Black hole spontaneous scalarisation with a positive cosmological constant

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Physics

Letters

B

www.elsevier.com/locate/physletb

Black

hole

spontaneous

scalarisation

with

a

positive

cosmological

constant

Yves Brihaye

a

,

∗

,

Carlos Herdeiro

b

,

Eugen Radu

b a_{Physique-Mathématique,UniversitedeMons-Hainaut,Mons,Belgium}

b_{DepartamentodeMatemáticadaUniversidadedeAveiroandCentreforResearchandDevelopmentinMathematicsandApplications(CIDMA),Campusde} Santiago,3810-183Aveiro,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received22October2019

Receivedinrevisedform21November2019 Accepted3January2020

Availableonline3February2020 Editor:M.Cvetiˇc

A scalar field non-minimally coupled to certain geometric [or matter] invariants which are sourced by [electro]vacuum black holes (BHs) may spontaneously grow around the latter, due to a tachyonic instability. This process is expected to lead to a new, dynamically preferred, equilibrium state: a scalarised BH. The most studied geometric [matter] source term for such spontaneous BHscalarisation is the Gauss-Bonnet quadratic curvature [Maxwell invariant]. This phenomenon has been mostly analysed for asymptotically flat spacetimes. Here we consider the impact of a positive cosmological constant, which introduces a cosmological horizon. The cosmological constant does not change the local conditions on the scalar coupling for a tachyonic instability of the scalar-free BHs to emerge. But it leaves a significant imprint on the possible new scalarised BHs. It is shown that no scalarised BH solutions exist, under a smoothness assumption, if the scalar field is confined between the BH and cosmological horizons. Admitting the scalar field can extend beyond the cosmological horizon, we construct new scalarised BHs. These are asymptotically de Sitter in the (matter) Einstein-Maxwell-scalar model, with only mild difference with respect to their asymptotically flat counterparts. But in the (geometric) extended-scalar-tensor-Gauss-Bonnet-scalar model, they have necessarily non-standard asymptotics, as the tachyonic instability dominates in the far field. This interpretation is supported by the analysis of a test tachyon on a de Sitter background.

©2020 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.

1. Introduction

The ground state of Einstein’s gravity with a positive cosmo-logicalconstant is

de Sitter (dS)

spacetime. Solutionsof Einstein’s gravity, orgeneralisationsthereof, withdS asymptoticsare of in-terest forvarious reasons.Firstly and foremost,observational ev-idence supports that our Universe is undergoing an accelerated expansion [1,2]. The simplest theoretical modelling of such ob-servations consistson assuming a smallpositive vacuumenergy,

i.e. a cosmologicalconstant



>

0,implyingthephysicalUniverse is asymptotically dS. Secondly, dS spacetime plays a central role in the theory of primordial inflation, the very rapid accelerated expansion in the early Universe, which is now part of the stan-dard cosmological model. Finally, from a theoretical perspective, theproposalofaholographic dualitybetweenquantumgravityin

*

Correspondingauthor.

E-mailaddress:Yves.Brihaye@umons.ac.be(Y. Brihaye).

dSspacetimeandaconformalfieldtheory ontheboundaryofdS spacetime[3,4] furtherstimulatedtheanalysisofasymptoticallydS spacetimes.

Withintheclassical solutionsofgravitatingfieldsin asymptot-ically dS spacetimes, the case of black holes (BHs) is especially interesting,asBHsare,inmanyways,thegravitationalatoms.One may wonder, for instance, how much dS asymptotics may spoil thecelebratedsimplicityofBHsin electrovacuumgeneral relativ-ity [5], wherefamouslyBHs haveno hair,inthesense they have no multipolar freedom.As in theasymptotically flat casebeyond electrovacuum [6], including additional degrees of freedom and couplings allows a richer landscape ofdS BHs. Letus give some examples.

Concerningscalar hair, a numberofno-hair resultsapplicable forrealscalarfieldsinasymptoticallyflatBHsstill holdfor



>

0 [7–10].Thiscovers,forinstance,modelswithapositive semidefi-nite,convexscalarpotential;orevennon-minimallycoupledcases, provided the scalar field potential is zero or quadratic [11]. BHs

https://doi.org/10.1016/j.physletb.2020.135269

0370-2693/©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

withscalar hair exist, nonetheless,if the scalar field potential is non-convex[9].Remarkably, foraconformallycoupledscalarfield witha quartic self-interaction potential there is an exact (closed form) hairy BH solution [12]. As another example, dS BHs with Skyrmehairhavebeenreportedin [15].Ontheflipside,somewhat unexpectedly, spherically symmetric boson stars, which are self-gravitating,massive,complexscalarfields [13], donotpossessdS generalisations [7],whichmaypreventtheexistenceof asymptoti-callydSBHswithsynchronisedhair [14].Turningnowtothecase of vector hair, dS BHs with Yang-Millshair have been discussed in [16,17], while dS BHs with (real) Proca hair are not possi-ble [10].Finally,sphaleronsand(non-Abelian)magneticmonopoles insidedSBHsarediscussedin [18].

The existence of a hairy BH solution does not guarantee per se any sort of dynamical viability of such solution, which is, of course, key for the physical relevance of the BH. But a quite generic dynamical mechanism to obtain new hairy BHs that co-exist andare dynamically preferredto the standard General Rel-ativity (GR) electrovacuum BH solutions of Einstein’s gravity has beenrecentlyunderscrutiny:thephenomenonofBH

spontaneous

scalarisation. This phenomenon is induced by non-minimal cou-plings which allow circumventing well-known no-hair theorems. Thenon-minimal coupling istypically betweenarealscalar field

φ

and some source term

I

, which can trigger a repulsive gravi-tational effect,via an effective tachyonicmass for

φ.

As a result, the GR solutions are unstable against scalar perturbations in re-gionswherethesourcetermissignificant,dynamicallydeveloping scalarhair,

i.e. spontaneously scalarising.

