Eur.Phys.J.CmanuscriptNo. (willbeinsertedbytheeditor) Constraints on the braneworld from compact stars R.Gonza´lezFelipea,1,2,D.ManrezaParetb,3,A.Pe´rezMart´ınezc,4,5 1ISEL-InstitutoSuperiordeEngenhariadeLisboa,InstitutoPolite´cnicodeLisboa,RuaConselheiroEm´ıdioNavarro1959-007Lisboa,Portugal 2DepartamentodeF´ısicaandCentrodeF´ısicaTeo´ricadePart´ıculas-CFTP,InstitutoSuperiorTe´cnico,UniversidadedeLisboa,AvenidaRovisco Pais,1049-001Lisboa,Portugal 3DepartamentodeF´ısicaGeneral,FacultaddeF´ısica,UniversidaddelaHabana,LaHabana,10400,Cuba 4InstitutodeCiberne´tica,Matema´ticayF´ısica(ICIMAF),LaHabana,10400,Cuba 5InstitutodeCienciasNucleares,UniversidadNacionalAuto´nomadeMe´xico,ApartadoPostal70-543,DistritoFederal04510,Mexico 6 1 Received:date/Accepted:date 0 2 y Abstract Accordingtothebraneworldidea,ordinarymat- In the Randall-Sundrum type-II brane model, the bulk a terisconfinedona3-dimensionalspace(brane)thatisem- geometry is curved and the brane is endowed with a ten- M bedded in a higher-dimensional space-time where gravity sion that is fine-tuned against the bulk cosmological con- 6 propagates. In this work, after reviewing the limits com- stant to ensure a flat Minkowski space-time in the brane. 1 ingfromgeneralrelativity,finitenessofpressureandcausal- Thebranetensionλ relatesthePlanckmasses,M andM , P 5 ityonthebrane,wederiveobservationalconstraintsonthe in four and five dimensions, respectively, via the equation ] c braneworldparametersfromtheexistenceofstablecompact λ =3M56/(4πMP2),whereMP=1.22×1019 GeV.Success- q stars.Theanalysisiscarriedoutbysolvingnumericallythe fulbigbangnucleosynthesisrequiresthatthechangeinthe - r brane-modifiedTolman-Oppenheimer-Volkoffequations,us- expansionrateduetothenewtermsintheFriedmannequa- g ing different representative equations of state to describe tionbesufficientlysmallatscales∼O(MeV).Amorestrin- [ matter in the star interior. The cases of normal dense mat- gentboundcanbeobtainedbyrequiringthetheorytoreduce 2 ter,purequarkmatterandhybridmatterareconsidered. toNewtoniangravityonscaleslargerthan1mm[4]. v ModificationstoEinsteingravityarealsorelevantinthe 3 vicinityofmassivecompactobjectsasblackholesandneu- 7 9 tron(quark,hybrid)stars.Compactstarsarethereforeaspe- 1 Introduction 1 cial laboratory to look for possible modifications of gen- 0 eral relativity (GR) and to test extra dimensions [9]. From . Braneworld ideas and, in particular, the Randall-Sundrum 1 theexistenceandstabilityofsuchastrophysicalobjects,one 0 (RS) models [1, 2] have been extensively investigated dur- expectsadditionalconstraintsontheparametersofalterna- 6 ingthelastdecade,mainlymotivatedbythedevelopmentof tive theories of gravity [10–17]. In particular, it has been 1 stringtheory.Aremarkablefeatureofthebraneworldisthat : shown that neutron stars (NS) put a lower bound on the v it modifies the Einstein equations locally and non-locally, brane tension λ, which is stronger than the bound coming i leadingtoaneffectiveenergy-momentumtensor.Thesemod- X from big bang nucleosynthesis, although weaker than the ifications have important consequences for cosmology [3], r experimentalNewtonlawlimit[10].Furthermore,thewell- a sincesignificantdeviationsfromEinsteingravitycouldhave knowncompactnesslimitGM/R≤4/9[18],obtainedinGR occurredatveryhighenergiesintheearlyuniverse.Inpar- byrequiringthefinitenessofpressureatthecentreofauni- ticular, in brane cosmology, the expansion rate of the uni- formstar,isreducedbyhigh-energy5Dgravityeffects. verseH scaleswiththeenergydensityρ asH∝ρ,whereas The macroscopic properties of compact stars also cru- this dependence is H ∝ρ1/2 in standard cosmology. This cially depend on the constituent matter in the star interior. high-energy behaviour, which is generic and not specific Moreprecisely,starmassesandradiiaredeterminedbythe toRSbraneworldscenarios,mayaffectearlyuniversephe- equationofstate(EoS)ofmatter,i.e.bytherelationp(ρ)be- nomena,suchasinflation[4]andthegenerationofthecos- tweenthepressureandtheenergydensityinsidethestar.In mologicalbaryonasymmetry[5–8]. theabsenceofauniqueframeworktodescribethephysicsof ae-mail:[email protected] compactstars,severalapproachescanbeadoptedforthede- be-mail:dmanreza@fisica.uh.cu terminationoftheEoS.Forinstance,itcanbereconstructed ce-mail:[email protected] frommass-radiusmeasurements[19–21].Alternatively,one 2 can resort to theoretical calculations based on chiral effec- i.e., stars with a hadronic outer region surrounding a quark tivefieldtheoriesandobtainstringentconstraintsfortheNS (ormixedhadron-quark)innercore. radii[22],whencombinedwithmassmeasurements. The study of the macroscopic