Astrophys Space SciDOI 10.1007/s10509-011-0663-4

O R I G I NA L A RT I C L E

Anisotropic dark energy stars

Cristian R. Ghezzi

Received: 19 January 2011 / Accepted: 21 February 2011© Springer Science+Business Media B.V. 2011

Abstract A model of compact object coupled to inhomo-geneous anisotropic dark energy is studied. It is assumeda variable dark energy that suffers a phase transition at acritical density. The anisotropic �-Tolman-Oppenheimer-Volkoff equations are integrated to know the structure ofthese objects. The anisotropy is concentrated on a thin shellwhere the phase transition takes place, while the rest of thestar remains isotropic. The family of solutions obtained de-pends on the coupling parameter between the dark energyand the fermionic matter. The solutions share several fea-tures in common with the gravastar model. There is a criticalcoupling parameter that gives non-singular black hole solu-tions. The mass-radius relations are studied as well as theinternal structure of the compact objects. The hydrodynamicstability of the models is analyzed using a standard test fromthe mass-radius relation. For each permissible value of thecoupling parameter there is a maximum mass, so the exis-tence of black holes is unavoidable within this model.

Keywords Dark energy stars · Gravastars · Black holes ·Compact objects

1 Introduction

A star that consumed its nuclear fuel, if massive enough, willend its life as a compact object or a black hole. The blackholes described by the Einstein theory of gravity contain sin-gularities. However quantum effects must be taken into ac-count at high curvature values, or short distances comparedwith the Planck length scale (Novikov and Frolov 1989).

C.R. Ghezzi (�)Departamento de Física, Universidad Nacional del Sur, Av. Alem1253, 8000 Bahía Blanca, Buenos Aires, Argentinae-mail: ghezzi@uns.edu.ar

The quantum effects can render an inner portion of the blackhole as a de Sitter spacetime. This was considered by sev-eral researchers, see for example: Gliner (1966); Gliner andDymnikova (1975); Poisson and Israel (1988), and refer-ences therein. Recently, Nicolini (2009), and Nicolini et al.(2006), obtained regular black hole solutions with an innerde Sitter portion through a non-commutative geometric ap-proach to quantum gravity.

Analogous concepts were extended to a set of modelswhich replace the whole black hole region with a de Sit-ter spacetime (see Chapline et al. 2001 2003; Mazur andMottola 2001, 2004). In fact, Chapline et al. (2001), built amodel of dark energy stars based on the analogy between asuperfluid condensate near its critical point with the neigh-borhood of an event horizon. The dark energy stars have ade Sitter spacetime matched with a Schwarzschild exteriorspacetime. The surface of phase transition is closely locatedabove the Schwarzschild radius.

The model of Mazur and Mottola (2001) (MM) is con-ceptually different from the dark energy stars, but possessseveral common features inspired in it. Their model is called“gravitational vacuum stars” (gravastars). They consideredthat independently of the matter that composes the gravi-tating object, as some critical limit is reached, the interiorspacetime suffers a gravitational Bose-Einstein condensa-tion (Mazur and Mottola 2001, 2004). The general rela-tivity is not valid in the zone of coexisting phases, but itis valid macroscopically out of the transition zone (Chap-line et al., and Mazur and Mottola). The net effect of thecondensation is that the mean value of the vacuum stress-energy tensor changes from a nearly zero value to a non-zero value. The mean value of the vacuum stress-energy ten-sor has the form: 〈Tμν〉 = 〈ρvac〉gμν (see Weinberg 1989;Dymnikova 2000), which behaves like a cosmological con-stant term with � = 8πGc−2〈ρvac〉. This is equivalent to

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dark energy of the cosmological constant type, and behaveslike a fluid with an equation of state: P̂ = −ρ c2. The MMmodel is a static, spherical symmetric, five layer solutionof the Einstein equations. It was built cutting and pastingthree different exact solutions to the Einstein field equa-tions. It has an interior de Sitter spacetime, while the exte-rior is Schwarzschild. The intermediate zone is a thick shellof matter that satisfies all the energy conditions. The threesolutions are matched by two thin anisotropic layers withdistributions of surface tension and surface energy density,that violate the strong energy conditions. The solution doesnot possess an event horizon, but has a compactness that isclose to one (the black hole compactness).

Several researchers have analyzed the gravastar solutionsusing semi-analytic or analytic methods. For example, Cat-toën et al. (2005) proposed a thick shell anisotropic gravastarmodel which has the advantage of smoothing out the largepressure jump present in the original MM model. Dym-nikova (2000, 2002) found new analytic solutions of thegravastar type, G-lumps, and �-black holes. Related newsolutions of nonsingular black holes were also found byMbonye and Kazanas (2005). Their solutions have an in-terior de Sitter geometry containing matter. Chirenti andRezzolla (2007) found thick and thin shell gravastar solu-tions and studied their stability. Chan et al. (2009) studiedanisotropic phantom energy stars; while Lobo (2006) stud-ied gravastar solutions with a quintessence-like equation ofstate. Nicolini et al. (2006) obtained “mini-gravastar” andregular black hole solutions.