Variousexpressionsof

I

havebeenconsideredintheliterature, that fallroughly into twotypes:

I

is a geometricinvariant, such as the Gauss-Bonnet invariant [19–35], the Ricci scalar for non-conformallyinvariantBHs [36],ortheChern-Simonsinvariant [37]; or

I

isa“matter”invariant,suchastheMaxwell

F

2 _{term[38–45].} Thisphenomenon isactually notexclusive ofscalarfields [46].It wouldbethereforeinterestingtounderstandtheimpactofa pos-itivecosmologicalconstantinthisphenomenon,andifitcanlead, dynamicallytohairyBHs ina dSUniverse.Workinthisdirection wasreportedin [47].

The goal of this paper is to assess the impact of a positive cosmological constant in the BH spontaneous scalarisation phe-nomenon, considering the two paradigmatic cases in the litera-ture, butaugmented with



>

0. We shallthen focus onBHs in Einstein-Maxwell-scalar- (EMS-) and extended-Scalar-Tensor-Gauss-Bonnet-scalar-(eSTGB-) models,which,for



=

0,both allow for BH scalarisation to occur. As we shall see, the impact of the positive cosmologicalconstant issubstantially differentin thetwocases,whichisrelatedtothe natureofthetachyonic in-stability,which forthematter modelis asymptotically quenched, leading to scalarised asymptotically dS charged BHs, but for the geometricmodel itisnot, leadingto anon-asymptoticallydS ge-ometry.

Thispaperisorganisedasfollows.Insection2we discussthe generalframework, introduce thetwo models andthe ansatz for the fields, discuss the conditions for scalarisation to occur and scalarised BHs to exist, providing the choice of the non-minimal couplingthatshall beusedinour work.Wealso analysethe be-haviour of a tachyonic scalar field on dS spacetime that will be relevantforourresults.Weendthissectionwithano-gotheorem forsmooth scalarhairconfinedbetweentheBHandcosmological horizon.In sections3and4wedescribe, respectively,thematter andthegeometricmodel.Ineachcasewestartwiththe construc-tion of the zero modes, the scalar clouds on the scalar-free BH, andthendiscusssome propertiesofthenon-linear scalarisedBH solutions.Section5providessomefinalremarks.

2. Thegeneralframework 2.1. Models and ansatz

The considerations in this work apply to a family of models describedbythefollowingaction(setting

c

=

G

=

1):

S

= −

1 16

π



d4x

√

−

g



R

−

2



−

2

(

∇φ)

2

−

f

(φ)

I

(ψ

;

g

)



,

(2.1) where R is the Ricci scalar,



>

0 is the cosmological constant,

φ

is arealscalar field, f

(φ)

isthe

coupling function and

I

isthe

source term. The lattermaydependonly onthe spacetimemetric

gμν oralsoonextramatterfields,collectivelydenotedby

ψ

.The correspondingequationofmotionforthescalarfieldandthe met-rictensorread

2φ =

f,φ

I

4

,

(2.2) Rμν

−

1 2gμν

+ 

gμν

=

2Tμν

,

where Tμν

=

T (φ) μν

+

Tμν(ψ )

.

(2.3) Here, T(φ)μν

= ∂

μ

φ∂

ν

φ

−

12gμν

(

∇φ)

2 _{is the scalar field} energy-momentumtensor,whereas Tμν is (ψ ) theenergy-momentumtensor associated with the source term inthe action (2.1). These equa-tions must,ofcourse,be supplementedwiththose describingthe dynamicsofthematterfields

ψ

,iftheyarepresent.

To be more concrete, we shall focus on two specific models within the family (2.1), corresponding totwo differentchoicesof sourceterm

I

.Theseare:

i) a

“matter” source:

I = L

M

≡

Fμν F μν , with

ψ

=

Aμ and Fμν

=

∂

μ Aν

− ∂

ν Aμ,

ii) a

geometric source:

I = L

G B

≡

R2

−

4Rμν Rμν

+

Rμνρσ Rμνρσ .

We shall refer to these models, respectively, as the Einstein-Maxwell-Scalar-(EMS-)modelandtheextended

Scalar-Tensor-Gauss-Bonnet- model (eSTGB-). For the former model, the

equationsofmotion (2.2)-(2.3) aresupplementedbytheMaxwell equationsfortheelectromagneticfield

∂

μ

(

√

−

g f

(φ)

Fμν

)

=

0

,

(2.4)

whiletheenergy-momentumtensorassociatedtothesourceterm reads T(ψ )μν

=

f

(φ)



FμρFνρ

−

1 4gμνFρσF ρσ



.

(2.5)

For the lattermodel, no extra matter fields are present(ψ

=

0), and the energy-momentumtensor associatedto the source term reads T(ψ )_μν

= −

2

α

Pμγ να

∇

α

∇

γf

(φ) ,

(2.6) where Pαβμν

= −

1 4

ε

αβρσR ρσ γδ

_ε

μνγδ

=

Rαβμν

+

gανRβμ

−

gαμRβν

+

gβμRαν

−

gβνRαμ

+

R 2



gαμgβν

−

gανgβμ



.

Inordertofindsolutionsofthemodel (2.1),whateverits con-crete realisation, an appropriate, sufficiently generalansatz must be chosen. In asymptotically dS spacetimes different coordinate

systemsservedifferentpurposes;wechoosestaticcoordinates.The advantageofthese(simple)coordinatesistheirindependenceona certain“time”coordinate,whichisaadapted totheKillingvector fieldwhichistimelike inthestaticpatch. Thiscoordinatesystem iscomputationallyconvenient,sincetherelevantequationsof mo-tion in our problemreduce to ordinary differential equations; it hides,however, thecosmologicalexpansion andthefact that the spacetimeisnotstationary.Themetricansatzinstaticcoordinates isoftheform

ds2

= −

e−2δ(r)N

(

r

)

dt2

+

dr

2

N

(

r

)

+

r

2

₍

_d

_θ

2

₊

_sin2

_θ

_d

_ϕ

2

_{) ,}

_(2.7) whereaconvenientparametrisationofthemetricfunction

N

(

r

)

is N

(

r

)

≡

1

−

2m

(

r

)

r

−



r2

3

.

(2.8)

Emptyde Sitterspacetime corresponds to

δ(

r

)

=

0 and m

(

r

)

=

0. Ithasacosmologicalhorizonatr

=

√

3/.TheSchwarzschild-de Sitter(SdS)solution,ontheotherhand,whichrepresentsaneutral BHinanacceleratingUniversehas

δ(

r

)

=

0

,

m

(

r

)

=

M

=

constant

.