stellar properties is car- ried out through the solution of the Tolman–Oppenheimer- Inouranalysis,thenon-local“dark”components[3,13] Volkoff(TOV)equations,properlymodifiedtoincludelocal arisingfromthebulkWeyltensoraremodelledviathesim- ple linear proportionality relation P = wU between the andnon-localbulkeffects.Thelocalandnon-localbulkcor- darkenergyU anddarkpressureP.Suchafunctionalform rections to the energy-momentum tensor turn out to play a crucialroleinthestabilityofcompactstarsonthebraneand follows,forinstance,fromtherequirementthatthevacuum inestablishingagreementwiththeobservationalconstraints. on the brane admits a one-parameter group of conformal The present work extends previous studies (e.g. [14]) not motions and the field equations are invariant with respect onlybyincludingintheanalysisthecompactnessandcausal- to the Lie group of homologous transformations [23]. The itylimitsinthebraneworld,butalsobyconsideringthefull aboverelationhasalsobeenusedin[11]tostudythejunc- parameterspaceforthebranetensionλ andthewcoefficient tionconditionsbetweentheinteriorandexteriorofstaticand oftheassumeddarkEoS. spherically symmetric stars on the brane. Recently, it has Thepaperisorganisedasfollows.InSect.2wepresent been employed in the study of the mass-radius relation of and briefly discuss the TOV equations on the brane. The somehadronicstars,hybridstars(HS)andquarkstars(QS) brane-modifiedcompactnesslimitsaresummarisedinSect.3 in the braneworld [14]. In a different context, this type of and a new causality limit is derived requiring the sublu- state-likerelationiscommonlyassumedincosmologytode- minality of the EoS for matter inside the compact star. In scribethemattercontentoftheuniverseatdifferentepochs Sect.4,wepresentthebraneTOVsolutionsforseveralrep- (w=1/3forradiationdomination,w=0foramatterdom- resentativeEoS.Thepredictedmass-radiusrelationsarethen inateduniverse,andw=−1foracosmologicalconstant). comparedwiththeobservationalconstraints.Finally,ourcon- It is pertinent to stress the limitations of our approach. cludingremarksaregiveninSect.5. We admittedly consider a linear relation between the dark energyU anddarkpressureP thatappearintheeffective 4Dequations.Thesequantitiescomefromthe5DWeylten- 2 TOVequationsonthebrane sor, which cannot be determined solely from the 4D equa- tions.TheparticularchoiceP =wU isthereforearestric- Thefieldequationsinducedonthebranehavetheform[3] tion.Thealternativewaystoproceedwouldbeeithertocon- sider an infinite number of functional dependences P(U) G =R −1Rg =κ2T +6κ2S −E , (1) µν µν µν µν µν µν 2 λ or,moreappropriately,toremovetheambiguitybysolving thefull5DEinsteinequations.Thelatterhasprovenavery whereG istheusualEinsteintensor,κ2=8πG,andT µν µν demandingtasksofar. isthestandardenergy-momentumtensor.1 Foraspherically The purpose of the present work is two-fold. First we symmetricstar,themetricinstaticcoordinatesisgivenby study the limits coming from general relativity, finiteness ds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2(dθ2+sin2θdφ2). (2) ofpressureandcausality,appliedtocompactobjectsinthe braneworld. In particular, assuming a star interior with a The tensors S and E encode the local and non-local µν µν vanishing dark pressure and a non-vanishing dark density, bulkcorrections,respectively.Foraperfectfluid,theexpres- andconsideringapurecausalEoS,weshallderiveabrane- sionsforT andS aregivenby[3] µν µν modified causality limit that depends on the brane tension, andwhichismorerestrictiveasλ decreases.Secondly,we Tµν =ρuµuν+p(gµν+uµuν) (3) studycompactstarsintheRStype-IIbraneworldinorderto and establishlimitsontheparametersofthemodel. 1 1 Ouranalysisisbasedonastrophysicalobservationsand S = ρ2u u + ρ(ρ+2p)(g +u u ), (4) µν µ ν µν µ ν 12 12 weuserepresentativeEoStodescribematterinthestarin- terior.Inparticular,fordensenuclearmatter,weshallcon- where uµ is the four-velocity of the fluid. The tensor E µν sidertheanalyticalrepresentationgivenin[24]fortheuni- reducestotheform fiedBrussels-MontrealEoSmodels,whicharebasedonthe nuclear energy-density functional theory with generalized Eµν =−κ62λ (cid:2)Uuµuν+Prµrν Skyrmeeffectiveforces.Forquarkmatter,weshallemploy (cid:21) (5) 1 thesimplephenomenologicalparametrisationgivenin[25], + (U −P)(gµν+uµuν) , 3 which includes QCD and strange quark mass corrections. We shall also consider a hybrid EoS to study hybrid stars, 1Hereafter,weuseasystemofnaturalunitswithc=1. 3 for the case of a static spherical symmetry. Here, r is a TointegratetheTOVequations,weneedappropriateini- µ unit radial vector, U is the non-local energy density (dark tialconditionsatthecentreofthestar.Asingeneralrelativ- radiation) and P is the non-local pressure (dark pressure) ity,weassumethattheenclosedmassiszeroatthecentre, onthebrane.FromEqs.