = 5/4 orbit or light ring. Both potentials enjoy the presence of 1 c = 5/3.75 a potential well. This fact means that DES also possesses c = 5/3.25 a second (stable) the light ring like CDS [46]. From this c = 5/3 Schwarzchild BH (c=1) fact, we may observe that DES possesses an unstable 0 light ring, one of the conditions to produce the gravi- 4 6 8 10 tational echoes. It also possesses a stable light ring at r the minimum of the potential well. The difference between CDS and DES is the peak of the potential for the FIG. 7. Redshift for interior (top) and exterior (bottom) so- same compactness. The potential well of DES is higher lutions in different compactness (M = 5, R = 9). (deeper) than CDS, which corresponds to the echo time. The echo time for DES is longer than CDS because of this. It means that the gravitational waves need more the bottom panels of Fig. 8, it can be seen that the echo time to travel in the potential well of DES than in the time becomes larger for larger c. On the other hand, the potential well of CDS. larger c will expedite the echoes. In Fig. 10, we show the effective potential for different c when C =0.44 in the top panel and for different C for c = 1.0002 in the bottom panel for DES. The top panel B. Effective potential shows that the peak of the potential decreases and the width of the potential well increases, but it is not sig- CDS enjoys the presence of a potential well in between nificant. Then for the bottom panel, we can infer that the light ring at r =3M and the center of the star r =0 the depth of potential well increases as the compactness [35, 46]. It means that CDS has a second light ring at the increases. When the compactness approaches the black minimum of the potential. In the previous subsection, we hole’s compactness (C = 0.5), the effective potential re- have shown the echo time and echo frequency for both sembles the black hole’s effective potential. The gravi- CDS and DES. We could infer that there is a similar tational waves take more time to travel in the potential feature regarding the gravitational echoes on both star well with a higher c because the potential is relatively models. Hence, DES must enjoy a potential well as CDS, wide. While in higher C, the gravitational waves take yet with different conditions due to the phantom field’s more time to travel since the potential well is deep. existence. For lative Schwarzschild exterior solution, the tortoise coordinate is given by We focus on the axial perturbation of DES (5) and its exterior solution (14). For a spherically symmetric space- r r∗ = r +2m ln 1 , (38) time (4), we may apply the wave equation as shown in 2m − Appendix B. To use wave equation (B2) for space-time metric (4), we need to transform The inverse of Eq. (38) that or r(r∗) can be obtained using the Lambert W function. The effective potential v(r) 2U(r), ξ(r) 2U(r)+2λ(r) 2χ(r). (36) of CDS in tortoise coordinate is shown in Fig. 11. The → →− − 10

1.0 0.5 CDS CDS 0.8 DES, c =1.2626 0.4 DES, c =1.2626 . ] DES, c =1.0002 DES, c =1.0002

3 0.6 0.3

[10

echo 0.4 f 0.2

echo 0.2 0.1

0.0 0.0 0.33 0.36 0.39 0.42 0.45 0.33 0.36 0.39 0.42 0.45 C C 5 0.16 C=0.42 C=0.42 4 C=0.44 C=0.44 ] 0.12

3 m [10 0.08

echo

2 f

echo 0.04 1 0 0.0 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 c c

FIG. 8. Echo time (ˆτecho = τecho/m) and echo frequency as a function of C (top) and as a function of c (bottom).

5 5 DES c=1.2626 CDS 4 c=1.0941 4 0.4 c=1.0002

3 0.2 3

eff

V V 0.0 2 2 0 3 6 9 1 1 0 0 0 4 8 12 16 20 0 2 4 6 8 r r 4 5 Phantom BH C=0.42 Schwarzschild BH 4 C=0.44 3 C=0.48 3 2

eff

V V 2 1 1 0 0 0 4 8 12 16 20 0 2 4 6 8 r r

2 FIG. 9. Effective potential (Vêff = Veff m ) of CDS and DES FIG. 10. Effective potential of DES for different c and C. for C = 0.44 and c = 1.2626.

diﬀerence between CDS and Schwarzschild black holes’ 11

0.16 tar comes as a serious alternative to a gravitational collapse system in which it has no singularity within it. 0.12 This star brings new interest in the gravitational system. Gravastar is introduced by Mazur and Mottola [28] as a dark, cold, and compact object with a de Sitter eff 0.08 condensate phase covered up by a thin shell. The thin V shell consists of an ultra-relativistic matter directly ad- 0.04 jacent with a total vacuum spacetime or Schwarzschild spacetime. However, it is presented in Ref. [50] that gravastar may not have a thin shell when it possesses an 0.0 -10 0 10 20 30 40 anisotropic structure. Gravastar is thermodynamically r stable with vanishing entropy in the interior region. 0.16 * Specifically, gravastar possesses three different seg- ments with three different equations of state,