(2.9) Inthe caseof the EMS-model, we shallbe interested in elec-trically charged BHs. Then, an ansatz for the electromagnetic 4-potential must be set. We shall restrict ourselves to a purely electricgaugepotential,

A

=

V

(

r

)

dt

.

(2.10) Thechoices

δ(

r

)

=

0

,

m

(

r

)

=

M

−

Q 2 2r

,

V

(

r

)

=

Q r

,

(2.11) yieldtheReissner-Nordström-deSitter(RNdS)BH,where

M and Q

arethegravitationalmassandthetotalelectriccharge,respectively (whosedefinitionissubtleforadSbackground[51]).Adiscussion ofthissolutioncanbefoundin [52,53].Finallyinallcasesweshall considerthescalarfieldisafunctionof

r only:

φ

= φ(

r

) .

(2.12)

Withtheansatz(2.7),(2.10) and(2.12) weaimatfinding non-singular,asymptotically dS spacetimescontaining aBH.The func-tion N

(

r

)

willhave(at least)two zeros,corresponding tothe BH horizon at r

=

rh

>

0 and the cosmological horizon located at r

=

rc

>

rh

>

0. Both these hypersurfaces are merely coordinate

singularities, where all curvature invariants are finite. A nonsin-gularextension acrossbothofthemcanbe found.Bothfunctions

N

(

r

)

and

e

−2δ(r) _{are strictly positive betweenthesehorizons.We} shallalso assume that all matter fields (together withtheir first and second derivatives) are smooth at both BH and cosmologi-calhorizons.Outsidethecosmologicalhorizon,

N

(

r

)

changessign, suchthat r becomes a timelikecoordinate.Toassurestandard dS asymptotics,werequire

m

(

r

)

→

M asymptotically outsidethe cos-mologicalhorizon, wheretheconstant M is theBH mass,ascan beprovenbyusingthequasilocalformalismandapproachin[48].1 Moreover,weassumethatthemetricfunction

δ(

r

)

vanishesinthe farfield,decayingfasterthan1/r3.The matterfield(s)asymptotic behaviour,onthe otherhand,willresultfromthe fieldequations

1 _{Forthispurpose,theaction(2.1) issupplementedwithaboundarycounterterm,} theBHmassbeingcomputedoutsidethehorizon,atfuture/pastinfinity.

and,asweshallsee,itwillnotalwaysbecompatiblewiththe as-sumedstandarddSasymptotics.

Both the eventandthecosmological horizonshavetheir own thermodynamical properties.Forexample, theHawking tempera-ture,

T

H andhorizonarea AH ofeachhorizonis,

T(_Hh,c)

=

1 4

π

e −δ(r)

_|

_N

₍

_r

₎

_|

r=rh,rc

,

A(_Hh,c)

=

4

π

r2

|

r=rh,rc

.

(2.13) Generically

T

(Hc)

=

T (c)

H ;thustwohorizonsarenotinthermal

equi-librium.

2.2. Conditions for scalarisation and scalarised BHs; choice of f

(φ)

Themechanismallowing foradynamical evolutionbetweena scalar-free BH and a scalarised one is, in principle, the same as forthecaseofasymptoticallyflatBHs.Thishasbeendescribed in variousreferences,

e.g. [

19,38],butweshall brieflyspellit outto keepthispaperself-contained.

Weassumethatthemodeladmits

scalar-free solutions;

thatis,

φ

=

0 isasolutionof(2.2).Thisimpliesthecondition

df d

φ

φ=0

=

0

.

(2.14)

The BHsolution with

φ

=

0 is a standard

-electrovacuum

solu-tion of Einstein’s gravity. For the two models we shall be inter-ested,thescalar-freesolutioniseithertheRNdSBHortheSdSBH. We also assume the model admits scalarised solutions, with

φ

=

0.Thesesolutionsformafamily,thatcanbelabelledbyan ex-traparameter(say,thevalueofthescalarfieldatthehorizon)that iscontinuously connectedtothe scalar-freesolution,approaching itastheextra parameterapproachesthevalue forthescalar-free

-electrovacuum

solution. One can further impose that the lat-tersolutionisunstableagainst scalarperturbations, suchthatthe scalarisedsolutionisdynamicallypreferred.Consideringasmall-φ expansionofthecouplingfunction(sinceoneisdealingwitha lin-earanalysisin

φ)

f

(φ)

=

f

|

φ=0

+

1 2 d2_f d

φ

2

_φ=0

φ

2

₊

_O

_(φ

3

_{) ,}

_(2.15) thelinearisedformof(2.2) reads

(

2 −

μ

2_eff

)φ

=

0

,

where

μ

2_eff

=

1

4 d2_f d

φ

2

φ=0

I

.

(2.16) Thus, thescalar-free solutionisunstable if

μ

2

eff

<

0; thatis there is a tachyonic instability triggered by a negative effective mass

squaredofthescalarfield.

Taking into account ourspecific models, we note that for the RNdSBH,

I

=

F_μνFμν

= −

Q 2 r4

<

0

,

(2.17) whereasforaSdSBH,

I

=

R2

−

4RμνRμν

+

RμνρσRaμνρσ

=

48M2 r6

+

8 3



2

_>

₀

_.

_(2.18)

Now weneedaspecificchoiceofthecouplingfunction f

(φ).

We shallfocusonaquadraticcouplingfunction,thesimplestfunction thatcontainsthenecessarytermin (2.15):

f

(φ)

=

a0

−

αφ

2

.

(2.19) The first constant is taken asa0

=

1 for the EMS- model and an arbitraryvalueforthe eSTGB-case.The second constant,

α

, defines the sign of d2_f

_(φ)/

_d

_φ

2_{, and hence that of}

_μ

2

eff. In fact,

μ

2

eff

= −

α

I/2.

From (2.16)-(2.19),theexistence ofatachyonic in-stabilityrequires

α

<

0 for EMS

−

and

α

>

0 for eSTGB

− .