Due to the large compactness of the gravastars it could bedifficult to distinguish a gravastar from a black hole. Somearguments against the existence of gravastars, that could beverifiable through observations, were given by Broderickand Narayan (2007). On the other hand, positive detectionfeatures must include high energy particles emitted from thesurface of the dark energy stars, as pointed out by Barbi-eri and Chapline (2004). The model of dark energy stars, orgravastars, is an interesting model which could alleviate theblack holes singularities and it is worth to explore its char-acteristics.

A fluid of the cosmological constant type has necessar-ily a constant density in the absence of matter or otherfields (Dymnikova 2000). The reason is that for a sourceof the type Tμν = (8πG)−1�gμν , the conservation equa-tion T μν ;ν = 0, implies a constant �. Thus, in order toform a condensate the dark energy must be coupled tothe matter. On one hand, this can be achieved by a di-rect proportionality between the dark energy and matter.Another possibility is to consider a pure inhomogeneousanisotropic vacuum like term (see Mazur and Mottola 2001;Dymnikova 2000). In this work these two possibilities arecombined. Thus, a set of compact objects made of fermionicmatter coupled to inhomogeneous, anisotropic, dark energyis studied in this paper.

The obtained solutions form a one-parameter family,where the parameter is the proportionality constant betweenthe dark energy and matter. The solutions converge on amodel of the gravastar type as the parameter approaches afinite value. In addition, a new method is reported for the nu-meric integration of the anisotropic Tolman-Oppenheimer-Volkoff equations with cosmological constant (�-TOV)over the whole star.

In the gravastar and dark energy star models the shell islocated above the event horizon. The exact position of theshell can not be given as a boundary condition of the �-TOV equations (without relaxing some other boundary con-dition), because the coordinates of the Schwarzschild radiusare determined by the full solution. In this work, the positionof the thin shell is obtained self-consistently as a result of thenumeric integration. The mass-radius relation and structureof the stable models are estimated and compared with theTolman-Oppenheimer-Volkoff neutron stars.

The paper is organized as follows: in Sect. 2 the Ein-stein equations and the notation are introduced. The �-TOVanisotropic equations are derived. In Sect. 3 the equation ofstate for the matter and dark energy are given. In this sec-tion, a surface tension at the shell is obtained as function ofthe anisotropy. The junction and boundary conditions for the�-TOV equations are discussed. In Sect. 4, the numericalalgorithm is explained. In Sect. 5, the results are discussed.The paper ends in Sect. 6 with some final remarks.

2 Einstein equations

The Einstein equations are:

Rμν − 1

2gμνR = 8πG

c4T μν, (1)

where R denotes the scalar curvature, Rμν is the Ricci ten-sor, and T μν is the energy momentum tensor.

Assuming spherical symmetry the line element in stan-dard coordinates (Weinberg 1972) is:

ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ + sin2θdφ2). (2)

The energy-momentum tensor is composed of matter withmass-energy density δ and pressure P , plus dark energy withdensity ρde, radial pressure Pde(r), and tangential pressurePde(t):

T 00 = δc2 + ρde c2 (3)

T 11 = −(

P + Pde(r)

)(4)

T 22 = −(

P + Pde(t)

) = T 33 (5)

T 01 = T 1

0 = 0. (6)

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In terms of the (variable) cosmological constant the darkenergy density is: ρde = �c2/8πG. The energy density iswritten as the rest mass density plus the internal energy den-sity of the gas: δ = ρ (1 + ε/c2) (see Shapiro and Teukolsky1983).

The dark energy radial pressure is proportional to thedark energy density:

Pde(r) = −ρdec2. (7)

The variable dark energy density is assumed to be pro-portional to the mass density ρde = α ρ, where α is a non-negative constant (see Sect. 3). To sum up, the subindex“de” (dark energy) will not be used from now on: Pde(r) =Pr and Pde(t) = Pt , except for the dark energy density ρde.