0.12 I. Interior 0 r < R0, ρ = p, ≤ − II. Shell R0

eff 0.08 III. Exterior R

V The ﬁrst segment is the interior from the center r =0 to 0.04 the radius of R0 with de Sitter phase. In this region, the radius satisﬁes 0.0 -20 0 20 40 60 r < R = Rs =2m, (41)

r* where Rs = 2m is the Schwarzschild radius. It is seen that in this circumstance, Buchdahl’s inequality is vio- FIG. 11. Effective potential of CDS in tortoise coordinate for lated. On the other hand, this denotes that the ultra- C = 0.44. compact object is formed. The third segment is the vacuum Schwarzschild solution, which is asymptotically flat. To find the second segment, but we need to assume that effective potentials is that CDS’ potential tends to infin- grr 1 at first to gain ρ = +p. We refer to read the ity at the center that corresponds with the centrifugal ≪ 2 original paper [28] about the thin shell’s derivation. term l(l +1)/r while the potential of the black hole van- In this section, we are going to investigate the similar ishes at the horizon. Nonetheless, for the phantom black condition of gravastar to DES. Following [50], due to the hole, the tortoise coordinate is explicitly given by anisotropic structure of DES, we assume that the thin r c r r shell does not exist for this gravastar-like solution. For r∗ = F c, 1+ c;2+ c; , (39) −2m 1+ c 2 1 2m DES, the horizon radius is located at which is a complex function, and we cannot find the in- Rs =2C. (42) verse analytically. Hence, the effective potential of DES For Rs = R =2m, it will violate the Buchdahl inequality cannot be described in tortoise coordinate. However, for DES (13). The compactness C will be equal to one. since the effective potential in coordinate r of DES pos- This condition corresponds to the following space-time sesses similar properties with CDS, it is presumable that metric the potential of DES possesses similar properties. Fur- c −c thermore, one can also employ the Pöschl-Teller poten- 1 r2 1 1 r2 ds2 = 1 dt2 + 1 dr2 tial to delve the wavefront of gravitational echoes [48] of − 4 − R2 4 4 − R2 which this is left for further study. − 1 r2 1 c + 1 r2(dθ2 + sin2 θdφ2). (43) 4 − R2 V. GRAVASTAR-LIKE CONDITION OF DARK If we set c = 1, metric (43) will reduce to dS metric as the ENERGY STAR interior of the gravastar. No singularity is present in this space-time metric. It would be fascinating to delve into There is a unique and fascinating condition for CDS. It the symmetry in (43) that we leave for the future works. could describe a simple gravitational vacuum condensate Furthermore, the negative pressure will be produced as star or famously called by gravastar [28, 29]. As we know, given by black holes are believed to be the final stage of the gravi- c−1 tational collapse. However, it possesses an event horizon 3C 1 r2 p = p = 1 . (44) where the physics that we know no longer exists. Gravas- t −4πR2 4 − R2 12

Hence, now p is already negative without the anisotropic radial pressure. The anisotropy comes from the phan- factor. Now, the anisotropy is given as follows tom field contribution. We have checked the existence of the dark energy represented by the phantom field using c−2 2D2r2 1 r2 energy conditions analysis. We know that the existence ∆= 1 . (45) R6 4 − R2 of dark energy should violate SEC. We have found that SEC is not thoroughly violated in all ranges of compact- Since radial pressure pr = pt ∆ is still also negative, it is ness, except in the Buchdahl limit or C = 4/9. It could − evident that all pressures are negative and depend on co- be seen that for C < 4/9, the phantom field does not fill ordinate r in the star’s interior. This kind of gravastar is, the center of the star. Hence, the dark energy modeled however, quite distinct from the gravastar constructed in by the phantom field in DES does not fill up a whole [28]. The energy density is proportional to the tangential star. However, in principle, the enormous pressure in pressure where it is given by the center of the star at Buchdahl limit may lead to the