(2.20) Observe that

α

is dimensionless for the EMS- model and has dimension

[

length

]

2 fortheeSTGB-model.2

Solving (2.16) on the

-electrovacuum

BH spacetimes and the above coupling function is an eigenvalue problem. The solu-tionsthatobeytheappropriateboundaryconditionsdescribezero modesor

scalar clouds.

Foreachchoice of

I

,they existfora spe-cific(discrete)setofglobalcharges.Theselinearzeromodesmark theonset oftheinstability triggeredby thescalar field perturba-tionandthebranchingofftowardsanewfamilyoffullynon-linear solutionsdescribingscalarisedBHs.

Ensuring the above instability of the scalar-free solutions can one reallyguarantee the existence ofa newset ofscalarised so-lutions?Although thiscan only be done by explicitly computing thelatter, some Bekenstein-typeidentitiesputconstraints onthe modelsthatcanhavescalarisedsolutions.Letusprovidethree ex-amples.

Asa firstexample,weintegrate eq. (2.2) alongahypersurface

V bounded bytheBHhorizonandthecosmologicalhorizon.Since thecontributionoftheboundaryterms vanishesforsmooth con-figurations,thisresultsintheidentity



V

d4x

√

−

g f,φ

I

=

0

.

(2.21) Assuming that the source term

I

does not change the sign be-tweentheBHandcosmologicalhorizons,whichistrueinthetest fieldlimitforthespecificmodelsdescribedabove,thisidentity im-pliesthat f,φ,whichequals

−

2

α

φ

forchoice (2.19),hastochange sign in the interval rh

<

r

<

rc fornon-trivial scalar fields to be

possible.Thus, thenumberofnodes

k

∈ N

0 ofthescalar field in betweenthe two horizons must be

k



1. In thiswork, for sim-plicity, we shall focus on solutions with the minimal number of nodes,

k

=

1.Ontheother hand,the eq. (2.21) excludesthe exis-tenceofregularsolutionsforseveralusualchoicesofthecoupling function,3e.g. f

(φ)

=

eαφ _{or f}

_(φ)

₌

_α

_φ

2n+1 _(with

_{n an}

_integer).

Asa second example,wemultiply eq. (2.2) by f,φ.After inte-gratingbypartsandusingthedivergencetheorem,thisresult is



V d4x

√

−

g



f,φφ

(

∇φ)

2

+

1 4f 2 ,φ

I



=

0

.

(2.22)

Again,ifthesourceterm

I

doesnotchangethesignbetweenthe BHandcosmologicalhorizonthisidentity requires f,φφ and

I

to havetheoppositesigninsomeintervalbetweenthetwohorizons, for a non-trivial scalar field profile to exist. From (2.16) and for our coupling thisis precisely the requirement that

μ

2

eff is nega-tive. Thus, a non-tachyonicscalar field with

μ

2

eff

>

0 everywhere 2 _{Inthisworkweshallplotvariousquantitieswhichareinvariantunderascaling} oftheradialcoordinater→ λr (withλ>0),andfortheeSTGB-model,also

α

→ α/λ2_{(andvariousglobalquantitiesscalingaccordingly).}

3 _{Thisobservation,togetherwiththeresultsinSection2.4,provideapartial} ex-planationforthenegativeresultsreportedinRef. [47].Theabsenceofsolutions therealsoforaquadraticcouplingfunction(2.19),canpresumablybeattributedto achoiceoftheinputparametersinthenumericalapproachoutsidethedomainof existenceofsolutions.

cannot yield scalar hair (at least asa test field on the standard

-electrovacuum

BHs).

Athird,related,exampleisfoundbymultiplying(2.2) by

φ,

the integrationresultingin



V d4x

√

−

g



(

∇φ)

2

+

1 4

φ

f,φ

I



=

0

.

(2.23)

Similarly,thisnowimpliesthat

φ

f,φ and

I

musthavetheopposite sign somewherein theinterval

r

h

<

r

<

rc.For ourcouplingthis

leadstothesameconclusionastheidentity (2.22).

2.3. A tachyon on dS spacetime

From the above discussion, a scalar field must have a tachy-onic behaviour somewhere in betweenthe BH andcosmological horizon,forscalarhairtoexist.Whatistheasymptoticbehaviour, beyondthecosmologicalhorizon,ofsuchatachyon?Thisquestion, whichimpactsonourfindingsofthenextsections,canbetackled byconsideringthemassiveKlein-Gordonequation,

(2

−

μ

2

_)φ

₌

_0, with

μ

2

₌

_{constant, asa test field on an empty de Sitter} space-time. Aclosed formsolutioncan be found,whichconsistsof the sumoftwomodes:

φ (

r

)

=

1 rPu



r rc



+

s rQu



r rc



,

where u

≡

3

χ

−

1 2 and

χ

≡

1

−

4

μ

2 3



.

(2.24)

Here, Pu,

Q

u areLegendrefunctionsand

s is

anarbitraryconstant.

Bothtermsintheabovesolutiondivergeat

r

=

0;

Q

u

(

r

/

rc

)

also

di-vergesatthecosmologicalhorizon,locatedat

r

=

rc

=

√

3/.Thus, inwhatfollowswetake

s

=

0.Then,thesolutioninthe neighbour-hoodofthecosmologicalhorizonexpandsas

φ (

r

)

=

1

rc

−

μ

2

2

(

r

−

rc

)

+

O

(

r

−

rc

)

2

_.

_(2.25)

For

r

rc,ontheotherhand,theapproximateformof

φ (

r

)

is

φ (

r

)



c₊r−32(1+χ)

+

_c−r−32(1−χ)

_,

where c_±

≡

r 1±3χ 2 c





∓

3χ 2



√

π

21±23χ

_



1∓3χ 2

 .

(2.26)

For a tachyonic field

μ

2

_<

_{0 and}

_χ

_>

_{1; thus}

_{φ (}

_r

₎

_{diverges as} r

→ ∞

.Letusstressthisconclusion:atachyonictestfield(solely dependingon

r)

thatisregularatthecosmologicalhorizonis nec-essarilyasymptoticallydivergent,andthetestfieldapproximation breaksdown.