(1)and(5),weseethatstandard4D m(0)=0,andthat p(0)= p ,where p isthecentralpres- c c generalrelativityisrecoveredinthelimitλ →∞. sure.Atthestellarsurface,werequire p(R)=0,i.e.avan- Solving Einstein’s equations for a perfect fluid matter, ishingpressure,whichcorrespondstoastarmassm(R)=M. the following modified TOV equations are obtained on the AsforthedarkcomponentU,weshallassumethatU(0)= brane: 0.Weremarkthatdifferentinitialconditionsforthedarken- dm ergy density U can be chosen. They could be given either =4πr2ρ , (6) dr eff atthecentreofthestaroratitssurface.Inthelattercase,a dp dΦ shootingmethodisrequiredfortheintegrationofthesystem =−(ρ+p) , (7) dr dr ofequations[17].Theboundaryconditionchosenhere,i.e. dΦ 2Gm+κ2r3(cid:2)peff+(4P)/(κ4λ)(cid:3) avanishingdarkdensityatthecentre,isthesimplestchoice. = , (8) Notealsothatifw=−1/2,thevalueofU atthecentreis dr 2r(r−2Gm) obtaineddirectlyfromEq.(16). dU 1 dρ dP 6 =− κ4(ρ+p) −2 − P The system (12)-(15) must be supplemented with the dr 2 dr dr r Israel-Darmoisjunctionconditions[26,27]atthestellarsur- dΦ −(2P+4U) , (9) face.Onthebrane,thisleadstothematchingcondition dr (cid:20) (cid:21) where 4 p + P =0, (18) ρ =ρ + 6 U , p =p + 2 U . (10) eff κ4λ surf eff loc κ4λ eff loc κ4λ where[f] ≡ f(R+)−f(R−),andthesuperscripts+and surf Intheaboveexpressions, −refertoquantitiesdefinedoutsideandinsidethestar,re- spectively.Therequirementp(R)=0atthesurfacethenim- ρ2 pρ ρ2 ρ =ρ+ , p =p+ + , (11) plies loc loc 2λ λ 2λ κ4 denote the effective local matter density and pressure, re- ρ2(R)+U−(R)+2P−(R)=U+(R)+2P+(R). (19) 4 spectively.Inordertosolvethesystemofdifferentialequa- tions(6)-(9),anequationofstate p(ρ)formatterandare- IntheabsenceofanyWeylstressintheinterior,wehave lation P(U) are required. As explained before, we shall P−=U−=0andEq.(19)impliesthattheexteriormust assumethesimpleststate-likerelationP =wU.Withthis be non-Schwarzschild, provided that the energy density ρ choice,Eqs.(6)-(9)become does not vanish at the surface. The same conclusion holds dm ifw=−1/2,sinceinthiscaseU−(R)+2P−(R)=0.On dr =4πr2ρeff, (12) theotherhand,ifρ(R)(cid:54)=0andtheexteriorisSchwarzschild dp dΦ (P+=U+=0),thennecessarilyU−(R)+2P−(R)(cid:54)=0 =−(ρ+p) , (13) in the star interior. Note that the condition P =U =0 is dr dr dΦ 2Gm+κ2r3(cid:2)peff+(4wU)/(κ4λ)(cid:3) the condition for a perfect fluid without non-local or Weyl = , (14) corrections. While for the exterior this corresponds to the dr 2r(r−2Gm) Schwarzschild solution, this is not the case for the star in- dU 2 (cid:20)κ4 dρ 3w =− (ρ+p) + U terior, where local high-energy corrections are present. A dr 1+2w 4 dr r consistent version of the Schwarzschild interior metric in (cid:21) dΦ thecontextofthebraneworld,includinglocalandnon-local +(w+2)U , (15) dr bulkterms,hasbeenfoundin[28].Regardingthematching conditions,ithasbeenshownthattheexteriorSchwarzschild wherethelastequationholdsforw(cid:54)=−1/2.Thecasew= −1/2shouldbetreatedseparately.SolvingforU inEq.(9) solutioniscompatiblewithastellardistributionmadeofreg- ularmatter[29]. andusing(8),weobtainforw=−1/2, WealsonotethattheTOVequationsinthebraneworld κ4(ρ+p)2 2Gm+κ2r3p admit asymptotically flat exterior solutions different from U(r)= loc , (16) 6v2 2(3Gm−r)+κ2r3p theSchwarzschildsolution.Inwhatfollows,ouranalysisis s loc restricted to solutions in the star interior with generic dark wherev isthespeedofsound, s energy and dark pressure components obeying the EoS re- dp lationP=wU.Possibleexteriorsolutionsarenotconsid- v2= . (17) s dρ eredinthiswork. 4 compactstarinthebraneworld,whencomparedtothegen- 3 eralrelativitycase. Anotherimportantlimitcomesfromrequiringthepres- 2.5 sureinsidethecompactobjecttobefinite.Sincethepressure decreaseswiththeradiusr,thisconditionisequivalenttothe 2 requirementofapositiveandfinitepressureatthecentreof thestar.Thisgivestheconstraint[10] (cid:2) M M/1.5 Schwarzschildlimit GM 4 1+7ρ/4λ+5ρ2/8λ2 p<∞(GR) R ≤ 9 (1+ρ/λ)2(1+ρ/2λ). (23) 1 λ=102 SolvingforMtofindthemaximummass,weobtain λ=103 √ 0.5 λ=104 M(R)= 4 πR2(cid:2)−2√πGλR 9G 00 5 10 15 20 25 +(cid:113)Gλ(5+4πGλR2)cos(cid:18)1tan−1x(cid:19)(cid:21), (24) R[km] 3 Fig.1 Generalrelativityandpressurefinitenesslimitsonthebrane, where ainssuunmitisnogfaMsteaVri/nfmte3ri.orwithP=U =0.Thebranetensionλisgiven (cid:0)125G−1λ−1+264πR2+144π2GλR4(cid:1)1/2 x= √ . (25) 2 πR(3+4πGλR2) 3 Compactnesslimitsinthebraneworld Inthelimitλ →∞,werecoverthepressurefinitenesslimit ofGR, In this section, we revisit the compactness limits on a uni- 4R formstarcomingfromgeneralrelativityandtherequirement M(R)= . (26) 9G of the finiteness of pressure. We shall also derive a brane- ThecurvedefinedbyEq.