Proca stars: Gravitating Bose-Einstein condensates of massive spin 1 particles

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Proca

stars:

Gravitating

Bose–Einstein

condensates

of

massive

spin

1

particles

Richard Brito

a

,

∗

,

Vitor Cardoso

a

,

b

,

Carlos

A.R. Herdeiro

c

,

Eugen Radu

c

a_{CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico–IST,UniversidadedeLisboa–UL,AvenidaRoviscoPais1,1049,Lisboa,Portugal} b_{PerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2J2W9,Canada}

c_{DepartamentodeFísicadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received24October2015

Receivedinrevisedform16November2015 Accepted17November2015

Availableonline28November2015 Editor:M.Trodden

We establish that massive complex Abelian vector fields (mass

μ

) can form gravitating solitons, when minimally coupled to Einstein’s gravity. Such Procastars (PSs) have a stationary, everywhere regular

and asymptotically flat geometry. The Proca field, however, possesses a harmonic time dependence (frequency w), realizing Wheeler’s concept of geons for an Abelian spin 1 field. We obtain PSs with both a spherically symmetric (static) and an axially symmetric (stationary) line element. The latter form a countable number of families labelled by an integer m∈ Z+. PSs, like (scalar) boson stars, carry a conserved Noether charge, and are akin to the latter in many ways. In particular, both types of stars exist for a limited range of frequencies and there is a maximal ADM mass, Mmax, attained

for an intermediate frequency. For spherically symmetric PSs (rotating PSs with m=1,2,3), Mmax

1.058M2

Pl/

μ

(Mmax1.568,

2

.337,

3

.247 M

2

Pl/

μ

), slightly larger values than those for (mini-)boson stars.

We establish perturbative stability for a subset of solutions in the spherical case and anticipate a similar conclusion for fundamental modes in the rotating case. The discovery of PSs opens many avenues of research, reconsidering five decades of work on (scalar) boson stars, in particular as possible dark matter candidates.

©2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

Introduction. Accordingtothelatestcosmologicaldata[1],

∼

26% oftheUniverse’senergycontent isdarkmatter(DM). The funda-mentalnature ofDM, however,is unknown. A widespread view-point is that DMconsists of weakly interactive massive particles (WIMPs);popularcandidatesarethelightestsupersymmetric par-ticles,withmasses



GeV, such asthe neutralino [2]. Despiteits successinmodelingstructureformation,someshortcomingsofthe WIMPsDMmodelariseinsmallscales,suchasthe“missing satel-lite” [3,4] and the “cuspy core” [5] problems. These issues may eventually be solved within the WIMP paradigm; but they sug-gest considering alternative models. Different proposals, some of whichhavebeenclaimedtosolvetheseproblems(see[6,7]for re-views),introduce light orultra-light bosonic particles/fields,with masses



eV, whichmayform, gravitationally,macroscopicBose– Einsteincondensates.Eventhoughsuchproposalshaveessentially a phenomenological character, ultra-light particles (axions)are a natural ingredient of the Peccei–Quinn mechanism to solve the

*

Correspondingauthor.

E-mailaddress:[email protected](R. Brito).

strong CP problem [8] and may be motivated at a fundamental levelbystringtheoreticalconstructions- theaxiverse[9].

Gravitationally-bound bosonic structures are thus relevant in the context of DM searches. The study of (scalar) Bose–Einstein condensatesasDMcandidatesisoftenperformedinaNewtonian limit.Inthefullyrelativisticregime,suchmodelsyieldgravitating solitons: (scalar) bosonstars (SBSs). These objects, initially pro-posedasa(scalar)realization ofWheeler’sgeon idea[10,11],have found avariety ofapplications, fromblackholemimickers in as-trophysics,toparticlemodelsinTeVgravityscenarios (see [12,13]

forreviews).

Inlightofrecentproposalsadvocatingmassivespin1particles asa DMingredient [14–17],the studyofthecorresponding self-gravitatingstructuresandtheirdynamicsisofspecialimportance.1

In this letter we show that much like massive spin 0 particles, massivespin1particlescanclusteraseverywheresmooth, asymp-totically flat lumps ofenergy undertheir own weight, producing gravitatingsolitonswedubProcastars (PSs).