In the presence of a BH, one mayexpect thisasymptotic be-haviour toremain, againifoneassumesregularity atthe cosmo-logicalhorizon, ifthescalarfield hasaneffectivetachyonicmass, asymptotically. This is corroborated by the numerical results in the next Sections. Although in our models

μ

2

eff is a function of r, theexistence(orabsence)ofanasymptotictachyonicbehaviour intheregion

r

rc willsourcea deviationfromstandardde

Sit-terasymptotics.The

r

=

0 singularityof (2.24),ontheotherhand, becomesirrelevantinthepresenceofaBHhorizon.

2.4. No smooth scalar hair confined within the cosmological horizon

We have seen that, on the one hand, a tachyonic behaviour is required for the scalar field to be non-trivial in between the

BHand thecosmological horizon;on theother hand,an asymp-totictachyonicbehaviourwillpotentiallyleadtodivergences.One mayask,thus, ifonecould confinethe non-trivialscalarentirely withintheBHandcosmologicalhorizon,thusexcising the poten-tialpathologicalbehaviour.

Ifsuchconfined scalarfield issmooth, not onlyitvanishes at thecosmologicalhorizon,butitsderivatives,andinparticularthe firstderivative,alsovanishtherein.Then,onecanshowthatfora largeclass ofmodels,

φ (

rc

)

=

0

= φ



(

rc

)

implythat

φ

≡

0 forthe

whole region rh

<

r

<

rc. The proof goes asfollows. Fora scalar

fieldwith

k nodes

in

r

h

<

r

<

rc,theassumption

φ (

rc

)

=

0 implies

theexistenceof(atleast)

k local

extremaofitsprofile.Recall

k



1. Let

r

0 bethelargestrootoftheequation

φ



(

r

)

=

0 (r0

<

rc).Then,

integratingthescalarfieldequation(2.2) between

r

0 and

r

c yields

e−δNr2

φ



r_rc 0

=

1 4 rc



r0 dr e−δr2f,φ

I

.

(2.27)

The left hand side of (2.27) vanishes. Indeed, a smooth configu-rationhas N

(

rc

)

e−δ(rc)

φ



(

rc

)

→

0; moreover, both N and e−δ are

finiteat

r

0,where

φ



(

r0

)

=

0.However, fortheEMS-modeland alsoforthetestfieldlimitoftheeSTGB-model,theintegrandof therighthandsidedoesnotchangethesigninthat

r-interval.

We concludethat

φ

≡

0 fortheconsidered

r-range.

Theargumentcan easilybe extended forall interval

r

h

<

r

<

rc,yielding the

adver-tisedresult.

3. ThescalarisedEMS-



blackholes 3.1. The zero modes

FortheEMS-model,thescalar-freesolutionistheRNdSBH, givenby (2.7), (2.8) and (2.10) with (2.11) and

φ

=

0.Letusfirst considerthe zeromodesof thescalar field perturbations. Inthis paperweonlyconsidersphericalmodes.

Thesmall-φ limit of thescalar field equation (2.2) on a fixed RNdSbackgroundgives

(

r2N

φ



)



−

α

Q

2

r2

φ

=

0

.

(3.28)

For



=

0,(3.28) admitsanexact,closedformsolutionintermof aLegendrefunction [38]

φ (

r

)

=

Pu

1

+

2Q 2

₍

_r

₋

_r h

)

r

(

r_h2

−

Q2

₎



,

where u

≡

√

4

α

+

1

−

1 2

.

(3.29) The leading behaviour of this solution as the asymptotically flat regionisapproachedis

φ (

r

→ ∞) =

2F1

1

−

√

4

α

+

1 2

,

1

+

√

4

α

+

1 2

,

1

;

Q2 Q2

₋

_r2 h



+

O



1 r



.

(3.30)

Allowing a generic value of

φ (

r

→ ∞)

, there is a continuum of zeromodesolutions,aslongas [38]

α

<

α

max

≡ −

1

4

.

(3.31)

The asymptoticvalue of

φ

is fixed by the ratio Q

/

M. Requiring, for a given

α

, that the scalar field vanishes asymptotically (i.e.

φ (

r

→ ∞)

=

0),only a discreteset ofvaluesof Q

/

M is allowed, correspondingtosolutionswithdifferentnodenumber.

No exact solution of (3.28) appears to exist for



=

0. In the neighbourhoodoftheBHhorizon,however,anapproximate (regu-lar)solutioncanbeexpressedasapowerseriesin

(

r

−

rh

),

as

φ (

r

)

= φ

h

+

α

Q2rh

(

r2c

+

rcrh

+

r2_h

)φ

h

(

rc

−

rh

)

rh

[−

rcr2_h

(

rc

+

2rh

)

+

Q2

(

r2c

+

2rcrh

+

3r_h2

)

]

× (

r

−

rh

)

+

O

(

r

−

rh

)

2

,

(3.32)

where

φ

h isthevalue ofthescalarfieldattheBHhorizon,afree

parameter.Asimilarexpressionholdsintheneighbourhoodofthe cosmologicalhorizon,with

r

h and

r

c interchangedand

φ

hreplaced

bythevalueofthescalarfieldatthecosmologicalhorizon,

φ

c.4

Performinga numericalintegration inthe region betweenthe BHandcosmologicalhorizons,ournumericalresultsindicatethat foragivenRNdSbackground,asspecified

e.g. by

thedimensionless ratios ( Q

/

M, rc

/

rh), solutionswhich areregular atboth horizons

exist fora discrete set of

α

, beinglabelled by the node number

k

>

0.Usingthesesolutions,theboundarydataatthe cosmologi-calhorizonisfixed;wethenintegratefromthehorizonoutwards, extendingthesolutionstotheasymptoticregion

r

→ ∞

.Forlarge

r, anapproximateformsolutioncanbefoundasapowerseriesin 1/r, withtheleadingordertermsbeing

φ (

r

)

= φ

_∞

+

φ

3

r3

+ . . . ,

(3.33)

where

φ

_∞ and

φ

3areconstantsfixedbythenumerics.