(24)ispresentedinFig.1for modifiedcausalitylimitfortheexistenceofstablestars. AssumingauniformdensityandP=U =0,thehigh- different values of the brane tension (dot-dashed lines). As inthecaseoftheSchwarzschildlimit,high-energybraneworld energy brane corrections are local. In this case, an astro- corrections are significant for λ (cid:46)104 MeV/fm3. We also physical lower limit on λ, independent of the Newton-law note that for a given radius R the maximum mass allowed and cosmological limits, can be established for all uniform bythebound(24)islowerthanthatcomingfrom thelimit stars[10]: inEq.(21). GMρ A third relevant constraint on the maximum star mass λ ≥ , (20) R−2GM comesfromcausality,i.e.fromrequiringthesubluminality oftheEoS,2i.e.v2≤1.Theboundobtainedfromthiscondi- withρ =3M/(4πR3).Forapositivebranetension,thisim- s tioniscontrolledbythestiffnessofthematterEoS,andsev- plies R>2GM, so that the Schwarzschild radius remains eral limits have been derived in the literature [31–33] (see alimitingradius.Belowtheastrophysicallimit(20),stable also[19]forareview).Inparticular,astringentcausallimit neutronstarscannotexistonthebrane.Thisboundismuch hasbeenobtainedin[33],basedonthe“minimumperiod” stronger than the cosmological nucleosynthesis constraint, EoS butturnsouttobeweakerthantheNewton-lawlowerbound. (cid:26) From Eq. (20), we find the following upper bound for the p= 0, ρ <ρs, (27) mass: ρ−ρs, ρ ≥ρs, 2R2(cid:18)(cid:113) (cid:19) whereρsisthesurfaceenergydensity.ThisEoScorresponds M(R)= πλ(3G−1+4πλR2)−2πλR . (21) tomaximalstiffnessathighdensitiesandminimalstiffness 3 at low densities, thus supporting the largest mass with the Asexpected,inthelimitλ→∞,werecovertheSchwarzschild smallestradius.Ingeneralrelativity,suchEoSalsoimplies limitofGR: that the maximum value of the mass scales with the radius asM∝R.ThenumericalintegrationoftheTOVequations 1R M(R)= . (22) forvariousinitialvaluesofthecentralpressurep thenleads 2G c totheGRrelation[33] The astrophysical bound given by Eq. (21) is illustrated in Fig.1fordifferentvaluesofλ (solidlines).Ascanbeseen R(cid:39)2.82GM. (28) from the figure, brane corrections to the compactness limit 2Forsomecaveatsonthespeedofsoundandtherequirementofcausal- wouldbesignificantforλ (cid:46)104MeV/fm3,leadingtoaless ityseee.g.[30]. 5 Asweshallshownext,thiscompactnesslimitismodifiedin 4 thebraneworld. causality(GR),vs≤1 Firstwenoticethat,fromEqs.(10)and(11),andusing 3.5 λ=102,w=0 theconservationequationsforρ andU,aneffectivespeed λ=103 λ=104 ofsound[3]isobtainedas 3 dp v2(1+ρ/λ)+(ρ+p)/λ+v2 2.5 v2 ≡ eff = s U , (29) (cid:2) s,eff dρeff 1+ρ/λ+3v2U M/M 2 where 1.5 8 U 1 v2 = . (30) U 3κ4λ ρ+p 0.5 As expected, in the limit λ →∞, we have v =v . Note s,eff s 0 alsothatv2 isnotnecessarilyapositivenumber,sincethe 0 2 4 6 8 10 12 14 16 18 20 U R[km] Weyl energy density can be negative. Moreover, when the radiation term vU dominates over the matter components, 4 √ vo2sn,eeffh(cid:39)asvv2s2s+,effp(cid:39)/ρ1+/31..Ontheotherhand,ifv2U (cid:28)ρ/λ then 3.5 cλλa==usa11l00i23ty,w(G=R0),vs≤1/ 3 LetusconsiderthecaseofaSchwarzschildexteriorand 3 λ=104 a star interior with P =0 (i.e. w=0) and U (cid:54)=0. In this 2.5 case,theboundarycondition(19)yieldsanegativedarkra- diationdensityatthestarsurface.WeobtainU =−κ4ρs2/4, /M(cid:2) 2 so that v2 =−2ρ /(3λ) at the surface. From Eq. (29) we M U s 1.5 thenfind v2(1+ρ /λ)+ρ /(3λ) 1 v2 = s s s . (31) s,eff 1−ρs/λ 0.5 Requiringvs,eff≤1,theconstraint 0 0 2 4 6 8 10 12 14 16 18 20 R[km] (4/3+v2)ρ λ > s s, (32) 1−v2 Fig.2 Causalityconstraintonthebrane,obtainedfromthecausalEoS s ofEq.(27)(topplot)andEq.(33)(bottomplot).Astarinteriorwith withv2<1,isobtained. P=0(i.e.w=0)andU (cid:54)=0isassumed.Thebranetensionλisgiven s inunitsofMeV/fm3. Next we illustrate how the corrections due to brane ef- fectsleadtomodificationsoftheGRcompactnesslimitgiven in Eq. (28). Solving numerically the system of equations Table1 Causality√limitinthebraneworldfortwocausalEoS(with (12)-(15),fromthestarcentreuntilthesurface,wecande- vs≤1andvs≤1/ 3),assumingavanishingdarkradiationpressure (P=0)andanon-zerodarkenergydensityU inthestarinterior.The terminethemaximumstarmass.Theresultsarepresentedin minimumradiusisapproximatelydescribedbyastraightlinewitha Fig. 2 (upper plot) for different values of the brane tension slopethatvarieswiththebranetension. λ and taking w=0. Deviations from the causality limit in √ GR (black solid line) occur for λ (cid:46)104 MeV/fm3, as can vs≤1 vs≤1/ 3 λ [MeV/fm3] R/(GM) R/(GM) beseenfromthecurvesformaximalmassesdepictedinthe 10 3.77 4.96 figure. In this case, for the star radii range of interest, the 102 3.45 4.64 maximummass(minimalradius)canbewellapproximated 103 3.06 4.13 byastraightline,M(cid:39)αR/G,withaslopeα thatincreases 104 2.83 3.73 ∞(GR) 2.82 3.68 asthebranetensionλ increases,reachingtheGRlimit(28) whenλ →∞.Thevaluesofα−1arepresentedinthesecond columnofTable1fordifferentvaluesofλ. Ithasbeenrecentlyconjecturedthatthespeedofsound coupled theories, and is respected by several strongly cou- √ shouldsatisfytheboundv ≤1/ 3inanymedium[34].It pled theories. If such a bound actually holds for the speed s iswellknownthatthisboundissaturatedinconformaltheo- of sound, it would modify the causality limit for compact ries,includingnon-interactingmasslessgasesforwhichp= astrophysical objects. In particular, it has been pointed out ρ/3. The limit also applies to non-relativistic and weakly thattheexistenceofneutronstarswithM∼2M ,combined (cid:12) 6 with the knowledge of the EoS of hadronic matter at low relation predicts a maximum mass M . While typically max densities,isinstrongtensionwiththisbound[34]. M (cid:46)2M for hybrid and strange quark EoS, maximum √ max (cid:12) Toillustratetheimplicationsoftheboundv ≤1/ 3for massesupto2.5M canbereachedwithnormalmatterEoS. s (cid:12) the stability of compact stars, let us consider the minimal From the astrophysical viewpoint, an important aspect causalEoS inthestudyofstellarconfigurationsistheirstability.Anec- (cid:26) essary condition for the stability of a compact star is given 0, ρ <ρ p= s (33) bytheso-calledstaticcriterion (ρ−ρ )/3, ρ ≥ρ , s s dM whichsaturatesthebound.3Thenumericalintegrationofthe >0. (35) dρ c TOVequationsofGRimpliesthecausalityrelation In other words, a compact star is stable if its mass M in- R(cid:39)3.68GM, (34) creaseswithgrowingcentraldensityρ =ρ(p ).Although c c thestabilityanalysisisusuallyquiteinvolved,somesimple whichismorerestrictivethatthebound(28).Theinclusion criteriacanbeformulatedbasedonthemass-radiusconfig- ofbranecorrectionsleadstomodificationstothislimit.For the case of a star interior with P =0 and U (cid:54)=0, the re- urationsandtheirstabilitywithrespecttoradialoscillations (see e.g. [35] and references therein). At each extremum sults are shown in Fig. 2 (lower plot). Once again, signifi- (critical point) of the M(R) curve, it is assumed that only cantdeviationsfromthecausalitylimitofGRtakeplacefor λ (cid:46)104 MeV/fm3. As before, the minimum radius is well oneradialmodechangesitsstabilityfromstabletounstable or, vice versa, from unstable to stable. Furthermore, at any approximatedbyastraightlinewhoseslopevarieswiththe criticalpoint,amodebecomesunstable(stable)ifandonly branetension.TheresultsaregiveninthelastcolumnofTa- if the curve bends counterclockwise (clockwise). Finally, a ble1.Wenotethat,foranygivenvalueofλ,theconstraint √ mode with an even (odd) number of radial nodes is said to obtainedfromrequiringv ≤1/ 3isalwaysstrongerthan s changeitsstabilityifandonlyifdR/dρ >0(dR/dρ <0) theonepreviouslyfoundundertheassumptionv ≤1. c c s atthecriticalpoint.Usingtheabovecriteria,thestabilityof agivenstellarconfigurationcanbeeasilychecked. 4 BraneTOVsolutionsandobservationalconstraints InordertoconfrontdifferentEoSwithcurrentmassand radius measurements, we shall use the following observa- Duetotheuncertaintiesinthedescriptionofthemany-body tionalconstraints.ForNSradii,weadopttherange interactionsandthenuclearsymmetryenergy,aswellasour lack of knowledge of the precise nature of strong interac- 7.6km≤R≤13.9km, (36) tions,theEoSofdensematterabovethenuclearsaturation where the lower bound follows from the measurement of density (ρ =2.7×1014g/cm3 (cid:39)150 MeV/fm3) is largely 0 theNSradiususingthethermalspectrafromquiescentlow- unknown.Dependingonthemattercompositionintheneu- massX-raybinariesinsideglobularclusters[36]andtheup- tronstarinterior,threemaintypesofequationsofstatehave per limit is taken from the analysis of [22], based on the beencommonlyused,namely,EoSfornormaldensematter, chiraleffectivetheory.Forthestarmasses,weconsiderthe purequarkmatterorhybridmatter. limits Inthecaseofnormaldensematter,neutronstarsaresup- portedagainstgravitationalcollapsebyneutrondegeneracy. 