Anisotropic Dark Matter Stars
AAnisotropic Dark Matter Stars
P.H.R.S. Moraes, ∗ G. Panotopoulos, † and I. Lopes ‡ Universidade de S˜ao Paulo (USP), Instituto de Astronomia,Geof´ısica e Ciˆencias Atmosf´ericas (IAG), Rua do Mat˜ao 1226,Cidade Universit´aria, 05508-090 S˜ao Paulo, SP, Brazil Centro de Astrof´ısica e Gravita¸c˜ao-CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico-IST,Universidade de Lisboa-UL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
The properties of exotic stars are investigated. In particular, we study objects made entirely ofdark matter and we take into account intrinsic anisotropies which have been ignored so far. Weobtain exact analytical solutions to the structure equations and we we show that those solutions i)are well behaved within General Relativity and ii) are capable of describing realistic astrophysicalconfigurations.
PACS numbers:
I. INTRODUCTION
Dark matter (DM) has certainly been one of the great-est mysteries of Physics. An important evidence of its ex-istence came from the analysis of rotation curves of spiralgalaxies by V. Rubin and collaborators in the 70’s of thelast century [1–3]. DM is thought to be a kind of matterthat does not interact electromagnetically and thereforecannot be seen, which is why it is called dark . However,it interacts gravitationally. In the case of spiral galaxies,it causes their rotation curves to be significantly higherthan one would expect by measuring only the gravita-tional field of luminous matter.It also has a fundamental role in the formation of galax-ies and large-scale structures in the universe [4–7]. Actu-ally it is believed that when baryonic matter decoupledfrom radiation at redshift z ∼ ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] and an example of them would be neutron stars. Theycan also bend light, causing gravitational microlensingeffects, that have been detected for some time [21–25].It should also be quoted that DM gravitational ef-fects could be understood as purely geometrical effects ofextended gravity theories [26]. Rotation curves [27–29]and even structure formation [30–34] have been explainedthrough the extended gravity channel.Here, in the present article, based on some of the sev-eral studies that empirically prove DM existence [35], weare going to stick to the standard approach, consideringDM exists and is non-baryonic.The Bose-Einstein condensate is a possibility in theDM particle scenario [36–40], and it was recently shownthat could exist in space by the Cold Atom Laboratoryorbiting Earth on board the International Space Station[41].The weakly interacting massive particles (so-calledWIMPs) [42–44] are among the best motivated DM par-ticle candidates. WIMPS interact through a feeble newforce and gravity as predicted by supersymmetry amongother theories [45, 46]. If they were in thermal equi-librium in the early universe they annihilated with oneanother so that a predictable number of them remainstoday [47].There may exist DM stars (DMSs) [48, 49] poweredby WIMP DM annihilation [50, 51]. In regions of highDM density, such as the Galactic center, the capture andannihilation of WIMP DM by stars has the potential tosignificantly alter their evolution [52–55]. In Reference[56] it was shown that WIMPs accreted onto neutronstars may provide a mechanism to seed strangelets incompact objects for WIMP masses above a few GeV. Thiseffect may trigger a conversion of most of the star into astrange star. Recall that neutron stars are pulsars, high-density stars with large rotation frequency rates locatedin the core of supernovae remnants [57, 58]. Some modelspredicted that strange stars could form inside these starsdue to the brake neutrons into their constituent quarks[59]. Due to a matter of stability, a portion ( ∼ / a r X i v : . [ g r- q c ] J a n of these quarks is converted to strange quarks and theresulting matter is known as strange quark matter.Neutron stars are expected to efficiently captureWIMPs due to their strong gravitational field. The anni-hilation of DM in the center of these stars could lead todetectable effects on their surface temperature, speciallyif they are in the center of our Galaxy [60].In [61], Kurita and Nakano investigated the collapse ofclusters of WIMPs in the core of Sun-like