The components of the Einstein equations are (see Wein-berg 1972; Ghezzi 2005):

T 00 : 8π(δ + ρde)/c

2 = e−λ

(λ′

r− 1

r2

)+ 1

r2(8)

T 11 : 8π(P + Pr)/c

4 = e−λ

(ν′

r+ 1

r2

)− 1

r2(9)

T 22 = T 3

3 :

8π(P + Pt)/c4

= e−λ

(ν′′

2− 1

4λ′ν′ + 1

4(ν′)2 + (ν′ − λ′)/2r

). (10)

An integral of (8) is:

e−λ = 1 − 2mG

rc2. (11)

The mass-energy up to the radius r ′ is:

m(r ′) = 4π

∫ r ′

0(δ + ρde) r2dr. (12)

In the special case of a constant dark energy density: m =m′ + 1

6 (�c2

G)r3 , where m′ is the integral (12) performed over

the mass-energy density of the matter alone.Subtracting (9) from (8) it is

e−λ

r

d(λ + ν)

dr= −8π

c2

(δ + P/c2). (13)

At the surface of the star the right side of the equation iszero, so λ + ν is independent of r . To get an asymptotic flatsolution it should be λ, ν → 0 (so λ + ν → 0) as r → ∞,thus:

λ = −ν for r ≥ rs, (14)

implies gtt = g−1rr for r ≥ rs , where rs is the surface radius.

It can be seen from (11) that m(r) → 0, as r → 0, in order to

get a regular metric at the center. So, an integration constantin (12) was set to zero. Equation (9) can be cast as:

1

2ν′ = 4π(P + Pr)r

3/c2 + mG/c4

r(r − 2mG/c2). (15)

2.1 Hydrostatic equilibrium structure equations

The hydrostatic equilibrium equation can be obtained fromthe Einstein equations:

P̂,r + a,r

a(δc2 + P) + 2

r(Pr − Pt ) = 0,

where the notation a = eν, is used, and P̂ = P + Pr is thetotal radial pressure. Rearranging the terms the anisotropic�-Tolman-Oppenheimer-Volkov (�-TOV) equation is ob-tained:

dP̂

dr= −(δc2 + P)

(mG

c2 + 4πG

c4 P̂ r3)

r(r − 2mG

c2

) + 2�P

r, (16)

where

�P = Pt − Pr,

is the anisotropic term. This equation and its solutions werefirst studied by Bowers and Liang (1974). Only for com-pleteness, it is possible to rewrite the equation above formatter and uncoupled dark energy:

dP̂

dr= 2

�P

r

− (δc2 + P)

(m′Gc2 + 4πG

c4 Pr3 − 8πG

3c2 ρder3)

r(r − 2m′G

c2 − 8πG

3c2 ρder3) ,

or as function of the cosmological constant � = 8πGρde/c2:

dP̂

dr= 2

�P

r

− (δc2 + P)

(m′Gc2 + 4πG

c4 Pr3 − 13�r3

)

r(r − 2m′G

c2 − 13�r3

) .

This equation reduces to the well known Tolman-Oppenhei-mer-Volkov (TOV) equations when �P = 0, and � = 0 (seeOppenheimer and Volkoff 1939). In this paper, (16) is nu-merically integrated.

3 Equation of state

The equations obtained above must be supplemented withequations of state for the matter and for the dark energy.

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3.1 Equation of state for the matter

A gas of neutrons at zero temperature has been consideredhere. This has led to a direct comparison of the results withthe Oppenheimer and Volkoff (1939) neutron star model.The energy per unit mass of the matter is given by (Shapiroand Teukolsky 1983):

δ c2 = m4nc

5

�3

1

8π2

[x

√1 + x2

(1 + 2x2)

− ln(x +

√1 + x2

)]. (17)

This gives the total energy of the gas, including the rest massenergy density ρ, measured in [erg cm−3]. The Fermi x pa-

rameter is: x = p/mnc, where p = ( 3h3

8πn)−1/3 is the mo-

mentum of the particles, and n is the number density of neu-trons. It can be written:

x =(

ρ

ρ0

)1/3

, (18)

with:

ρ0 = m4nc

3

3π2�3= 6.106 × 1015 [g cm−3]. (19)

The pressure is given by (Shapiro and Teukolsky 1983):

P = m4nc

5

�3

1

8π2

[x

√1 + x2

(2x2/3 − 1

)

+ ln(x +

√1 + x2

)], (20)

measured in [dyn cm−2]. In this work the matter is assumedisotropic.

3.2 Equation of State for the dark energy

The dark energy is assumed to be proportional to the restmass if its density is above a certain critical value ρc, i.e.:

ρde ={

α ρ for ρ ≥ ρc

0 for ρ < ρc,(21)

where α is a nonnegative proportionality constant (herecalled “coupling parameter”).