1.08M ≤M≤2.05M , (37) (cid:12) (cid:12) The EoS takes into account nucleon-nucleon interactions and is characterised by a vanishing pressure at null densi- where the conservative lower bound comes from the ex- ties. For hybrid stars, the EoS is usually softened at high pected range for the gravitational NS birth masses [37, 38] densitiesbyaddinghadronicorpurequarkmatteratthein- andthemaximumvalueisobtainedfromrecentradio-timing ner core of the star, which leads to a phase transition at a observationsofthepulsarPSRJ0348+0432[39]. givencriticaldensity.NormalandhybridEoSdonotleadto In the braneworld, the macroscopic properties of stable stringentboundsforthestarradii,whichinprinciplecanbe stellar configurations are controlled not only by the matter large (∼100 km). On the other hand, pure quark stars are EoSbutalsobyhigh-energybraneeffects,characterisedin conjectured to be mainly composed of strange quark mat- our setup by the brane tension and the assumed dark EoS. ter(SQM)inthegroundstate,withavanishingpressureat Next we analyse the implications of these effects on the non-zerodensities.Forsuchstarsthemaximumradiusisnot mass-radius relations of stable compact stars for the three solarge(∼10km).InallthreeEoScases,themass-radius classes of EoS mentioned above. In our analysis, we shall take into account the stability criteria as well as the physi- 3Note that the simple EoS of the well-known MIT bag model (with callyplausibleconditionofcausalityv ≤1,basedonthe masslessquarksandnostrongcouplingconstant)hasalsothisform.In s,eff thiscase,theparameterρsisassociatedtothebagconstant,ρs≡4B. effectivespeedofsounddefinedinEq.(29). 7 Table2 ParametersaiusedinEq.(38)fortheEoSBSk19,BSk20and BSk21[24]. 3 observationalrange BSk19 BSk20 BSk21 PSRJ0348+0432 2.5 NSBSk21(GR) a 3.916 4.078 4.857 λ=5×102,w=0 1 λ=103 a2 7.701 7.587 6.981 2 λ=5×103 a3 0.00858 0.00839 0.00706 λ=104 a4 0.22114 0.21695 0.19351 M(cid:2) λ=5×104 a5 3.269 3.614 4.085 M/1.5 a 11.964 11.942 12.065 6 a 13.349 13.751 10.521 7 1 a 1.3683 1.3373 1.5905 8 a 3.254 3.606 4.104 9 a −12.953 −22.996 −28.726 0.5 10 a 0.9237 1.6229 2.0845 11 a 6.20 4.88 4.89 12 0 a13 14.383 14.274 14.302 0 2 4 6 8 10 12 14 16 18 20 a 16.693 23.560 22.881 R[km] 14 a −1.0514 −1.5564 −1.7690 15 a16 2.486 2.095 0.989 3 a17 15.362 15.294 15.313 NSBSk21(GR) a 0.085 0.084 0.091 λ=5×102,w=0 a18 6.23 6.36 4.68 2.5 λ=103 19 λ=5×103 a20 11.68 11.67 11.65 λ=104 a21 −0.029 −0.042 −0.086 2 λ=5×104 a 20.1 14.8 10.0 22 a23 14.19 14.18 14.15 M(cid:2) /1.5 M 1 4.1 Neutronstars Letusfirststudytheexampleofaneutronstar.Todescribe 0.5 thecrustandthecoreofthestar,weshalluseananalytical representationoftheBrussels-MontrealunifiedEoSforcold 0 1 2 3 4 5 6 7 8 9 nuclear matter, referred to as models BSk19, BSk20, and ρc/ρ0 BSk21[40–42].Weconsiderthefollowingparametrisation Fig. 3 Top plot: Mass-radius relation for the NS BSk21 EoS. The of p(ρ)[24] curves are given for different values of the brane tension λ (in MeV/fm3), assuming a vanishing dark pressure in the star interior a +a ξ+a ξ3 ζ = 1 2 3 {exp[a (ξ−a )]+1}−1 (w=0)Bottomplot:Themassofthestarversusthecentralenergy 1+a4ξ 5 6 densityρc(inunitsofthenuclearsaturationdensityρ0). +(a +a ξ){exp[a (a −ξ)]+1}−1 7 8 9 6 +(a +a ξ){exp[a (a −ξ)]+1}−1 (38) straints of Eqs. (36) and (37), where the dark grey band is 10 11 12 13 +(a +a ξ){exp[a (a −ξ)]+1}−1 the mass range of the pulsar PSR J0348+0432 [39]. In the 14 15 16 17 plots,thesmallcirclesdenotethemaximummassM (in a a max + 18 + 21 , unitsofthesolarmassM )obtainedbyimposingthestabil- 1+[a (ξ−a )]2 1+[a (ξ−a )]2 (cid:12) 19 20 22 23 ity criterion alone, while the diamond markers indicate the where ξ =log (ρ/g.cm−3) and ζ =log (p/dyn.cm−2). maximumstarmassatwhichtheEoSsubluminalitybound 10 10 The parameters a (i=1,...,23) for the three models are v ≤1isviolated.Asλ increases,thisboundapproaches i s,eff giveninTable2.Inwhatfollows,werestrictouranalysisto the usual GR causality constraint vs ≤1, so that for larger themodelBSk21. values of λ the stability condition is more restrictive than WeproceedtosolvethesystemofbraneTOVequations thecausalitybound. for different initial values of the central pressure p in or- As can be seen from the top panel of Fig. 3, the max- c der to determine the mass-radius relation for such neutron imum mass and the corresponding star radius decrease as stars. We consider first the case of a dark EoS with w=0. λ decreases,sothatrequiringagreementwithobservational TheresultsarepresentedinFigs.3and4,wherethestable constraints leads to a lower bound on the brane tension, mass-radius configurations, i.e. those that verify the stabil- λ (cid:38) 8×102 MeV/fm3. The star radii lie in the range 8– itycriteriaandobeythecausalitylimitv ≤1,aregiven. 