stars and theconsequent possible formation of mini-black holes, whichwould generate gravitational wave emission.The aforementioned Bose-Einstein condensate has alsobeen considered as the DM modeling for stars. On thisregard, one can consult e.g. [62–69]. In particular,in [68, 69], DMSs were investigated in the Starobinskymodel of gravity [70]. It has been shown in [69] thatDMSs have smaller radius and are slightly more massivein Starobinsky gravity.In the present article we will assume a boson star asour model for DMS. A wide variety of boson stars havebeen proposed and investigated in the literature [71–75](for some recent references on this subject, one can check[76–79]). Our DMS will be modeled from the equationof state (EoS) proposed in [75] (check also [80]). It isinteresting to mention that some proposals for detectingboson stars were reported in [81–86].The environment inside DMSs is expected to be ex-tremely dense, specially when neutron star-like objectsare under consideration. Under such conditions of ex-treme density, anisotropy is expected to appear [87–90].Anisotropy in neutron stars has been investigated inthe literature. The hydrostatic features were firstly ap-proached in [91], where it was shown that deviations fromisotropy would entail changes in the star maximum mass.This approach was extended in [92] to also cover theproblem of stability under radial and non-radial pulsa-tions. The effects of anisotropy on slowly rotating neu-tron stars was studied in [93]. In [94], anisotropic neutronstars were also considered in the framework of Starobin-sky gravity. Further studies of anisotropic neutron starscan be seen in References [95–98].To the best knowledge of the present authors,anisotropy has not yet been considered in DMSs. Suchan investigation is the main goal of the present article.The plan of our work is the following: in the next sectionwe briefly summarize the structure equations describinghydrostatic equilibrium of anisotropic stars. In SectionIII we present the exact analytical solution and we showthat it is well behaved and realistic within General Rel-ativity. Finally, we finish our work in Section 4 with theconcluding remarks. II. RELATIVISTIC STARS WITHANISOTROPIC MATTER
Within General Relativity the starting point is Ein-stein’s field equations G µν = R µν − R g µν = 8 πT µν . (1)In (1), G µν is the Einstein tensor, R µν is the Ricci tensor, R is the Ricci scalar, g µν is the metric tensor, we setNewton’s constant G and the speed of light, c , to 1, whilefor anisotropic matter the stress-energy tensor, T µν , hasthe form T µν = Diag( − ρ, p r , p t , p t ) , (2)with ρ being the energy density, p r the radial pressureand p t the tangential pressure.In order to find interior solutions describing hydro-static equilibrium of relativistic stars, we integrate thestructure equations including the presence of a non-vanishing anisotropic factor [99, 100]: m (cid:48) ( r ) = 4 πr ρ ( r ) , (3) ν (cid:48) ( r ) = 2 m ( r ) + 4 πr p r ( r ) r [1 − m ( r ) /r ] , (4) p (cid:48) r ( r ) = − [ ρ ( r ) + p r ( r )] m ( r ) + 4 πr p ( r ) r [1 − m ( r ) /r ] + 2∆ r , (5)where m ( r ) and ν ( r ) are the components of the metrictensor assuming static, spherically symmetric solutionsin Schwarzschild-like coordinates, ( t, r, θ, φ ), ds = − e ν dt + 11 − m ( r ) /r dr + r ( dθ + sin θ d φ ) , (6)and ∆ ≡ p t − p r is the anisotropic factor. All quanti-ties depend on the radial coordinate r only, and a primedenotes differentiation with respect to r . Clearly, setting∆ = 0 we recover the usual Tolman-Oppenheimer-Volkoffequations [101, 102] for isotropic matter.Moreover we impose at the center of the star, r = 0,the following initial conditions m (0) = 0 , (7) p (0) = p c , (8)with p c being the central pressure. Upon matching withthe exterior vacuum solution ( T µν = 0, Schwarzschildgeometry) at the surface of the star, r = R , the followingboundary conditions must be satisfied p ( R ) = 0 , (9) m ( R ) = M, (10) e ν ( R ) = 1 − MR , (11)with R being the radius of the star, and M being itsmass. III. ANISOTROPIC DARK MATTER STARS:EXACT ANALYTICAL SOLUTION
Boson stars are self-gravitating clumps made of eitherspin-zero fields, called scalar boson stars [72] or vectorbosons, called Proca stars [103, 104]. The maximummass for scalar boson stars in non-interacting systemswas found in [105, 106], while in [75, 107] it was pointedout that self-interactions can cause significant changes.A complex scalar field, Φ, minimally coupled to gravityis described by the Einstein-Klein-Gordon action [108] S = (cid:90) d x √− g (cid:18) R π + L M (cid:19) (12) L M = − g µν ∂ µ Φ ∂ ν Φ ∗ − V ( | Φ | ) (13)where g is the metric determinant, L M is the matterlagrangian and V is the self-interaction scalar potential.For static spherically symmetric solutions we make forthe scalar field the ansatz [108]Φ( r, t ) = φ ( r ) exp ( − iωt ) , (14)where the oscillation frequency ω is a real parameter.Although the scalar field itself depends on time, itsstress-energy tensor is time independent and the Ein-stein’s field equations take the usual form for a fluid,for which the energy density is computed to be [109, 110] ρ = ω e − ν φ + e − λ φ (cid:48) + V ( φ ) , (15)while the radial and tangential pressures are found to be[109, 110] p r = ω e − ν φ + e − λ φ (cid:48) − V ( φ ) , (16) p t = ω e − ν φ − e − λ φ (cid:48) − V ( φ ) . (17)Clearly, a boson star is anisotropic since the two pres-sures are different. Under certain conditions, however,the anisotropy may be ignored and the system can betreated as an isotropic object. A concrete model of theform V ( | Φ | ) = m | Φ | + λ | Φ | , (18)with m being the mass of the scalar field and λ being theself-interaction coupling constant, was studied e.g. in[80], in which the authors considered the following EoS[75]: p r = ρ (cid:18)(cid:114) ρρ − (cid:19) , (19)where ρ is a constant given by ρ = m λ . (20)This EoS describes the boson stars that are approxi-mately isotropic provided that the condition λ π (cid:29) m (21) holds [80].In the two extreme limits we recover the well-knownresults p r ≈ ρ ρ , ρ (cid:28) ρ , (22)for diluted stars [39], and p r ≈ ρ , ρ (cid:29) ρ (23)in the ultra relativistic limit.In the first extreme limit, any model, irrespectively ofthe form of the potential, will be described by the samepolytropic EoS, with index n = 1 and γ = 2. In thepresent work we propose to investigate the properties ofrelativistic stars made of anisotropic exotic matter char-acterized by the polytropic EoS p r = Kρ , K = z/B (24)where z is a dimensionless number while B has dimensionof pressure and it is of the order of the energy density ofneutron stars and quark stars, B (cid:39) (150 M eV ) .In the case of stars with anisotropic matter there arefive unknown quantities in total and only three differ-ential equations. Therefore, we are free to impose twoconditions. Given the EoS, the simplest thing to do isto assume a certain profile for the energy density. In thefollowing we shall consider the ansatz ρ ( r ) = ρ c (cid:18) − r R (cid:19) , (25)which ensures that the energy density starts from a finitevalue at the origin, which is the central value ρ c , and itmonotonically decreases with r , until it vanishes at thesurface of the star.Now, all the other quantities may be computed oneby one using the structure equations and the EoS. Inparticular, the radial pressure is immediately computedmaking use of the EoS, while the mass function is com-puted using the tt component of the field equations, andit is given by m ( r ) = 4 π (cid:90) r dxx ρ ( x ) = 4 πρ c r (cid:18) − r R (cid:19) . (26)The temporal metric component ν is computed makinguse of the radial field equation as follows ν ( r ) = log (1 − M/R ) − (cid:90) rR dx m ( x ) + 4 πx p r ( x ) x [1 − m ( x ) /x ] . (27)Finally, the anisotropic factor is computed making useof the conservation of energy∆( r ) = r (cid:26) [ p r ( r ) + ρ ( r )] m ( r ) + 4 πr p r ( r ) r [1 − m ( r ) /r ) + p (cid:48) r ( r )] (cid:27) , (28)while the tangential pressure is computed from p t ( r ) = p r ( r ) + ∆( r ).Next we shall investigate the behavior as well as theviability of the solutions we just found. A. Causality, stability and energy conditions