In this case the radial dark pressure is:

Pr ={

−ρde c2 for ρ ≥ ρc

0 for ρ < ρc.(22)

Observe that the equation of state of the matter plusthe dark energy is analogous to the MIT bag model for

hadrons (see Glendenning 2000, pp. 323–324). The dark en-ergy equation of state given above, corresponds to the bagconstant contribution of the MIT bag model. The bag con-stant is taken as density dependent in this work. The MITbag model is used in astrophysical models of hybrid starswith a deconfined phase of quarks at the core of the star. Forisospin symmetric nuclear matter the quark-hadron phasetransition must occur at a critical mass density of about twoto five times the saturation nuclear density, although its cer-tain value is unknown. But recently was considered that forisospin asymmetric nuclear matter and temperatures of sev-eral MeV the onset of the transition to quark matter couldhappen already around saturation nuclear density, or evensmaller densities, depending on the bag constant. This donot contradict accelerator physics, mainly because dynami-cal timescales in collisions are very short (10−23 s), com-pared to weak time processes (10−6–10−8 s), and muchshorter than dynamical timescales in explosive events likesupernova explosions, which are of the order of ms. Sostrangeness can be produced and maintained in weak equi-librium in astrophysical environments (Sagert et al. 2009).This means that it is possible to consider a sub-saturationcritical density for the deconfinement transition in a com-pact object.

The model studied here can be thought as an extrapola-tion of an hybrid star model, where the bag constant takesarbitrarily large values. In the spirit of the original gravas-tar model it is assumed that the vacuum phase transition isinduced (or enhanced) by a gravitational field condensationeffect. As will be discussed below, this lead to solutions thatinterpolate between normal neutron stars and gravastars.

According to this considerations, it is set the critical den-sity: ρc = 2×1014 g cm−3. The dependence of the solutionson ρc will not be discussed here, for simplicity, because thesolutions are qualitatively similar but with lower maximummass (if ρc is larger).

The transition in the vacuum energy density is assumedas enhanced by the gravitational Bose-Einstein condensation(Chapline et al. 2003). As the negative pressure of the darkenergy helps to maintain a larger mass compact object, thedark energy density will be assumed without upper boundin order to get the maximum masses allowed by the presentmodel.

The tangential dark energy pressure and the radial darkenergy pressure differ only at the shell of phase transition:Pr �= Pt , at ρ = ρc. Thus, the dark energy is in generalanisotropic at the shell of phase transition. The tangentialpressure Pt is determined by the field equations.

3.2.1 Anisotropy and surface tension

There is a surface tension at the interface of phase transi-tion. This is expected, in general, at the interface between

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two substances. In the absence of matter, the equations forpure anisotropic, inhomogeneous, dark energy (Dymnikova2000) are obtained from (16) and (22):

Pr = −ρdec2,

Pt = Pr + r

2

dPr

dr.

(23)

A surface tension can be defined as:

σ = (�P ) δr, (24)

measured in dyn/cm. Here δr is the width of the shell, whichis expected to be of Planckian thickness lP ∼ 10−33 cm.With these definitions, (23) can be written as:

[Pr ] = 2σ/r, (25)

where [Pr ] = P +r − P −

r , is the jump in the dark radial pres-sure, P ±

r is the radial pressure evaluated above/below theshell, respectively, and 2σ/r is evaluated at the shell’s ra-dius r. Equation (25) is analogous to a Young-Laplace equa-tion for the spherical interface between two substances (Lan-dau and Lifshitz 1959). The anisotropy (surface tension)compensates the large dark energy pressure gradient. From(25) above, it follows that a positive surface tension im-plies a larger radial dark pressure from above the thin shell.1

There is a net radial pressure compressing the shell. In thiscase, the tangential dark pressure is larger than the radial oneon the shell (see (24)). On the contrary, a negative surfacetension, gives a net radial dark pressure directed outwardly,and a tangential dark pressure lower than the radial one. Themodel analyzed in this paper has positive surface tension.

The surface tension is not directly calculated (and it isnot necessary) in the algorithm, but it could be found withan indirect calculation. It can be observed that it remainsfinite. This is inferred a posteriori from the numeric re-sults, using the Israel-Lanczos (Israel 1966) thin shell junc-tion conditions. As observed by Mazur and Mottola (2001),the coefficient gtt of the metric is continuous, but its firstderivative and the metric coefficient grr are in general dis-continuous, due to the discontinuity in the equation of state.The outwardly directed normal vector to the interface isn = (

√grr )

−1∂r , and the extrinsic curvature is defined asKb

a = ∇anb , where a, b, are indices of the hypersurface co-

ordinate system. The discontinuities in the extrinsic curva-ture as a function of the surface energy density η and the

1In fluid dynamics (Landau and Lifshitz 1959) (25) is commonly writ-ten −[Pr ] = 2σ ′/r , with σ ′ = −σ . Thus, what is called a positivesurface tension σ here is a negative surface tension in the textbooks,and vice versa.

surface tension σ are (Mazur and Mottola 2004):

[Ktt ] =

[(√

grr )−1

2gtt

dgtt

dr

]

= 4πG

c4(η + 2σ),

[Kθθ ] = [Kφ

φ ] =[

(√

grr )−1

r

]

= −4πG

c4η.