13 km. Furthermore, from the bottom panel of Fig. 4, we s,eff Theshadedgreyareascorrespondtotheobservationalcon- concludethatρ (cid:46)9ρ .Wealsonoticethatthemass-radius c 0 8 3 3 observationalrange PSRJ0348+0432 2.5 NSBSk21(GR) 2.5 λ=6×103,w=0 w=1 2 w=−1 2 w=−0.6 (cid:2) w=−0.51 M(cid:2) M/M1.5 /max1.5 M 1 1 NSBSk21,w=0 w=5 0.5 0.5 w=−1 w=−0.7 w=−0.5 0 0 0 2 4 6 8 10 12 14 16 18 20 103 104 105 106 R[km] λ[MeV/fm3] 3 3 2.5 2.5 2 2 (cid:2) (cid:2) M M/M1.5 /max1.5 M NSBSk21(GR) 1 λ=6×103,w=0 1 w=1 NSBSk21,λ=5×102 w=−1 λ=103 0.5 w=−0.6 0.5 λ=5×103 w=−0.51 λ=104 λ=5×104 0 0 1 2 3 4 5 6 7 8 9 10 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ρc/ρ0 w Fig. 4 Top plot: Mass-radius relation for the NS BSk21 EoS. The Fig.5 Topplot:Maximumneutronstarmassasafunctionofthebrane curvesaregivenfordifferentvaluesofthedarkEoSparameterwand tensionfordifferentvaluesofthedarkEoSparameterw.Bottomplot: λ =6×103 MeV/fm3.Bottomplot:Themassofthestarversusthe Maximum star mass as a function of the dark EoS parameter w for centralenergydensityρc(inunitsofρ0). differentvaluesofthebranetension. 4.2 Quarkstars curves bend clockwise for some negative values of w (see e.g.thecurvesw=−0.6andw=−0.51inthetoppanelof It has been conjectured that strange quark matter could be Fig.4).Inthesecases,themaximumstarmassisrestricted the true ground state of strongly interacting matter at zero by the speed-of-sound condition v ≤1, since the stabil- s,eff pressure and temperature [43]. If this possibility actually ityconditionisviolatedatmuchhighervaluesofM andR. exists,itwouldopenthewindowfortheexistenceofcom- Forcomparison,wehavealsoindicatedwithcrosses(×)the pact stellar objects totally composed of SQM [44]. To de- mass-radiusconfigurationatwhichtheGRcausalitycondi- scribetheEoSinsidesuchstars,weconsiderthesimplephe- tionv ≤1isviolatedinsuchcases. s nomenologicalparametrisationgiveninRef.[25].Itconsists InFig.5,themaximumstarmassisgivenasafunction ofapowerseriesexpansioninthequarkchemicalpotential ofthebranetension(toppanel)andthedarkEoSparameter µ, w(bottompanel).Fromthefigureweconcludethatthemax- imumstarmasspredictedforthistypeofEoSiscompatible ρ = 9 a µ4− 3 a µ2+B, withobservationsprovidedthatλ (cid:38)6×102 MeV/fm3.For 4π2 4 4π2 2 (39) w(cid:38)−0.1, the value of Mmax remains practically constant p= 3 a µ4− 3 a µ2−B. with the variation of w, depending only on the value of λ. 4π2 4 4π2 2 Asimilarbehaviourisobservedforw(cid:46)−0.5.However,for Besides the bag parameter B, this parametrisation con- −0.3<w<−0.1, the maximum mass is quite sensitive to tainstwoadditionalparameters,a anda ,whichareinde- 4 2 w,asitbecomesevidentinthebottompanelofFig.5. pendentof µ andarerelatedtotheQCDandstrangequark 9 mass(andcolorsuperconductivity)corrections,respectively. 3 Note that for three-flavour quark matter consisting of free observationalrange massless quarks one has a4 =1 and a2 =0. In this case, PSRJ0348+0432 Eqs.(39)coincidewiththewell-knownMITbagmodelEoS, 2.5 QS,B=60(GR) λ=5×102,w=0 i.e.ρ(p)=3p+4B[45].Inournumericalcalculations,how- λ=103 ever,weconsiderthemorerealisticvaluesa4=0.7anda2= 2 λλ==510×4103 (180MeV)2[25].Furthermore,wetakeB=60MeV/fm3.4 (cid:2) λ=5×104 M We follow the same procedure as before, i.e. we inte- M/1.5 gratethebrane-modifiedTOVequationsfordifferentinitial values of the central pressure p and determine the mass- 1 c radius relation for the corresponding quark star configura- tion. Our results are presented in Figs. 6 and 7. As shown 0.5 inthetoppanelofFig.6,themaximummassandthecorre- spondingstarradiusdecreaseasλ decreasesandagreement 00 2 4 6 8 10 12 14 16 18 20 withobservationalconstraintsimposesthelowerboundλ (cid:38) R[km] 4×103 MeV/fm3, for w=0. The allowed star radii are in 3 therange8–10km.Moreover,fromthebottompanelofthe QS,B=60(GR) figure, we conclude that the central energy density for the λ=5×102,w=0 maximummassconfigurationsisboundedbyρ (cid:46)11ρ . 2.5 λ=103 c 0 λ=5×103 In Fig. 7, we present the mass-radius relation for the λ=104 case of λ = 6×103 MeV/fm3 and different values of w. 2 λ=5×104 We notice that for certain negative values of w the mass- (cid:2) M radiuscurvesbendclockwise,reachingthemaximummass M/1.5 at relatively high central densities, ρ ∼40ρ , bounded by c 0 1 therequirementofsubluminalityoftheEoS.Althoughsuch density values are higher than the maximum central den- sity of typical bare strange stars, they are below the crit- 0.5 ical density required for the formation of a stable charm- quark star. We recall that a c-quark can be created via the 0 1 2 3 4 5 6 7 8 9 10 11 12 weakreactionu+d→c+d.Sincethecharmquarkmassis ρc/ρ0 m (cid:39)1.275GeV[46],theproductionofaquarkcrequires c Fig.6 Topplot:Mass-radiusrelationforaquarkstar,usingthephe- ρ ≥ρcrit,c=9m4c/(4π2)(cid:39)1.4×1017g/cm3(cid:39)5.2×102ρ0. nomenologicalEoSinEq.