Anoutstandingfactisthat,differentlyfromthe



=

0 case, so-lutionswith

φ

∞

=

0 werenotfound.Thatis,thescalarfielddoes not vanishasymptotically.Thisnumericalfinding agrees withthe analysisinsection2.3.Indeed,fortheMaxwellcase, theeffective tachyonicmassvanishesinthefarfieldregion,

cf. (

2.17),andthus (2.26) reducesto(3.33).Thebehaviourof

φ

_∞,aswellasthe varia-tionofthecriticalvalueof

α

astheBHchargetomassratio

Q

/

M

isvaried, isillustrated inFig. 1(leftpanel) fortwovaluesof the ratio

r

c

/

rh.

3.2. The non-linear solutions

Letusnowconsiderthenon-linearsolutionsthatbifurcatefrom theRNdSfamilyatthescalarclouds.Theansatz (2.7),(2.8), (2.10) and(2.12) yieldsthefollowingsetofcoupledordinarydifferential equations5_: m

=

r 2_N

_φ

2 2

+

e2δ_r2_V2 2 f

(φ)

,

δ



₊

_r

_φ

2

₌

₀

_,

_(3.34)

(

f

(φ)

eδr2V

)



=

0

,

(

e−δr2N

φ



)



=

e δ_r2 2 df

(φ)

d

φ

V 2

_.

_(3.35) Theelectricpotentialcanbeeliminatedfromtheaboveequations noticingtheexistenceofafirstintegral,

V

=

e−δ Q

r2_f

_(φ)

,

(3.36)

where Q is an integration constant interpreted as the electric charge.

Thesystemofequation(3.34)-(3.35) willbesolvednumerically. To doso, we first findthe approximate formof the solutions at

4 _{Inthenumericswehavesetφ}

h=1 withoutanylossofgenerality.

5 _{Thereisalsoanextraequation,whichisaconstraint,andcanbederivedfrom} (3.34)-(3.35).

Fig. 1. (Leftpanel)Asymptoticvalueofthescalarfield(φ∞)(mainplot)andcriticalvalueof

α

(inset)vs. thechargetomassratiofordSscalarcloudsontheRNdSbackground, fortwoillustrativevaluesofrc/rh.(Rightpanel)RadialprofilesforthemetricfunctionsandelectrostaticpotentialofatypicalEMS-BHwith>0.

Fig. 2. Normalisehorizonarea(leftpanel)andscalarfieldvalueatthehorizon(rightpanel)vs. thechargetomassratioforscalarisedEMS-BHs,forafixedvalueofrc/rh

anddifferentvaluesofthecouplingconstant

α

.Therightpanelalsoshowsthevalueofthemetricfunctione−δr_{atthehorizon.}

theboundaryofthedomainofintegration.Firstly,closetotheBH horizon,therelevantfunctionsareapproximatedas:

m

(

r

)

=

rh 2

−



r2 h 6

+

m1

(

r

−

rh

)

+ . . . ,

δ(

r

)

= δ

h

+ δ

1

(

r

−

rh

)

+ . . . ,

(3.37)

φ (

r

)

= φ

h

+ φ

1

(

r

−

rh

)

+ . . . ,

V

(

r

)

=

Vh

+

v1

(

r

−

rh

)

+ . . . .

These expressions depend on the following set of constants:

rh

,

,

m1

,

δ

h

,

δ

1

,

φ

h

,

φ

1

,

Vh

,

v

1.Thefieldequationsrelatethese pa-rameters.Weobtain: m1

=

Q2 2r2_h

(

1

−

αφ

2 h

)

,

v1

= −

e−δh_Q

(

1

−

αφ

2 h

)

r2h

,

φ

1

=

αφ

he2δ0rhv21 1

−

2m1

− 

r2_h

,

δ

1

= −φ

12rh

.

(3.38)

Thus,the independentparameters are

r

h

,

,

φ

h

,

δ

h

,

V

h,which

de-termineallothers.Asimilarexpression holdsatthecosmological horizonwhichislocatedat

r

=

rc

>

rh,introducingthenew

inde-pendentparameters

φ (

rc

),

δ(

rc

),

V

(

rc

).

Also,one findsthe

follow-ingasymptoticsofthesolutionsinthefarfield: m

(

r

)

=

M

−

Q 2 2r

(

1

−

α

2

_φ

2 ∞

)

+ . . . ,

δ(

r

)

=

3q 2 s 2r6

+ . . . ,

(3.39)

φ (

r

)

= φ∞

+

qs r3

+ . . . ,

V

(

r

)

=

V∞

+

Q

(

1

−

αφ

2 ∞

)

r

+ . . . ,

which introduces the new independent parameters M

,

Q

,

V

0

,

qs

,

φ

∞.6

The field equationsforthismodel (andalso themodelin the next section)have been solved by theNewton-Raphson method, with an adaptive mesh selection procedure, with the solver de-scribed in [49]. The solutions are found in two steps: first, by integratingfrom

r

h to

r

c,andthenfromthecosmologicalhorizon

toinfinity(theregioninsidetheBHhorizonisnotconsidered, al-thoughitcouldbestudiedfollowing [50]).Inourapproach,both

r

h

and

r

c areinputparameter,thecorrespondingvalueof



resulting

fromthenumericaloutput.Inthefollowing,weshallexhibitsome illustrativesolutions,whichreflectthemostrelevantpropertiesof thedomainofexistencestudied.

The profileof atypical scalarised RNdSBH isshownin Fig.1

(right panel). Onechecks that N

(

r

)

vanishes both at theBH and cosmologicalhorizons;thescalarfieldstartsatapositivevalueat the BHhorizonandisnegative atthe cosmologicalhorizon, pos-sessing precisely one node; moreover it does not approach zero asymptotically.Onealsoobservesthatboththemassfunction

m

(

r

)

(whichismonotonicallyincreasing)andthemetricfunction

e

−δ(r) appear to converge for large r suggesting a smooth solution is asymptoticallyattained.

Considering now a more global perspective on the full set of computed solutions, the emerging picture has some similarities with that found for the



=

0 EMS model [38,41], and can be summarised asfollows - seeFig.2. Foreach

α

<

α

max,a branch offullynon-linearsolutionsbifurcatesfromaRNdSBHwitha

par-6 _{ThevalueofoneoftheparametersV}

h, V(rc), V∞canbefixedviaagauge transformation.

