Radial excitations of current-carrying vortices

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Radial excitations of current-carrying vortices

Betti Hartmann, Florent Michel, Patrick Peter

To cite this version:

Betti Hartmann, Florent Michel, Patrick Peter.

Radial excitations of current-carrying vortices.

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb 1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

Radial

excitations

of

current-carrying

vortices

Betti Hartmann

a

,

Florent Michel

b

,

Patrick Peter

c

a_{InstitutodeFísicade SãoCarlos(IFSC),UniversidadedeSãoPaulo(USP),CP 369,13560-970,SãoCarlos,SP,Brazil} b_{LaboratoiredePhysiqueThéorique,CNRS,Univ.Paris-Sud,UniversitéParis-Saclay,91405 Orsay,France}

c_{Institutd’AstrophysiquedeParis(GRεCO),UMR7095CNRS,SorbonneUniversités,UPMCUniversitéParis 6,InstitutLagrangedeParis,98 bisboulevardArago,}

75014Paris,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received17November2016

Receivedinrevisedform19January2017 Accepted6February2017

Availableonlinexxxx Editor:A.Ringwald

WereportontheexistenceofanewtypeofcosmicstringsolutionsintheWittenmodelwithU(1)×U(1)

symmetry. Thesesolutions are superconducting with radially excited condensates that exist for both gaugeandungaugedcurrents.Ourresultssuggestthatthesenewconfigurationscanbemacroscopically stable, but microscopically unstable to radial perturbations. Nevertheless, they mighthave important consequencesforthenetworkevolutionandparticleemission.Wediscusstheseeffectsandtheirpossible signatures.Wealsocommentonanalogieswithnon-relativisticcondensedmattersystemswherethese solutionsmaybeobservable.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

0. Introduction

Lineartopologicaldefects,e.g.,vortexlinesincondensedmatter physics[1,2],superfluidvortices(seee.g.[3]),non-Abelianvortices (see e.g. [4]), also frequently appear assolutions of highenergy field theorieswhere they are called cosmic strings[5]; these in-clude grand unified (GUT) or superstring theories. Such strings yieldmanycosmologicalandastrophysicalconsequences[6,7]that havenotall beenyetfullyinvestigated.Manymicroscopicmodels havebeendiscussed,providingdirectcontactwiththeunderlying fieldtheory[8].

Eveninthesimplestmodelssuch asthepopular U

(

1

)

Abelian Higgs model with a Mexican hat potential, there are no known analytical solutions, straight and static ones [9] having only been constructed numerically. These so-called Abelian–Higgs or Abrikosov–Nielsen–Olesen (ANO) vortices have a localized energy-momentum tensor and a quantized magnetic-like flux with the magnetic-like field pointing in the direction of the string axis. Numerical solutions were found in Refs. [10–12] andtheir grav-itationaleffectshavebeenstudiedinRefs.[11–17].

TheANOstringandfluxtubewidthsareinverselyproportional, respectively,totheHiggsandthegaugebosonmasses.Whenthese masses are equal, the strings saturate the Bogomolonyi–Prasad– Sommerfield(BPS) bound [18] implying that theenergy per unit lengthisproportionaltothetopologicalcharge.This,inturn,

guar-E-mailaddresses:bhartmann@ifsc.usp.br(B. Hartmann), florent.michel@th.u-psud.fr(F. Michel),peter@iap.fr(P. Peter).

antees stability; the fields then satisfy a set of coupled first or-derdifferentialequationswhosesolutionsarealsonotanalytically knownandthushavetobeconstructednumerically.Furthermore, itwasshownthatBPSstringsremainsoincurvedspace–time,i.e., theysatisfythesameenergyboundasinflatspace–time[14,15].

Cosmic strings howevercan be, andmore often than not are

[19] superconducting[20],carryingpersistentcurrentsthatcanbe spacelike(like anordinarycurrent),timelike(akintoacharge),or null(chiralorlightlike[21]).Currentsupto1020_{Amps(GUTcase)} can be induced by either bosonic orfermionic [22] charge carri-ers.Intheformercase,thescalarfieldthatundergoesspontaneous symmetry breaking, leading to theformation ofthe strings, cou-ples non-triviallyto asecond one. Forappropriate choicesof the self-couplings andvacuumexpectationvaluesofthe scalarfields, thesecondfieldcanformacondensateinthecoreofthestring.

While standard cosmic strings have energy per unit length

ε

equal to their tension

μ

, this ceases to be true in the presence ofacurrent.Therelationbetweentheenergyperunitlengthand thetensionofasuperconductingstringsolutionoftheU

(

1

)

×

U

(

1

)

modelhasbeendiscussedindetailin[23,24]andithasbeen sug-gestedthatthe equationofstate isoflogarithmicform[25].This hasbeenconfirmednumericallyin[26].Superconductingsolutions also exist in other models, such as the semilocal SU

(

2

)

global

×

U

(

1

)

local model[27,28]aswellasintheSU

(

2

)

local

×

U

(

1

)

local elec-troweakmodel[29].