(26)

Recall that σ , as defined in this paper, has contrary sign toits usual definition. The first condition given by (26) is re-lated to (25). An assumed jump in the radial pressure gives asurface tension through (25), while (26) gives a jump in thefirst derivative of gtt (there is no jump in grr , since it is as-sumed that η = 0). Conversely a jump in the first derivativeof gtt is proportional to a surface tension, and this could becalculated after obtaining the numeric solution (see the nextsection). Equation (25) is valid at the shell even when theisotropic fermion matter is included. In this case, the stress-energy tensor as a distribution valued tensor is,

Tαβ = �(l)T +αβ + �(−l)T −

αβ + δ(l)Sαβ, (27)

where the symbols have the usual meaning, and l denotes theproper distance along geodesics from the hypersurface (seePoisson 2004, for definitions). The surface stress-energytensor is:

Sab = ηuaub − σ (hab + uaub), (28)

in the present case is enough to set η = 0, and to choose σ

to cancel the term −c4 ([Kab] − [K]hab) /8π (the “left handside” of Einstein equations). Its components were given in(26). This is the standard procedure for dealing with thinshells in general relativity (Poisson 2004).

A word of caution is worth mentioning. It is possible toderive an expression, from (23) and (22), for the tangentialdark energy pressure at the shell, analogous to that obtainedby Poisson and Israel (1988):

Pt = −ρde c2 θ(a − r) + 1

2ρde c2 a δ(r − a), (29)

where a is the radius of the shell. They pointed out thatthis expression becomes singular at the event horizon. Infact, as they showed, a change in variables gives δ(r − a) =[grr (a)]1/2 δ(s), in terms of the proper distance s from theshell. So, if r = a is light-like the pressure becomes singu-lar, “even when considered as a distribution” (Poisson andIsrael 1988). This means that the thin shell of phase transi-tion cannot be located exactly at the event horizon, but canbe located elsewhere. Although this conclusion was origi-nally obtained by a precursor idea of the dark energy stel-lar models, it is, of course, applicable to the models studiedhere.

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4 Numeric algorithm

The discontinuity in the dark pressure is managed as fol-lows: the radial derivative of the total pressure in (16) is splitin two terms,

dP̂

dr= lim

δr→0

(δPr

δr+ δP

δr

), (30)

taking the limit with δr tending to zero. The difficulty to in-tegrate the �-TOV equations over the whole star is that thefirst term on the right side of (30) diverges at the shell, i.e.:limδr→0

δPr

δr|r=a → ∞. This term is associated with the sec-

ond term on the right side of (29). The solution, proposedhere, is to cancel that term with an appropriate amount ofanisotropy on the right side of (16). As explained, this isequivalent to set a finite surface tension at the shell (see(24)).

Now the set of equations to be integrated are (11), (12),and (16) (without the discontinuity), complemented with theequations of state described above. The boundary conditionfor (16) is P̂ = 0 at the surface of the star, and the initialcondition, a prescribed mass density at the center.

Briefly, the algorithm works as follows: first, it integratesthe equation

dP

dr= −(δc2 + P)

(mG

c2 + 4πG

c4 P̂ r3)

r(r − 2mG

c2

) , (31)

from the center of the star, with a given central mass den-sity. Next, it updates the values of P̂ , P , ρ, δ, ρde, and m.If the mass density is above zero, it goes a radial step furtherand repeat all. If it is zero the code ends. Of course, in thepractice δr is never zero, but equal to the radial step of theintegrator with δr � a. It has been checked that the resultsare robust respect to variations in δr .

Note that (31) is different from (16). The term dPr/dr

was canceled by a surface tension σ so that δPr = 2σ/r .Equations (11), (12), and (16) are integrated with a

Runge-Kutta algorithm of the fourth order. The employedcode HE05v2 is an update of the previous code HE05v1 (seeGhezzi 2005, for more details).

4.1 Caveats

Although the dark energy pressure derivative was canceled,it is not granted that the solutions will be smooth. The solv-ing procedure described above eliminates the delta Diracdistribution from (16), but smoothness was not imposed any-where in the remaining equations. As it will be shown, thereare solutions which are not smooth at the transition shell. Itis possible to integrate the non-smooth solutions performingthe calculation at zero order precision on the shell, and atfourth order in the rest of the star.

Fig. 1 Pressure profile for each star. The pressure tends to a constantnegative value inside the core as the critical parameter (αc = 0.5533)is approached. The pressure is measured in [g cm−3] and normalizedby the factor 1013 c2

Fig. 2 Mass density profile for each dark energy star. The mass densityis not smooth at the phase transition shell. The density is measured in[g cm−3] and normalized by the factor 1013

Another caveat is that the denominator of (16) is zeroat the event horizon, and the equation becomes singular.A larger numeric resolution is needed as the compactnessapproaches one. This problem is inherent to the field equa-tions, and of course, it is independent of the position of thetransition shell. However, it also includes the issue pointedout above: the transition shell cannot be located at the eventhorizon.