The radial and tangential speeds of sound, defined by c r ≡ dp r dρ , (29) c t ≡ dp t dρ , (30)should take values in the interval 0 < c r,t < ≡ c r (cid:18) ρp r (cid:19) , (31)should be larger than 4/3 [111].Finally, the solutions obtained here should be able todescribe realistic astrophysical configurations. Therefore,as a further check we investigate if the energy conditionsare fulfilled or not. To that end, the conditions [112–116] ρ ≥ , (32) ρ + p r,t ≥ , (33) ρ − p r,t ≥ , (34) E + ≡ ρ + p r + 2 p t ≥ , (35) E − ≡ ρ − p r − p t ≥ , (36)are investigated.Our main numerical results are summarized in Figures1, 2 and 3 below, assuming the following numerical valuesfor z , B and ρ c : B = 2 × − m pl , (37) ρ c = 14 B, (38) z = 0 . , (39)with m pl being the Planck mass. (27) and (28) corre-spond to a star with the following properties R = 12 km, (40) M = 2 . M (cid:12) , (41) C = 0 . , (42)where C = M/R is the compactness factor of the star.In particular, Fig. 1 shows the normalized anisotropicfactor ∆( r ) /B versus r/R . It vanishes both at the centerand at the surface of the star and it is positive throughoutthe object. The adiabatic index Γ versus r/R is shownin Fig. 2, where the Newtonian limit of 4 / / R Δ / B FIG. 1: Normalized anisotropic factor, ∆ /B , versus normal-ized radial coordinate r/R . / R Γ FIG. 2: Relativistic adiabatic index, Γ, versus normalizedradial coordinate r/R . The horizontal line corresponds to theNewtonian limit of 4 / / R m / M s un FIG. 3: Mass function (in solar masses) versus normal-ized radial coordinate r/R . The dashed curve correspondsto isotropic stars. / R M e t r i c P o t en t i a l s FIG. 4: The two metric components, e ν (lower curve) and1 / (1 − m/r ) (upper curve) versus normalized radial coordi-nate r/R . The dashed curves correspond to isotropic stars. / R X i / B FIG. 5: Energy density ρ/B (orange curve) radial pressure p r /B (blue curve) and tangential pressure p t /B (green curve)versus r/R . The dashed curves correspond to isotropic stars,where energy density lies above pressure. In the Figures 3-6 a comparison is made between starswith anisotropic matter and their isotropic counterpartswith the same EoS and the same radius. In particular,in Fig. 3 we show the mass functions versus r/R , whileFig. 4 shows the two metric potentials versus r/R . Fi-nally, Fig. 5 shows normalized energy density and pres-sures versus r/R , while in Fig. 6 we show the speeds ofsound, both radial (blue curve) and tangential (orangecurve), versus r/R .Clearly, causality is not violated as both sound speedstake values in the range (0 ,
1) throughout the star. More- over, the condition Γ > / / R c s FIG. 6: Radial (blue curve) and tangential (orange curve)sound speeds, c r , c t , versus normalized radial coordinate r/R .The dashed curve corresponds to isotropic stars. IV. CONCLUSIONS
In summary, in the present work we have studied ex-otic stars with anisotropic matter within General Rel-ativity. We have investigated in detail the propertiesof dark matter-type configurations, taking into accountthe presence of anisotropies. Exact analytic expressionsfor all the quantities of interest, such as mass function,anisotropic factor, relativistic index, speed of sound etc,have been found. Causality, stability criteria and energyconditions are also discussed. It is found that the solu-tions obtained here are well-behaved solutions capable ofdescribing realistic astrophysical configurations. Finally,a direct comparison with their isotropic counterparts wasmade as well.
Acknowledgments
PHRSM thanks CAPES for financial support. Theauthors G. Panotopoulos and I. Lopes thank theFunda¸c˜ao para a Ciˆencia e Tecnologia (FCT), Portu-gal, for the financial support to the Center for Astro-physics and Gravitation-CENTRA, Instituto SuperiorT´ecnico, Universidade de Lisboa, through the ProjectNo. UIDB/00099/2020 and grant No. PTDC/FIS-AST/28920/2017. [1] V.C. Rubin et al., Astrophys. J. , L107 (1978).[2] V.C. Rubin et al., Astrophys. J. , 35 (1979).[3] C.J. Peterson et al., Astrophys. J. , 770 (1978). [4] W. Hu, Astrophys. J. , 485 (1998).[5] G.R. Blumenthal et al., Nature , 517 (1984).[6] C.S. Frenk et al., Astrophys. J. , 507 (1988). [7] J.M. Gelb and E. Bertschinger, Astrophys. J. , 467(1994).[8] B. Ryden, Introduction to Cosmology (Addison Wesley,San Francisco, USA, 2003).[9] S. Dodelson, Modern Cosmology (Academic Press, Am-sterdam, Netherlands, 2003).[10] Planck Collaboration, Astron. Astrophys. , A13(2016).[11] J. Aalbers et al., J. Cosm. Astrop. 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