Fig. 3. (Leftpanel)Theratio

α

/M2_{vs. thenormalisedcosmologicalconstantforthecriticalSdSBHthatsupportsasphericalcloudwithk=1 inthe}

eSTGB-model.The insetshowsthevalueofthescalarfieldatthecosmologicalhorizon.(Rightpanel)RadialprofilefunctionsforatypicalsolutionoftheEGBs-model.

ticularchargetomassratio

Q

/

M (and agivenratio

r

c

/

rh).Theleft

panelofFig.2exhibitsthisbifurcationinaBH(normalised) hori-zonarea diagram

vs. the

charge tomassratio.Onecanappreciate that,forafixedvalueof

Q

/

M, thescalarisedsolutionhasalarger BHhorizonthanthecorrespondingscalar-freesolution.Also, over-chargedsolutions exist,justas inthe



=

0 model. Eachbranch ofthe scalarised BHs can be specified by the value ofthe scalar field atthehorizon- Fig. 2 (rightpanel).Each such branchends atacritical,(likely)singular,configuration:thenumerics indicate theKretschmannscalarandthe horizontemperaturediverge, the BHhorizonareavanishes(with A(_Hc)stillfinite),whereasthemass parameter M stays finite. All thesefeatures resemble the



=

0 case.

Contrastingwiththe



=

0 case,thescalarisedBHsdonot ap-proach precisely the scalar-free solution as r

→ ∞

. Indeed, the scalar field does not vanish as r

→ ∞

, approaching a constant nonzerovalue,afeature anticipatedfromtheanalysisofthezero modes.7

4. ThescalarisedeSTGB-



blackholes 4.1. The zero modes

FortheeSTGB-model,thescalar-freesolutionistheSdSBH, givenby (2.7) and(2.8) with (2.9) and

φ

=

0.Increasingthevalue ofM in deSdSsolutionimpliesthatthecosmologicalhorizon (lo-catedatthelargestrootoftheequation

N

(

r

)

=

0)shrinksinsize, pulledinwardsby thegravitationalattraction oftheBH. Asa re-sultthere is a largest BH, the Nariai solution [54], which occurs when M

=

1/(3

√

).

Spaceswithlargervaluesof M are unphys-ical,containingnaked singularities.Letusagainfirstconsiderthe zeromodesofthescalarfieldperturbations.

Restrictingto the small-fieldlimit, equation (2.16) on theSdS backgroundbecomes

(

r2N

(

r

)φ



)



+

α

6

(

2r 6

₊

_r2 crh2

(

rc

+

rh

)

2

)

r4

₍

_r2 c

+

rcrh

+

r2h

)

2

φ

=

0

,

(4.40) wherewe haveeliminated the parameters M

,



infavour of the twohorizonsradii

r

h

,

rc.Theapproximateexpression ofa regular

solutionneartheBHhorizonreads

φ (

r

)

= φ

h

+

6

αφ

h

(

2r_h4

+

r2crh2

+

r4c

+

2rhr3c

)

r3_h

(

2r_h4

+

rcr_h3

−

rc4

−

2rc3rh

)

(

r

−

rh

)

+ . . . ,

(4.41)

7 _{Despitethisfact,usingtheapproachin[51],itcanbeshownthattheconstant} Q canstillbeidentifiedwiththetotalelectriccharge,asevaluatedatfuture/past infinity.

where

φ

histhearbitraryconstantcorrespondingtothescalarfield

value atthe horizon. A similarexpansion exists nearthe cosmo-logical horizon, which introduces another constant

φ (

rc

),

instead

of

φ

h.

Similarly to the case in section 3.1, solving the perturbation equation (4.40) can be viewedasan eigenvalueproblem: impos-ingsmoothnessforthescalarfield attheBHhorizon(r

=

rh)and

atthecosmologicalhorizon(r

=

rc) selectsadiscretesetof

back-groundconfigurations,specifiedbythedimensionlessratio

α

/

M2_. Foreachvalueofthisratio,adiscretesetofscalarprofilesisfound, labelledby the numberofnodes

k

>

0.For



=

0 these are dis-cussedin [19,34,55].Thedimensionlessratio

α

/

M2 andthescalar fieldvalueatthecosmologicalhorizonareshownagainstthe cos-mologicalconstantfor

k

=

1 scalarcloudsinFig.3(leftpanel).We remarkthatas



→

0,theratio

α

/

M2doesnotmatchthe thresh-oldvalueforthefundamentalmodein [19,34,55],whichhas

k

=

0, butratherthefirstexcitedstate,whichhas

k

=

1.

There is, however,a key difference betweenthe scalar clouds in thismodeland thosein both thescalar cloudsin the asymp-toticallyflateSTGBmodelandtheEMS-modeldiscussedinthe previoussection.Thescalarcloudsalwaysdiverge as

r

→ ∞

.That it,forlarge

r,

the leadingtermsoftheasymptoticsolutionofthe eq. (4.40) consistinthesumoftwomodes

φ (

r

)

=

c1r− 3 2(1+ √ 1+16α/9)

₊

_c 2r− 3 2(1− √ 1+16α/9)

_{+ . . . ,}

_(4.42) where c1 andc2 are two constants resulting from thenumerics. Thesolutions with

c

2

=

0 wouldpossesstherightasymptotic be-haviour;butthesedonotarisewhenintegratingfromthenearBH region. This behaviour is interpreted fromthe discussion in sec-tion2.3.Since, from (2.18),

μ

2

eff r→∞

_{→ −}

8

α

/

<

0 in theeSTGB- model,eq. (2.26) impliesthat thescalarfield necessarilydiverges asymptotically.WhiletheBHhorizonindeed‘cures’thesingularity inside the cosmologicalhorizon, no solutions with

μ

2

eff

<

0 exist whichare regularatboth horizons

and for

large

r.

Thus,the dis-cussionofzeromodesalreadyanticipatesthatBH scalarisationin the eSTGB-model willchange thede Sitterasymptotics. More-over,thetestfieldapproximationbreaksdownoutsidethe cosmo-logicalhorizon.

4.2. Including backreaction

Withtheansatz(2.7) and(2.12),asuitablecombinationofthe equations ofmotion leads to firstorder equationsfor the metric functions, m

=

F1

(

N

,

φ,

φ



),

δ



=

F2

(

N

,

φ,

φ



)

anda second order equationforthescalarfield,

φ



=

F3

(

N

,

φ,

φ



).