5 Discussions

A family of solutions is obtained, depending on α, for eachstellar central density. In order to analyze the structure of thestars, a model with central density: ρ = 2.86 × 1014 g cm−3,is singled out. The results of the calculations are shown inFigs. 1–4. Figures 1, 2, and 3, show the stellar structure ofthe set of dark energy stars. These stars are in the stable

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Table 1 Summary of solutions

Type w α Matter EOS Layers Anisotropy Converges to

a ≤ 1 ≥ αc (α ≤ α′c) neutrons 3 Yes Black hole

b > 1 ≥ αc (α > α′c) neutrons 3 Yes Black hole

c 1 = α′c (α′

c < αc) neutrons 3 Yes Gravastar, cosmological const. as r → 0

d ≤ 1 < min(α′c, αc) neutrons 3 Yes Gravastar

e > 1 α′c < α < αc neutrons 3 Yes Gravastar

f – 0 neutrons 0 No TOV star (Oppenheimer and Volkoff 1939)

– – – P = ρ/3 0 Yes Anisotropic star (Bowers and Liang 1974)

– 1 – P = const. 3 No DE star (Chapline et al. 2001)

– 1 – P = const. 5 Yes Gravastar (Mazur and Mottola 2001)

Fig. 3 Coefficient g−1rr = 1−2m(r)G/r c2, for dark energy stars with

different values of the parameter α. At the surface of the star g−1rr = gtt

Fig. 4 The compactness function: rsch/r , is plotted for several darkenergy stars. As the parameter α is increased, the compactness at thesurface tends to one, and the model approaches a regular black hole

branch of the solutions (see below). The different solutionswere obtained varying the coupling parameter from α = 0 toα = 0.5533.

The summary of possible solutions is given in Table 1,presented together with some well known solutions. Thetype (f) solutions are obtained setting α = 0, and corre-spond to the typical Tolman-Oppenheimer-Volkoff neutronstar model.

Black hole solutions are obtained with α ≥ αc = 0.5533.The critical value αc is obtained by interpolation and can-not be calculated exactly,2 as was explained above (seeSects. 3.2.1 and 4). The black hole solutions are charac-terized by a surface radius smaller than or equal to theSchwarzschild radius, or a larger or equal to one compact-ness. These are indicated as type (a) and (b) solutions inTable 1. The critical αc parameter depends on the particu-lar family of solutions considered, i.e.: on the central massdensity.

Figure 1 shows the pressure profile for each dark energystar of the family. The pressure tends to a negative constantinside the dark energy star as the critical parameter is ap-proached. The pressure is positive and decreasing with ra-dius outside the thin shell. Figure 2 shows the mass den-sity for each dark energy star. The density tends to a con-stant value, inside the shell, as the critical parameter is ap-proached (α → αc). The density profiles are non smooth forthis family, because there is a surface tension at the thin shellof phase transition (see (26)). This is expected in general,since the matter density need not be smooth across the shell.

In general, the total pressure (effective equation of state)can be written as:

P̂ = −w ρ c2, (32)

with

w =(

α − P

ρc2

). (33)

2In fact the interpolated value is αc = 0.553317, with a precision of theorder of 10 cm in the radial step.

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The value of w is bounded from above by α, and it dependson (α,ρ). For the analyzed model w = 0.5287 for α = αc .Thus, the total pressure is P̂ > −ρ c2, for the considered setof dark energy stars. Every family of solutions contains atleast a black hole, i.e.: the critical parameter αc is finite forall families of solutions. This was obtained by inspection ofthe numerical solutions.

For a given value of the central density (for a given familyof solutions), there exists an α = α′

c, so that the effectiveequation of state is of the cosmological constant type at thecenter of the star, i.e.: P̂ = −ρ c2 (do not confuse α′

c withαc). If αc > α′

c the sequence of solutions, as function of α,converge on a dark energy star of this kind (type (c)) beforeconverging on a black hole. On the contrary, if αc ≤ α′

c, thesequence converges first on a black hole.

The critical parameter α′c can be obtained exactly from

the neutron gas equation of state. For example: the Fermiparameter is x = (ρ/ρ0)

1/3 = 0.3604 for the solutions con-sidered here (with central density ρ = 2.86 × 1014 g cm−3).With w = 1, using (33) and (20), is obtained:

α′c = 1+ 3

8

[x

√1 + x2

(2x2

3−1

)+ ln

(x +

√1 + x2

)]/x3,

(34)

that gives α′c = 1.0248. The same calculation can be done

for any other central density. The value of αc must beobtained numerically, i.e.: it cannot be obtained withoutknowledge of the complete sequence of solutions. In thepresent case αc < α′

c so, the solutions converge on a blackhole before reaching a solution with an effective cosmolog-ical constant equation of state, at the center of the star. Theparameter space was swept looking for another family ofsolutions that contains a type (c) gravastar. This is not a triv-ial task, because w must be calculated for α = αc, while αc

vary between the different families. So, a complete set ofsimulations had to be performed for each central density. Itwas found that w(αc) increases with the central density ofthe models. In particular, a gravastar model of type (c) wasfound with a central density of ρ = 4.92 × 1016 g cm−3. Inthis case, the critical parameter for the type (c) gravastar isα′

c = 1.416, while the critical parameter for a black hole so-lution is αc = 1.5192. The compactness of the gravastar is0.9079. The structure of this star is shown in Fig. 5. Un-fortunately, this model is in the unstable branch of the so-lutions. Of course, as explained above, it is possible to findsolutions—stable or unstable—with larger compactness, butwith w �= 1.