c−1 production of a tunnel representing a formation of the 3(3m 2M) 1 r2 traversable wormhole [15]. ρ = − 1 , (46) 4πR3 4 − R2 We also have considered the causal conditions and stabilities of DES due to the convective motion and gravita- as well as with the charge density of DES tional cracking. Similar to CDS, DES with phantom dark c−1 energy does not satisfy the radial and tangential speeds’ 3D 1 r2 ρ = 1 . (47) causal conditions. Furthermore, DES is not stable due D −2πR3 4 − R2 to the convective motion and the gravitational cracking unlike CDS. The surface redshift has also been investi- We can see obviously that there is no singularity on the gated. We compared for both surface redshift predicted pressures, energy density and charge density. Indeed, by CDS and DES. The results show that the surface red- when c = 1, all pressures and energy density reduce to shift for DES is larger than that of CDS. It means that the ordinary gravastar. EoS parameters of this star are the presence of the phantom dark energy prolongs the now given as follows wavelength that is observed at infinity. − m 32πD2 R2 1 Because we have observed several remarkable features ω = 1 , r − 3m 2M − 3(3m 2M)R − r2 near the Buchdahl limit, it has been fascinating to inves- " − − # tigate DES as an ultra-compact object that might pro- m ω = . (48) duce gravitational echoes. There is an unstable light ring t −3m 2M − (photon sphere) on r = M(2c +1) for DES, so we com- It is also intriguing that, unlike the ordinary gravas- puted the echo time and echo frequency. For DES, there tar, EoS parameters is varied, dependent on mass M, is an imaginary contribution to the echo time. However, dark charge D and radius R. In this case, we can find the imaginary contributions coming from the integration ωr,t < 1 for arbitrary M,D,R which represents phan- from the surface to the photon sphere cancel each other. tom dark− energy. Moreover, the radial EoS is coordinate- We have found that the phantom field’s presence de- dependent. Hence, this is the condition of the interior of lays the gravitational echoes and makes the frequency dark gravastar. becomes bigger in every value of compactness. The unique solution in the exterior region with the Furthermore, the increase of c corresponds to the in- phantom field that approaches flat space-time as r creasing echo time or the decreasing echo frequency. Re- is a metric that is given in Eq. (14) and the phantom→ ∞ markably, the delay of the echoes also corresponds to the scalar field is given in Eq. (15). It occurs for the radius effective potential of the perturbed wave equation. We R < r. In Ref. [28], the thermodynamic analysis of have shown that DES enjoys a more profound (higher) gravastar is also investigated where the entropy in the potential well than CDS with the same C. However, we interior remains zero using the assumption of vanishing could not portray the potential well of DES in tortoise chemical potential. For our dark gravastar, it is obviously coordinate since this coordinate is a complex function seen that p + ρ = 0. However, there must be a dark of r and we could not find the inverse of r∗(r) analyti- charge contribution6 to the thermodynamic equation that cally. Hence, in gravitational echoes study, these features shall yield zero entropy in the interior region. However, obtained from DES compared to that of CDS might be we leave this analysis for future studies. probed in examining dark energy in compact astrophysical objects. We also have taken C = 1 on DES. This requirement VI. CONCLUSIONS is also the condition to obtain a simple gravastar from CDS. In this limit, it has been found that the interior A class of interior solutions of the Einstein field equa- space-time metric resembles a dS metric, although with tion with perfect fluid matter and phantom scalar field the power of c on the metric potentials as given in Eq. has been presented that we call as a dark energy star (43). Obviously, for c = 1, the dS space-time would (DES). DES has an anisotropic