1 _{InsomeDMliterature,axion-likeparticlesandhiddenphotonsarenow} typi-callyreferredtoasweaklyinteractingslimparticles(WISPs)[38].

http://dx.doi.org/10.1016/j.physletb.2015.11.051

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Einstein-(complex)-Procatheory. WeconsideronecomplexProca field, with mass

μ

(or equivalently, two real Proca fields with the same mass). It is described by the potential 1-form

A

, and field strength

F =

d

A

. We denote the corresponding complex conjugates by an overbar,

A

¯

and

F

¯

.The minimal Einstein-(com-plex)-Procamodelisdescribedbytheaction[18,19]:

S

=



d4x

√

−

g



R 16

π

G

−

1 4

F

αβ

F

¯

αβ

₋

1 2

μ

2

_A

α

A

¯

α



.

The Einstein and Proca field equations are, respectively, Gαβ

=

8

π

G Tαβ,

∇

α

F

αβ

=

μ

2

A

β, where the energy–momentum tensor

reads: T_αβ

= −

F

σ(α

F

¯

β)σ

−

1 4gαβFσ τ

F

¯

σ τ

+

μ

2



A(

α

Aβ)

¯

−

1 2gαβAσ

A

¯

σ



.

The Proca equationsimplythe Lorentz condition (which isnot a gaugechoice,butadynamicalrequirement):

∇

α

A

α

=

0

.

The global U

(1)

invariance of the action, under the transfor-mation

A

μ

→

eiα

A

μ, with

α

constant,impliestheexistenceofa

conserved 4-current, jα

=

₂i

 ¯F

αβ

_Aβ

₋

_F

αβ

_Aβ

¯



_{. From the Proca} equation,

∇

α jα

=

0. Thus a conserved Noether charge exists, Q ,

obtainedintegratingthe temporalcomponentofthe 4-currenton aspace-likeslice

:

Q

=

_d3_x

√

₋

_{g j}t_.

SphericallysymmetricPSs. Wefirstconsiderspherically symmet-ricsolutions,takingthefollowinglineelementform

ds2

= −

σ

2

₍

_r

₎

_N

₍

_r

₎

_dt2

₊

dr2

N

(

r

)

+

r

2_d

_

2

,

(1)

whereN

(

r

)

≡

1

−

2m(r

)/

r andtheProcapotentialansatz

A

=

e−i wt[ f

(

r

)

dt

+

ig

(

r

)

dr]

.

(2)

σ

(

r

),

m

(

r

),

f

(

r

),

g

(

r

)

areall realfunctionsoftheradialcoordinate only and w is a real frequency parameter. As for SBSs the har-monictimedependenceof

A

iscrucialandthecomplexnatureof

A

makes it compatible withthe time-independent lineelement. ButunlikeSBSstherearenowtwoindependentradialfunctionsfor the‘matter’field,makingtheanalysismoreinvolved(evenmoreso intherotatingcase).TheProcaequationsyield

d dr

r2

[

f

−

w g

]

σ



=

μ

2r2f

σ

N

,

w g

−

f 

₌

μ

2

σ

2N g w

,

where denotes radial derivative. Thesetwo equationsimplythe Lorentz condition constraint (which determines f

(

r

)

in terms of theremainingfunctions).TheessentialEinsteinequationsread

m

=

4

π

Gr2



(

f

−

w g

)

2 2

σ

2

+

1 2

μ

2



g2N

+

f 2 N

σ

2



,

σ



σ

=

4

π

Gr

μ

2



g2

+

f 2 N2

_σ

2



.

For the ansatz (1)–(2), the Noether charge reads Q

=

4π μ2 w

_∞

0 dr r2g

(

r

)

2

σ

(

r

)

N

(

r

),

andthe energydensitymeasured by astaticobserveris

−

T_tt

=

(

f 

₋

_{w g}

₎

2 2

σ

2

+

1 2

μ

2



g2N

+

f 2 N

σ

2



.