There is another type of solutions which are closer tothe MM gravastar model: the solutions with α → αc , whichhave a density profile nearly constant in a large portion ofthe stellar interior (see Figs. 1 and 2). The models withα → αc—but not reaching αc—have a large compactness

Fig. 5 Detail of the pressure and density profile with α = α′c and with

central density ρ = 4.92 × 1016 g cm−3. In this case the magnitudeof the total radial pressure at the center is: P̂ = −ρ c2. The physicalunits of the density and pressure (×c2) are: [1e16] gr cm−3, the scaleis linear

and are not contained in a black hole. In order to have an in-ner de Sitter metric, the matter density must be exactly zeroat the core; this can be checked from (11), (12), and (13).However, the numerical limit of the models as ρ, ρc → 0do not converge to the MM model: the density and pressureprofiles keep their shape roughly, but the radius of the darkenergy stars increase continuously as the density is lowered.It means that the radius of the stars tends to infinity as thecentral density tends to zero. The limit spacetime in this caseis Minkowski (vacuum). But there is another way around toobtain an MM gravastar, with an inner de Sitter metric, asthe limit of the numerical model: it is to change the order ofthe limits. First kept fixed ρde = constant in the obtained so-lution for α → αc . This is equivalent to leaving the dark en-ergy and shell radius frozen. After that, take the limit ρ → 0,to obtain an inner de Sitter spacetime. In the process the in-ner mass will be overestimated, so the result is a perturbedMM gravastar on the stable branch. These solutions werenot included in Table 1, but can be derived from type (d) or(e) solutions. It is not possible to know how this gravastarwill evolve with the present methods, so this issue will notbe further discussed here.

5.1 Energy conditions

The dark energy is defined as a fluid that violates the strongenergy condition (SEC). A distinction of all possible kindof fluids interior to a gravastar were classified in Chan et al.(2009), according to where the fluid violates the SEC condi-tion, or some of the null energy conditions (NEC).

Respect to an orthonormal basis, the SEC condition atthe shell reads: ηc2 − 2σ ≥ 0 and the NEC: ηc2 − σ ≥ 0.

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In the present case, as η = 0, and σ ≥ 0, the SEC and NECconditions are violated at the shell. So, the anisotropic thinshell is made of “repulsive phantom energy”, according tothe Chan et al. (2009) classification.

On the other hand, the possibilities for the inner fluid aremore exotic. Locally the SEC condition for the inner fluidis: (δ + ρde)c

2 + 3P̂ ≥ 0 (recall that the fluid is anisotropiconly at the shell), while if the NEC condition is satisfied:(δ + ρde)c

2 + P̂ ≥ 0. All the solutions found satisfies theweak energy condition (WEC) and the NEC.

The solution (b) (or (e)) is a black hole (or a gravas-tar) with dark energy in its interior. The dark energy vio-lates the SEC. Thus, the formation of an inner singularityis avoided, because SEC violation produce a repulsive effecton geodesics (Raychaudhuri theorem). The solutions of type(a) (or (c)–(d)) are black holes (or gravastars) divided in twosubsets according to the energy conditions. There is a subsetwhich do not violate the SEC: this is an attractive normalfluid; while the second subset has a repulsive dark energycore.

5.2 Compactness

The compactness is defined as the quotient between theSchwarzschild radius and the radius of a sphere. It is a func-tion of the radius and the parameter α. Figure 4 shows thecompactness function for the considered family. It can beseen that the compactness of the star surface tends to one asα → αc. If α ≥ αc the compactness is larger or equal to one,and the shell is contained within a black hole. The solutionswith α ≥ αc are black holes, indicated as type (a) or (b) inTable 1.

5.3 Radial stability

Figure 6 shows the mass-radius relations for dark energystars, with a coupling parameter that varies from α = 0 toα = 0.53. There are two maxima in the mass-radius curve.The first one is at the density of phase transition, where thecurve is non smooth. The second maximum is due to thematter equation of state, like in the usual TOV neutron stars.There is a valley in the curves because at low masses the re-pulsive effect of the dark energy is more important than itsweight. However, at larger masses the weight overcomes thenegative pressure effect and the masses of the dark energystars become larger than fiducial TOV stars with the sameradius.