structure with negative be reproduced. The obtained space-time metric has no 13 singularity as expected for dS space-time. Unlike the or- The above equation the gives dinary gravastar from CDS, the dark gravastar pressures dφ 2 f 2c−1 are still anisotropic and coordinate-dependent in the in- = . (A2) terior. Therefore, the radial EoS is still dependent on the dt r2 sin2 θ coordinate r and parameters M,D, and R, even though the tangential EoS parameter is coordinate-independent. Then using the following geodesic equation, Since dark gravastar contains an anisotropic structure, d2γλ dγµ dγν we might not assume that there is a thin shell. The exte- +Γλ =0, (A3) dτ 2 µν dτ dτ rior space-time is a black hole with a phantom field with an asymptotically flat condition. This dark gravastar we can find possesses a distinct structure as gravastar constructed 2 dφ cf 2c−1f ′ from CDS. However, dark gravastar similarly possesses = − . (A4) dt r sin2 θ(crf ′ rf ′ 2f) no singularity in the space-time metric, pressures, energy − − density, and charge density. Then using Eqs. (A2) and (A4), one can get For future works, it will be interesting to examine the waveform of the gravitational echoes in DES using a nu- r = M(2c + 1). (A5) merical method or to use approximation from Pöschl- Teller potential as CDS since they have a similar feature This photon sphere radius reduces to the Schwarzschild of the effective potential. However, the obstacle to find one when D = 0 or on the other hand, c = 1. the tortoise coordinate for the phantom black hole should be resolved first. Moreover, the study of dark gravastar with a thin shell and its thermodynamics will also be Appendix B: Axial Perturbation possible. In the end, it will be fascinating also to study the gravitational echoes from dark gravastar obtained from DES. Since the future gravitational waves detectors In [46], it is given that for spherically symmetric met- will be available soon, examining ultra-compact DES and ric, dark gravastar will be possibly a new way to probe locally ds2 = ev(r)dt2 + eξ(r)dr2 + r2 dθ2 + sin2 θdφ2 , (B1) dark energy in compact astrophysical objects. − the single wave equation for a field Ψs,l(r∗,t) is given as follows 2 2 ACKNOWLEDGMENTS ∂ ∂ + V (r) Ψ (r∗,t)=0, (B2) ∂t2 − ∂r2 s,l s,l ∗ A. S. is partly supported by DRPM UI’s grants No: (ξ−v)/2 where the tortoise coordinate dr∗ = e dr has been NKB-1368/UN2.RST/HKP.05.00/2020 and No: NKB- used. The general effective radial potential of spin s is 1647/UN2.RST/HKP.05.00/2020. given as l(l + 1) 1 s2 V (r)= ev(r) + − [v′(r) ξ′(r)] s,l r2 2reξ(r) − Appendix A: Photon Sphere of Dark Energy Star 4π [P (r) ρ(r)] . (B3) − − The unstable light ring or photon sphere is an area of space where gravity is extreme in which photons are P (r),ρ(r) are the pressure and energy density of the per- forced to travel in orbits. For the Schwarzschild black fect fluid of the star, respectively. The azimuthal quan- hole, photon sphere is located at r = 3M or r = 3r /2 tum number shall satisfy l s where s is the spin of the s ≥ where r is the Schwarzschild radius. The exterior solu- perturbed field where s = 0, 1, 2 refer to the scalar, s ± ± tion of DES is not similar to the Schwarzschild solution. vector and tensor field, respectively. However, they both have spherical symmetry. The ex- For a gravitational perturbation on the Schwarzschild terior solution of DES is given by Eq. (14) while the black hole, one can use the ansatz Ψ2,l(r∗,t) = −iωst phantom field is given by Eq. (15). e ψ2,l(r) to obtain the following wave equation For a photon traveling at a constant radius r, dr = 0 2 d ψ2,l 2 and ds = 0. It is convenient also to rotate the coordinate 2 + [ωs V2,l(r)]ψ2,l =0. (B4) system such that θ is constant. For instance, we may dr∗ − choose θ = π/2, so dθ = 0. Setting these conditions to The effective potential for Schwarzschild black hole is the exterior metric (14), we find then given by 1 2M c 2 2 1−c 2 2 2m V BH = 1 [l(l + 1)r 6M]. (B5) f dt = r f sin θdφ , f =1 . (A1) 2,l r3 − r − − r 14