Finally,PSssatisfythevirialrelation:

∞



0 dr r2

σ



μ

2



g2

−

f 2

₍

_4N

₋

₁

₎

σ

2_N2



−

(

w g

−

f

)

2

σ

2



=

0

,

isused totestthenumericalaccuracyoftheresults.Thisrelation can alsobe usedtorule outnon-gravitatingsolutions, i.e.

‘Proca-balls’withoutbackreaction.2

Asymptotic behaviours. An analysis of the field equations both nearthe originandatspatial infinity revealssmooth behaviours. Closetor

=

0 wefind f

(

r

)

=

f0

+

f0 6



μ

2

−

w2

σ

2 0

r2

+

O

(

r4

) ,

g

(

r

)

= −

f0w 3

σ

2 0 r

+

O

(

r3

) ,

m

(

r

)

=

4

π

G f 2 0

μ

2 6

σ

2 0 r3

+

O

(

r5

) ,

σ

(

r

)

=

σ

0

+

4

π

G f₀2

μ

2 2

σ

0 r2

+

O

(

r4

) ,

where f0,

σ

0 are constants, the values of f

(

r

)

and

σ

(

r

)

at the origin,respectively.Asr

→ ∞

,wefind f

(

r

)

=

c0 e−r  μ2_−w2 r

+ . . . ,

g

(

r

)

=

c0 w



μ

2

₋

_w2 e−rμ2_−w2 r

+ . . . ,

m

(

r

)

=

M

+ . . . ,

log

σ

(

r

)

= −

4

π

G c 2 0

μ

2 2

(

μ

2

₋

_w2

₎

3/2 e−2r  μ2_−w2 r

+ . . . ,

where M is the ADM mass and c0 is a constant; observe that

w

<

μ

,whichisaboundstatecondition.

Numerical results. The solutions that smoothly interpolate be-tweenthetwoaboveasymptoticbehavioursarefoundnumerically. In numerics–andinthe following–we set

μ

=

1,4

π

G

=

1, by using ascaled radial coordinater

→

r

μ

(togetherwithm

→

m

μ

,

w

→

w

/

μ

)andscaledpotentials f

→

f

√

4

π

G, g

→

g

√

4

π

G.The equationsaresolved byusingastandardRunge–KuttaODEsolver and implementing a shootingmethod interms of the parameter

f

(0).

In Fig. 1 we plotthe ADM mass, M, andthe Noether charge,

Q , asa function of theProca field frequency, w.As w

→

1, the mass andNoethercharge ofthesolutions vanish,but Q

/

M

→

1. Approachingthislimit,PSsbecomespatiallydilutedwithan effec-tivesizemuchlargerthantheirSchwarzschildradii,3 andtrivialize at w

=

1.Reducing w fromthismaximalvalue,PSsbecomemore compact; both M and Q follow a spiral, towards a central criti-calconfigurationlocatedaroundw



0.891.Themaximumofboth

M and Q occursfor wmax



0.875,withMmax



1.058

<

Qmax



2 _{It was recentlyshown that self-interactions can, however,allow such} non-backreacting‘Proca-balls’[39],inanalogywiththeMinkowskispaceQ -ballsfound inthescalarcase[40].

3 _{Itshouldbenoticedthatthesegravitationalsolitonsdonotpossessasurface,} sovariousdefinitionscouldbetakenfortheeffectiveradius.Foradiscussionon apossibledefinitionofaneffectiveradiusforSBSssee,e.g.[36].Foradefinition oftheeffectiveradiussuchthatitcontains98% ofthetotalmass,Procastarsat theminimalstablefrequencyarearoundfivestimeslargerthantheirSchwarzschild radius(seealso[27]).

Fig. 1. ADMmassM andtheNoetherchargeQ forsphericalPSsvs. theProcafield frequency w.TheinsetshowsM vs. theshootingparameterf(0),whichis analo-goustothecurveforM vs. thecentralvalueofthescalarfieldforSBSs.