(39).Thecurvesaregivenfordifferentval- Basedonthestabilityanalysis,ithasbeenfoundinthecon- uesofthebranetensionλ (inMeV/fm3),assumingavanishingdark textofGRthatcharm-quarkstarsareunstableagainstradial pressureinthestarinterior(w=0).Bottomplot:Themassofthequark oscillations[47].Inthemass-radiusplane,suchaninstabil- starversusthecentralenergydensityρc(inunitsofρ0). ityismanifestedintheinwardlyspirallingbehaviourofthe curves, and it is also confirmed through the calculation of 4.3 Hybridstars thestaroscillationfrequencies[47]. Finally,inFig.8,themaximumstarmassispresentedas Themaximumvaluesofthecentralenergydensitiesρ ,ob- c afunctionofthebranetension(toppanel)andthedarkEoS tainedfromEoSofpurelyhadronicmatter,aretypicallyin parameterw(bottompanel).Weseethatthemaximumstar the range 5ρ –10ρ . At such densities, one expects quark 0 0 mass predicted for the quark model is compatible with ob- degrees of freedom to play a relevant role inside compact servations provided that λ (cid:38)103 MeV/fm3. As in the case stars.Itisthennaturaltoconsiderthepossibilityofthefor- of pure neutron stars, for w(cid:38)−0.1, the value of Mmax re- mation of hybrid compact objects, i.e. stars that contain a mains essentially constant with the variation of w, and de- coreofquarkmatterwithacrustalofnuclearmatter.Wede- pends only on the value of λ. The same conclusion holds scribeahybridstarusingthefollowingcombinedEoS.For forw(cid:46)−0.5.Intherange−0.3<w<−0.1,however,the densities ρ (cid:46)4ρ , the EoS is modelled by Eq. (38), while 0 maximummassisquitesensitivetothevalueofw,asseen for the core of the hybrid star, corresponding to densities inthebottompanelofFig.8. ρ >4ρ ,weusethesimplephenomenologicalparametrisa- 0 tiongiveninEq.(39). 4Assuming massless quarks and neglecting the strong coupling con- Wedeterminethemass-radiusrelationofthehybridstar stant,thehypothesisthatthree-flavourquarkmatterhasanenergyper baryonlowerthanthatofordinarynucleiholdsfor59MeV/fm3(cid:46)B(cid:46) bytheintegrationofthebrane-modifiedTOVequationsfor 92MeV/fm3. differentinitialvaluesofthecentralpressure pc.Ourresults 10 3 3 observationalrange PSRJ0348+0432 2.5 QS,B=60(GR) 2.5 λ=6×103,w=0 w=1 2 w=−1 2 w=−0.6 (cid:2) w=−0.51 M(cid:2) M/M1.5 /max1.5 M 1 1 QS,B=60,w=0 w=5 0.5 0.5 w=−1 w=−0.7 w=−0.5 0 0 0 2 4 6 8 10 12 14 16 18 20 103 104 105 106 R[km] λ[MeV/fm3] 3 3 QS,B=60(GR) QS,B=60,λ=5×102 λ=6×103,w=0 λ=103 2.5 w=1 2.5 λ=5×103 w=−1 λ=104 w=−0.6 λ=5×104 2 w=−0.51 2 (cid:2) (cid:2) M M/M1.5 /max1.5 M 1 1 0.5 0.5 0 0 5 10 15 20 25 30 35 40 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ρc/ρ0 w Fig.7 Topplot:Mass-radiusrelationforaquarkstar,usingthephe- Fig.8 Topplot:Maximumquarkstarmassasafunctionofthebrane nomenologicalEoSinEq.(39).Thecurvesaregivenfordifferentval- tensionfordifferentvaluesofthedarkEoSparameterw.Bottomplot: uesofthedarkEoSparameterwandλ =6×103 MeV/fm3.Bottom Maximum star mass as a function of the dark EoS parameter w for plot:Themassofthequarkstarversusthecentralenergydensityρc differentvaluesofthebranetension. (inunitsofρ0). arepresentedinFigs.9and10.FromthetopplotofFig.9, we conclude that the maximum mass and the correspond- persistsforcertainnegativevaluesofw,reachingthemaxi- ing star radius decrease as λ decreases. Furthermore, it is mummassatrelativelyhighcentraldensities,ρ ∼45ρ ,as c 0 seen that for λ (cid:38)8×102 MeV/fm3, and taking w=0, the depictedinthebottomplotofFig.10. massesandradiiobtainedareinagreementwiththeobser- vational constraints. The maximum mass M ∼ 1.98M(cid:12) is InFig.11,themaximumstarmassisplottedasafunc- obtainedforGR(i.e.inthelimitλ →∞),andthisvalueis tionofλ (toppanel)andw(bottompanel).Weseethatthe consistent with the observational range of the pulsar PSR maximum star mass predicted for the hybrid star model is J0348+0432 [39]. This is to be compared with the maxi- compatible with observations provided λ (cid:38)103 MeV/fm3. mummassof2.29M(cid:12)attainedwiththeNSBSk21EoS(see Thebehaviourofthemaximummasswithwissimilartothe Fig.3).Thecentralenergydensitiesareρc(cid:46)6.5ρ0(bottom oneobtainedforpureneutronandquarkstars.Forw(cid:46)−0.5 panel of Fig. 9). The allowed star radii are in the range 8– and w(cid:38)−0.1, the value of M remains essentially con- max 13km,similartothoseobtainedforpureneutronstars. stant with the variation of w, depending only on the value In Fig. 10, we present the mass-radius relation for the of λ.Yet,themaximummassisverysensibletothe varia- case of λ =6×103 MeV/fm3 and different values of the tion of w in the range −0.3<w<−0.1 (as in the case of darkEoSparameterw.Asinthecaseofquarkstars,weno- pureneutronstars).Thisisclearlyseeninthebottomplotof tice that the clockwise bending of the mass-radius curves Fig.11.