Thesearethe equa-tionsusedinournumericalapproach,buttheexpressionforthe

F

i

arelongandunenlightening;we shallthereforenot includethem here.

Fig. 4. “Mass”(leftpanel),BHhorizonareaandthevalueofthemetricfunctione−δ(r)_{atthehorizon(rightpanel)for}

eSTGB-BHsvs. thescalarfieldattheBHhorizon,for differentvaluesofrc/rh.Thereddotsindicatethecriticalconfigurationswherethebranchesstoptoexist.

AsfortheEMS-model,theeSTGB-modelpossessesBH so-lutionswithanon-trivialscalarfieldwhichareinterpretedasthe non-linearrealisationsofthezeromodesdiscussedabove.The pro-fileofatypicalsolution isshowninFig.3(rightpanel).Comparing withthecorresponding profiles fortheEMS-case, displayed in Fig.1(rightpanel)oneboth observessimilaritiesanddifferences. Again, N

(

r

)

vanishes both atthe BH and cosmologicalhorizons; thescalar fieldstarts againat apositive value attheBH horizon andis negative at the cosmologicalhorizon, possessing precisely onenode.Again,itdoesnotapproachzeroasymptotically; indeed it diverges,although this isnot apparent in thedisplayed range. Butnowone observesthat themassfunction

m

(

r

)

growssteeply inthedisplayedrange,whereas themetricfunction

e

−δ(r) _appear to converge for large r. The solution extends smoothly through bothhorizons;both R and Kretschmannscalararefiniteas

r

→

rh

andr

→

rc.Indeed,one can check thisby obtaining apower

se-riesofthesolution,validclosetotheBH/cosmologicalhorizon.But asymptotically,thesolutionsdonotapproachdeSitterspacetime.

Conveyingamoreglobalperspectiveofthedomainofexistence ofthesesolutionsleads tothefollowingremarks.Similarlytothe



=

0 case, a branch of eSTGB- BHs bifurcates from any zero mode. In appropriate variables, these eSTGB- solutions form a line,startingfromthesmooth

-vacuum

limit,as

φ

→

0,and end-ing at a limiting solution - Fig. 4. The existence of this limiting solutioncanbeunderstoodbynoticingthat,similarlytothe



=

0 case [19–21], the nonlinearity associated with the Gauss-Bonnet termimpliesthatthederivativeofthescalarfieldat

r

=

rhsolvesa

secondorderequationintermsof

φ (

rh

),



and

α

(thesameholds

atthe cosmologicalhorizon). Then

φ



(

rh

)

becomes imaginary for

some criticalconfiguration,andasresultthe numericaliterations failtoconverge. The“mass” Mc

=

m

(

rc

),

BHhorizonarea andthe

valueofthemetricfunction

e

−δ(r)_{attheBHhorizonareshownin} Fig.4fortheeSTGB-BHsasafunctionofthescalarfieldatthe BHhorizon,with

φ (

rh

)

=

0 correspondingtotheSdSlimit.Thered

dotsmarkingthecriticalconfigurations.

As before, we first numerically integratedthe field equations betweenthe BH andcosmological horizon. Ina second step,the solutions were extended to the region r

>

rc. For all

configura-tionsweconsidered,thescalarfielddivergesfor

r

→ ∞

,afeature inherited from the test field limit. As a result, the mass func-tion diverges as m

(

r

)

∼

r32(1+

√

1+16α/9) _{which implies N}

₍

_r

₎

_∼

r12(1+3 √

1+16α/9)

_>

_r2_{.Thismeansthesolutionsdonotapproacha} dSspacetimeatfuture/pastinfinity.Thetachyonicscalarfield dom-inatesthebehaviourasymptotically.Thisis(likely)amanifestation ofthecosmologicalinstabilityineSTGBmodelsdiscussedin [56].

5. Furtherremarks

In thisworkwe havestudiedthe impactofa positive cosmo-logical constant on two paradigmatic models of BH spontaneous scalarisation.For



=

0,theirelectrovacuumBHsolutionsmay be-comespontaneouslyscalarised,duetoa tachyonicinstability trig-geredbyscalarperturbations[19–21,38].

Our study shows that the response of the two models, that share many features for



=

0, to a non-zero cosmological con-stant is quite different. While thesolutions of theEMS- model share thekey propertiesoftheir asymptotically flat counterparts, withmild differencesonly, theeSTGB-modeldiffers fromboth their flat spacetime counterpart and the EMS-. This difference can betracedtothedifferentasymptoticbehaviour ofthesource term

I

in the action (2.1). For both models, the scalar field ac-quires an

effective tachyonic

mass

μ

foraregion closetothe BH horizon. However, whilefor the EMS- thescalar field becomes massless asr

→ ∞

(the square of the effectivefield mass being proportional with Maxwell invariant F2_{), thisis not the casefor} theeSTGB-model.Inthelatter,

μ

2 _{approachesasymptotically a} negative value, being proportional to the Gauss-Bonnet invariant for dS spacetime. As a result, the scalar field diverges in the far field,whichresultsinnon-dSasymptoticsofthesolutions,despite thepresenceofacosmologicalhorizon.Atthesametime,the con-sidered configurations are regular in the region between the BH andcosmologicalhorizon.

Whiletheresultsinthisworkhavebeenfoundforaquadratic coupling ofthe scalarfield, we expect that the basicfeatures do not depend onthis specificchoice of thecouplingfunction. As a directionoffurtherresearch,itwouldbeinterestingtoinvestigate thestabilityoftheEMS-solutions.

Acknowledgements

This work is supported by the Fundação para a Ciência e a Tecnologia (FCT) project UID/MAT/04106/2019 (CIDMA), by CEN-TRA(FCT)strategic projectUID/FIS/00099/2013, by nationalfunds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from the projects PTDC/FIS-OUT/28407/2017andCERN/FIS-PAR/0027/2019.Thisworkhas fur-ther been supported by the European Union’s Horizon 2020 re-searchandinnovation(RISE)programmesH2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 andH2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.The authorswouldliketoacknowledge net-workingsupportbytheCOSTActionsCA16104andCA18108.

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