Fig. 2. Energyand Noetherchargedensities,electricfield(Er= |Frt|)andprofile

functions(inset)ofastablePSsolutionwithM=1.034,w=0.9,f(0)=0.061. 1.088. Weobserve that M

<

Q from w

=

1 downto almost the minimalallowedfrequency.InthelowerpartofthespiralM

>

Q ,

andthe binding energy 1

−

M

/

Q (in units

μ

=

1) is negative,4

correspondingtoaregionof“energyexcess”,wherethesolutions mustbeunstableagainstperturbations.

Weremarkthatforthe familyofsolutionsexhibitedinFig. 1,

f

(

r

)

hasonenode5 _andg

₍

_r

₎

_{isnodeless.Thesearethe}

fundamen-tal solutions. There are also excited solutions, with more nodes forthesefunctions,butwedidnot attempttostudytheminany systematicway.StudiesofSBSs suggestthat excitedsolutions are unstable [20]. In Fig. 2 we show the behaviour of physical and profilefunctionsforatypicalPSsolution.Inparticularobservethe energyconcentrationnearr

=

0.

Stability. Apositive binding energy–as thatobserved along the upperbranch ofFig. 1 – doesnot guarantee linear stability.For SBSs,a perturbative stabilityanalysisshowsthatthesolutionsare onlystablefromthemaximalfrequencyw

=

1 downuntilthe fre-quencyatwhichthemaximalmassisattained, wmax.Beyondthis

4 _{Notethat Q represents thenumber offrees bosonparticles,and so Qμis} themassoftheseQ freeparticles.ComparingQμwiththegravitationalmassM, yieldsthedifferenceinmass/energybetweenfreebosonparticlesand gravitation-allyboundbosonparticles.Itisthennaturalthatapositive1−M/(Qμ)isrequired forstability.

5 _{Theexistenceofanodeisnotobviousfromtheplots;butonecanshow} ana-lyticallythat,foranysphericalPS,f(r)changessignatleastonce.Thisdiffersfrom SBSs,forwhichfundamentalmodesarenodelessinthe(single)profilefunction.

Fig. 3. Fourierfrequencyoftheperturbationsasfunction ofthePS’stotalmass. ThecriticalmassatwhichPSsbecomeunstablecorrespondstothestar’smaximum mass.

pointanunstablemodedevelops[21,22].Ananalogousconclusion holds forPSs, aswenow show by considering their linearradial perturbations.

Weassumethatallperturbationshaveaharmonictime depen-denceoftheforme−it_,with

_

_{beingthecharacteristicvibrational}

frequenciesofthePS.Gaugefreedomallowstowritetheperturbed metricas: ds2

= −

σ

2

(

r

)

N

(

r

)



1

−

h0

(

r

)

e−it



dt2

+

dr 2



₁

₊

_h 1

(

r

)

e−it



N

(

r

)

+

r 2_d

_

2

_;

₍₃₎

thevectorfieldisperturbedas

A

=

e−i wt



f

(

r

)

+

e−it

f1

(

r

)

+

i

f2

(

r

)

r



dt

+



ig

(

r

)

+

e−it

g1

(

r

)

+

i

g2

(

r

)

r



dr



,

(4)

where h0, h1, f1, f2, g1 and g2 are radial perturbations around the background solution,and

is a small, bookkeeping parame-ter.Theperturbed Einstein–Procasystemcanbereducedtoapair ofsecondorderODE’s.Imposingregularityoftheperturbationsat the origin and at infinity, the resulting system is a two dimen-sionaleigenvalueproblemfor



andoneotherconstantwhichwe havechosen to be thevalue of h0 atthe origin. A numerical so-lution isthen obtained by a two dimensional shooting, withthe resultshowninFig. 3,whereweplot



2versusthePS’sADMmass aroundthemaximalmass.For



2

>

0 themodeispurelyrealand corresponds to stable normal modes. This occurs for w

>

wmax.

For



2

<

0 the mode isa purely positive imaginarynumber, in-dicating that these configurations are unstable (cf. Eqs. (3) and

(4)).Thus,incompleteagreementwiththeSBSscase[21–24],we can see that the star’s maximum mass corresponds to a transi-tionpointbetweenstableandunstableconfigurations.Forunstable configurationswithmass0.1% belowthethreshold,theinstability timescale

τ

is already smaller than

τ



100M. Thus, we expect theunstablebranchofPStosharesimilarpropertiestothose ob-served inthe scalarcase: configurationswhich reachthis branch will eitherquicklycollapseto blackholes ormigrateback tothe stablebranchviamassejection(the“gravitationalcooling” mech-anism)[23,25–27].