While for an interior solution (for example CDS), one ﬁnd l(l + 1) 6m(r) V Star = ev(r) 4π(P ρ) . (B6) 2,l r2 − r3 − −

[1] S. Perlmutter et al., Astrophys. J. 517, 565 (1999). [27] R. A. Konoplya, C. Posada, Z. Stuchl´ık, and A. Zhidenko, [2] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. Rev. D 100, 044027 (2019). Phys. D 15, 1753 (2006). [28] P. O. Mazur and E. Mottola, Proc. Nat. Acad. Sci., 111, [3] V.V. Kiselev, Class. Quantum Gravity 20, 1187 (2003). 9545 (2004). [4] S. Fernando, Gen. Relativ. Gravity 44, 1857 (2012). [29] P. O. Mazur and E. Mottola, Class. Quantum Grav. 32, [5] B. Toshmatov, Z. Stuchl´ık and B. Ahmedov, Eur. Phys. 215024 (2015). J. Plus 132, 98 (2017). [30] M. D. Danarianto and A. Sulaksono, Phys. Rev. D 100, [6] Z. Xu and J. Wang, Phys. Rev. D 95, 064015 (2017). 064042 (2019) [7] S. G. Ghosh, Eur. Phys. J. C 76, 222 (2016). [31] H. Hernández, L. A. Núñez and A. Vásquez-Ram´ırez, [8] M. F. A. R. Sakti, H. L. Prihadi, A. Suroso and F. P. Eur. Phys. J. C 78, 883 (2018). Zen, arXiv:1911.07569. [32] R. Chan, M. F. A. da Silva and J. F. V. da Rocha, Mod. [9] S. Chen, B. Wang and R. Su, Phys. Rev. D 77, 124011 Phys. Lett. A 24, 1137 (2009). (2008). [33] A. M. Setiawan and A. Sulaksono, Eur. Phys. J. C 79 [10] M. F. A. R. Sakti, A. Suroso and F. P. Zen, Eur. Phys. 755 (2019). J. Plus 134, 580 (2019). [34] H. Maulana and A. Sulaksono, Phys. Rev. D 100, 124014 [11] H. L. Prihadi, M. F. A. R. Sakti, G. Hikmawan and F. (2019). P. Zen, Int. J. Mod. Phys. D 29, 2050021 (2020). [35] P. Pani and V. Ferrari, Class. Quantum Grav. 35, [12] M. F. A. R. Sakti, A. Suroso and F. P. Zen, Ann. Phys. 15LT01 (2018). 413, 168062 (2020). [36] L. Mersini-Houghton and A. Kelleher, arXiv:0808.3419. [13] M. F. A. R. Sakti and F. P Zen, Phys. Dark Universe, [37] J. Enander and E. Mortsell, Phys. Lett. B 683, 7 (2010). 100778 (2021), In Press. [38] R. S. Conklin, B. Holdom and J. Ren, Phys. Rev. D 98, [14] P. Bhar and F. Rahaman, Eur. Phys. J. C 75, 41 (2015). 044021 (2018). [15] F. S. N. Lobo, Class. Quantum Grav. 23, 1525 (2006). [39] G. Darmois, Memorial des Sciences Mathematiques Stable dark energy stars XXV, Fasticule XXV. Paris, France: Gauthier-Villars; [16] P. Bhar, F. Rahaman, T. Manna and A. Banerjee, Eur. 1927. Chap. V. Phys. J. C 76, 708 (2016) . [40] W. Israel, Nuo. Cim. B, 44, 1 (1966), and erratum-ibid [17] D. Horvat and A. Marunović, Class. Quantum Grav. 30, 48, 463 (1967). 145006 (2013). [41] P. Bhar, F. Rahaman, S. Ray and V. Chatterjee, Eur. [18] O. Bertolami and J. Páramos, Phys. Rev. D 72, 123512 Phys. J. C 75, 190 (2015). (2005). [42] M. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988). [19] C. Vafa, arXiv:hep-th/0101218 [hep-th]. [43] M. Visser, Lorentzian Wormholes: From Einstein to [20] T. Okuda and T. Takayanagi, J. High Energy Phys. 03, Hawking (New York: American Institute of Physics) 062 (2006). [hep-th]. (1995). [21] G. Clement, J. C. Fabris, and M. E. Rodrigues, Phys. [44] S. Sushkov, Phys. Rev. D 71, 043520 (2005). Rev. D 79, 064021 (2009). [45] F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005). holes [46] A. Urbano and H. Veermäe, JCAP 04, 011 (2019). [22] R.R. Caldwell, Phys. Lett. B 545, 23 (2002). [47] M. Mannarelli and F. Tonelli, Phys. Rev. D 97, 123010 [23] Y-F. Cai, E. N. Saridakis, M. R.Setare, and J-Q Xia, (2018). Phys. Rep. 493, 1 (2010). [48] A. Testa and P. Pani, Phys. Rev. D 98, 044018 (2018). [24] F. Piazza and S. Tsujikawa, J. Cosmol. Astropart. Phys. [49] K. Lanczos, Ann. Phys. 379, 518 (1924). 07, 004 (2004). [50] C. Cattoen, T. Faber and M. Visser, Class. Quantum [25] S. S. Yazadjiev, Phys. Rev. D 83, 127501 (2011). Grav. 22, 4189 (2005). [26] S. Yazadjiev, Mod. Phys. Lett. A 20, 821 (2005).