The dark energy star of the family is initially (at α = 0)on the stable region near the first maximum, but its positionon the mass-radius curve is displaced to the left, to lowerradius, as α increases.

The criteria dM/dρ > 1 (dM/dρ < 1) is a necessarycondition for hydrodynamic stability (instability), respec-tively, but it is not a sufficient condition (Glendenning 2000).

Fig. 6 Mass radius relation for a set of solutions. The curve with α = 0corresponds to normal TOV neutron stars

However, it is possible to apply a Wheeler’s M(R) anal-ysis to the mass-radius relation. The method was devisedby Wheeler, and the precise conditions under which it isvalid, were given by Thorne (see Bardeen et al. 1966;Thorne 1965; Meltzer and Thorne 1966). It is useful, in thepresent case, to obtain the number of unstable radial modesfor each dark energy star in hydrostatic equilibrium with-out calculating the eigenfunctions and eigenfrequencies ofthe radial perturbations. The condition for the applicabilityof the method is satisfied: the microscopic equation of staterepresents cold catalyzed matter, i.e.: no changes in nuclearcomposition are considered.3 The method is applied as de-scribed in Meltzer and Thorne (1966): the graph of the massversus radius, is parameterized by the central density of theconfiguration (for a particular α value). As we move in thedirection of increasing density all configurations remain sta-ble until the first maximum value of M is reached. At thefirst extremal point, the fundamental mode of radial oscil-lation becomes unstable. At the second extremal point, an-other mode of radial oscillation changes stability. The direc-tion of stability change depends on the shape of the M(R)

curve at that extremal point. If the curve bends clockwise,with increasing density, then one previous unstable modebecomes stable. On the contrary, if the curve bends counter-clockwise, then one previous stable mode becomes unstable.The same criterion is applied for each succeeding extremalpoint.4

3For a more realistic equation of state the Wheeler M(R) analysis isapplicable if the adiabatic index satisfies a certain criteria (Bardeen etal. 1966).4This description of the Wheeler analysis is based on the paper ofBardeen et al. (1966). Another clear exposition is given in Thorne(1965).

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It is generally assumed that a cold star can be dynami-cally unstable against nonradial perturbations only if its low-est radial mode is also unstable (see Thorne 1965). Thus theradial stability analysis reveals the absolute stability and in-stability of the dark energy stars. According to this analysis,the stability and instability regions for α = 0.53 are indi-cated in the Fig. 6, with the letters “s” and “u” respectively.There appears an unstable band between two stable ones,starting at the density of phase transition and ending at thebottom of the valley. This is analogous to the unstable bandbetween the white dwarfs and the neutron stars (see Bardeenet al. 1966).

6 Final remarks

In this work, the �-TOV anisotropic equations were solvedfor compact objects composed by a Fermi gas coupled todark energy. The dark energy traces the matter distributionat high energy density, with a strength depending on the pa-rameter α. The main difference is that, in the original darkenergy stellar model, the core of the star is de Sitter andthe shell is located close, or replacing, the event horizon. Inthe solutions found in this paper, the position of the shell iscalculated from the model and not imposed as a boundarycondition. Moreover, unlike other gravastar models, the DEstars considered here contain matter in its core (an exceptionis the model of Mbonye and Kazanas 2005).

A simple numerical method was devised to integrate thestellar structure equations from the center of the star to itssurface, including the shell. The compactness of the stars,their structure and mass-radius relations were calculated forcomparison with the well known theoretical TOV model.

It is found that the dark energy plus anisotropic stresshelps to stabilize a neutron star, but there is yet a maximummass for stable DE stars.

It has been shown that the effective equation of state is:P̂ = −w ρ c2, with w = 1, w < 1, or w > 1, depending onthe density and the assumed value of α. There is a criticalparameter α′

c so that the total pressure, at the center of thestar, is of the cosmological constant type P̂ = −ρ c2.

There exists, also, a critical parameter αc which separatesdark energy stars from regular black hole solutions.

The mass-radius curves for the dark energy stars have amaximum mass, signaling the unavoidability of black holeformation for large enough mass. It is unlikely that someother EOS for the matter will avoid the BH formation. Theonly possibility left is a different interaction law betweendark energy and matter, for which we have no clues atpresent. This issue must be further studied.

One of the characteristic features of the mass-radius rela-tion for the dark energy stars is that there appears an insta-

bility gap between the two stable bands, which do not existfor TOV neutron stars. It seems probable that these unstablestars migrate to the stable branch at larger radius. The de-tection of this gap could imply a large interaction betweendark energy and matter, and could be a clear signature of thevalidity of the model. However, its negative detection doesnot rule out the existence of dark energy stars. In this case,their identification through the mass-radius curve could bemore difficult.

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