RotatingPSs. We now turn to rotating PSs. These solutions are foundwithanaxi-symmetricmetricansatz

ds2

= −

e2F0(r,θ )_dt2

+

_e2F1(r,θ )

₍

_dr2

+

_r2_d

_θ

2

₎

+

e2F2(r,θ )_r2_sin2

_θ



d

ϕ

−

W

(

r

, θ )

r dt



2

.

The Procafield ansatz is giveninterms ofanother four func-tions

(

Hi

,

V

)

whichdependalsoonr

,

θ

A

=



H1 r dr

+

H2d

θ

+

i H3sin

θ

d

ϕ

+

i V dt



ei(mϕ−wt)

,

(5)

with m

∈ Z

+. The Einstein–Proca equations are solved with the followingboundary conditions,whichwe have foundto be com-patible with an approximate construction of the solutions on the boundary of the domain of integration:

∂

rFi



_r=0

=

W



_r=0

=

Hi

|

r=0

=

V

|

r=0

=

0; also Fi



_r=∞

=

W



_r=∞

=

Hi

|

r=∞

=

V

|

r=∞

=

0; and

∂θ

Fi



_θ=0,π

= ∂

θW



_θ=0,π

=

Hi

|

θ=0,π

=

V

|

θ=0,π

=

0 (note

that for m

=

1 the conditions for H2

,

H3 are different, with

∂θ

H2



_θ=0,π

= ∂

θH3



_θ=0,π

=

0). All rotating PSs we have con-structedso farare symmetricw.r.t. a reflectionalong the equato-rialplane. Odd-paritycompositeconfigurations,however,are also likelytoexist.

As usual, the ADM mass M and angular momentum J are

readoff fromtheasymptoticexpansion, gtt

= −

1

+

2G M/r

+ . . .

, gϕt

= −

2G J sin2

θ/

r

+ . . .

.In analogywiththe SBSscase [28,29],

one can show that the angular momentum is a multiple of the Noethercharge, J

=

m Q ,butincontrasttotheSBSscase,the an-gularmomentumdensityT_{ϕ and}t theNoetherchargedensity jtare notproportional anylonger.Regularity oftheProcastarsimposes thatthenumberm isaninteger. Asaresult, thereisacountable numberof1-parametercontinuousfamilies ofaxisymmetric solu-tionsintermsofm (withm

=

0 thesphericallysymmetriclimit). However,foreachintegerazimuthalharmonicindex,m,thereisa continuoussetofsolutionsintermsof w.

The Einstein–Proca equations for this rotating case are quite involved and shall not be presented here. They are solved nu-merically, subjectto theabove boundaryconditions,byusingthe elliptic PDE solver FIDISOL [30] based on the Newton–Raphson procedure.6 InFig. 4weexhibit themassvs. angularmomentum, togetherwiththemassandtheNoethercharge vs. theProcafield frequency, for rotating PSs with m

=

1,2,3 (inset). One can see thatM and J arealwayspositivelycorrelated.7 Thephysical quan-titiesforthelargestmassPSsareexhibitedinthefollowingtable, for m

=

0 (spherical) and m

=

1,2,3 (rotating), where they are comparedwiththeresults intheliterature forthecorresponding largestmassSBSs(lastcolumn);oneobservesthatMmax isalways

smallerforthelatter.

wmax Mmax Qmax Jmax MmaxSBSs

m=0 0.875 1.058 1.088 0 0.633[10] m=1 0.839 1.568 1.626 1.626 1.315[29,31] m=2 0.767 2.337 2.461 4.921 2.216[31] m=3 0.667 3.247 3.489 10.467 not reported

Some physical properties of the rotating PSs mimic those of SBS:forexample,therotatingPSssolutions spatiallydelocalizeas

w

→

1 (similarly to the static case) and trivialize in that limit; also,form

=

1,neithertheenergydensitynortheNoethercharge densityvanishonthesymmetryaxis.Novelfeatures,however,also

6 _{Weuseascalingsimilartothatdescribedinthesphericalcase.Weestimate} thetypicalerrorofthesolutionstobesmallerthan1partin104_{.Inthespherical} casemuchhigheraccuraciesareobtained;e.g.,thevirialidentityistypicallyobeyed to1partin1012_.

7 _{Withincreasingm itbecomesincreasinglydifficult,numerically,tomovefurther} alongthespiral.

Fig. 4. M vs. J diagramforrotatingPSswithm

=

1,2,3.TheinsetshowsM (solid redlines)andQ (dashedgreenlines)vs. w,wherem=0 correspondstothe spher-icalPSsinFig. 1.

appear.Asanexample,thedistributionoftheenergy–momentum tensor (and Noether charge) is more intricate than the “mass torus”foundforrotatingSBSs[28]andexhibitsacomposite struc-ture with more than one torus; moreover, the existence of field linesleadstonew(qualitative)features.Theseaspectswillbe dis-cussedindetailelsewhere.

We have not attempted to studyin detail the stability of ro-tating PSs. For rotating SBSs, catastrophe theory arguments [32]

support a similar conclusion to that obtained in the static case: thesolutions arestablefromthemaximal frequencydowntothe point where the maximal mass is attained. We expect a similar conclusion toholdforrotatingPSs. Intherotatingcasethere can also occur ergoregion instabilities[33]; forSBSs ergoregionsonly appearintheunstablebranchofsolutionsandan analogous situ-ationistrueforPSs.8

Discussionand outlook. The discovery of PSs suggests various novel directions of research, besides the detailed analysis of the solutions presented herein; let us mention two. 1) In the scalar case,asingle(ratherthancomplex)realfieldallowstheexistence ofverylonglivedquasi-solitons–oscillatons;thesameistruefor theProcacase[27].2)CanoneaddablackholeinsideaPS,likefor othergravitatingsolitons?Whileforsphericallysymmetric config-urations onecanshowtheabsenceofblackholesolutions,thisis possibleinsidea spinningPS,againincompleteanalogywiththe SBScase[34–36].Thesesolutionscircumventstandardno-hair the-orems(e.g.[37])sinceaProcafieldoftheform(5),doesnotshare theisometriesofastationaryandaxi-symmetricspacetime.

A final remark relating PSs with DM phenomenology. Since ordinary matter, which constitutes only

∼

5% of the Universe’s energy content, is comprised ofover ten elementary particles, it is reasonable to admit that DM is composed of different kinds offundamentalentities,butwhich,whengravitationallyclustered intomacroscopiclumps,displaysomeuniversality.IfBose–Einstein condensates ofultra-light scalarfieldsare viableDMmodels, the existence ofPSs akintoSBSssuggeststheformermaybeanother

DMcomponent.

Acknowledgements

R.B. acknowledges financial supportfromthe FCT-IDPASC

pro-gram through the grant SFRH/BD/52047/2012, and from the

8 _Form

₌

_{1,2,3 anergo-region –actuallyanergo-torus–ariseson thefirst} branchfor w

=

0.745,0.671,0.648,respectively,thusforw smallerthanthe cor-respondingwmax.

Fundação Calouste Gulbenkian through the Programa Gulbenkian de Estímuloà Investigação Científica.V.C. acknowledges financial support provided under the European Union’s FP7 ERC Starting Grant “The dynamics of black holes: testing the limits of Ein-stein’s theory” grant agreement no. DyBHo-256667, and H2020 ERCConsolidatorGrant“Matterandstrong-fieldgravity:New fron-tiersin Einstein’s theory” grant agreement no. MaGRaTh-646597.

C.H. and E.R. thank funding from the FCT-IF programme and

the grants PTDC/FIS/116625/2010,

NRHEP-295189-FP7-PEOPLE-2011-IRSES.Thisresearch wassupportedinpartbythePerimeter InstituteforTheoreticalPhysics,andbytheGovernmentofCanada throughIndustry Canada andby theProvince of Ontariothrough theMinistryofEconomicDevelopment& Innovation.

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