In this Letter we report on a new type of solutions in the U

(

1

)

×

U

(

1

)

model: superconducting strings withradially excited condensates.Thesesolutions possessa finitenumberofnodes in the scalar field function associated to the unbroken U

(

1

)

sym-http://dx.doi.org/10.1016/j.physletb.2017.02.015

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

JID:PLB AID:32618 /SCO Doctopic: Theory [m5Gv1.3; v1.199; Prn:14/02/2017; 10:25] P.2 (1-6) 2 B. Hartmann et al. / Physics Letters B•••(••••)•••–•••

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metry.Radial excitationsof solitonic-likesolutions are quite well known: they appear in non-topological soliton systems such as Q -ballsandbosonstars[30]withan unbrokensymmetry aswell asintopologicalsolitonsystemssuchasmagneticmonopoles[31]

in which a continuous symmetry gets spontaneously broken. To ourknowledge they were so far never considered inthe present context.Yet, studyingthesolutions oftheU

(

1

)

×

U

(

1

)

model us-ing both analytical and numerical techniques, we find that they are a generic prediction. They are thus important to understand themathematical structure ofthe theory,andmayhave nontriv-ialconsequencesfortheevolutionofthestringnetworkaswellas theemissionofparticlesduringreconnectionbetweentwostrings.

1. Themodel

We study the Witten model of superconducting strings with bosonic currents [20]. This model contains two complex scalar fields

φ

and

σ

which are minimally coupled to two (different) U(1)gauge fields.Using ametricwithsignature

(+,

−,

−,

−)

,the Lagrangiandensityreads

L

= −

1 4FμνF μν

₋

1 4GμνG μν

₊

1 2Dμ

φ



Dμ

φ



∗

+

1 2Dμ

σ



Dμ

σ



∗

−

V

(φ,

σ

),

(1)

wherethepotentialisgivenby V

(φ,

σ

)

=

λ

1 4

(

|φ|

2

₋

_η

2 1

)

2

+

λ

2 4

|

σ

|

2

₍

_|

_σ

_|

2

₋

₂

_η

2 2

)

+

λ

3 2

|φ|

2

_|

_σ

_|

2

_,

(2) thegaugecovariantderivativesread

Dμ

φ

=



∂

μ

−

ieφBμ



φ ,

Dμ

σ

=



∂

μ

−

ieσAμ



σ

,

(3) andthefieldstrengthtensorsofthetwoU(1)gaugefields Aμ and Bμ are

Gμν

= ∂

μBν

− ∂

νBμ

,

Fμν

= ∂

μAν

− ∂

νAμ

.

(4)

Inthefollowingwewillusecylindricalcoordinates

(

t

,

r

,

θ,

z

)

and workwiththeansatz

Bμdxμ

=

1

eφ

[n

−

P

(

r

)

] d

θ , φ (

r

, θ )

=

η

1h

(

r

)

einθ

,

Aμdxμ

=

Az

(

r

)

dz

+

At

(

r

)

dt

,

σ

(

t

,

r

,

z

)

=

η

1f

(

r

)

ei(ωt−kz)

.

(5) From now on, we shall restrict attention to the caseeσ

=

0, for whichthe externalgauge field Aμ canbe setto zeroaswell; in fact, thisgauge field issourced by the currentflowingalong the stringbut hardlybackreacts on thestring microstructure[24].In Section4webrieflycommentonthecaseeσ

=

0.

Introducing the dimensionless coordinate x

:=

√

λ

1

η

1r, the equations ofmotion only depend on the dimensionless coupling constants

α

2

=

e 2 φ

λ

1

,

q

=

η

2

η

1

,

γ

i

=

λ

i

λ

1

(

i

=

2

,

3

),

(6) as well as the (also dimensionless) state parameter w

:= (

k2

₋

ω

2

_)/(λ

1

η

21

)

;theyread



P x





=

α

2P h 2 x

,

(7) 1 x



xh





=

h



P2 x2

+

h 2

₋

₁

₊

_γ

3f2



,

(8) 1 x



xf





=

f



w

+

γ

2



f2

−

q2



+

γ

3h2

,

(9)

whereaprimedenotesaderivativewithrespecttox.The bound-aryconditionsatthestringaxisx

=

0 andatinfinityread

P

(

x

)

|

x=0

=

n

,

h

(

x

)

|

x=0

=

0

,

f

(

x

)

|

x=0

=

0

,

lim x→∞P

(

x

)

=

0

,

xlim→∞h

(

x

)

=

1

,

f

(

x

)

x→∞

=

0



x−1/2



.

(10) The importantphysicalquantities ofthesolutions are theenergy perunitlength

ε

andthetension

μ

,given(inrescaledunits)by

ε

= −

2

π

xT_ttdx

,

μ

= −

2

π

xT_zzdx

,

(11)

respectively,withtheenergy-momentumtensor Tμν

= −

2

δ

L

δ

gμν

+

gμν

L

,

(12)

aswellasthe(rescaled)absolutevalue I ofthecurrent: I

=

2

π



|

w

|

xf2dx

.

(13)

Note that our model bears strong similarities with that of Ref.[32]providedonemakesthereplacements

η

1

→

1,

β

1

→

2

λ

1,

β

2

→ λ

2

/

2,

β



→ λ

3

/

2 and

α

→ λ

2

η

2₂

/

2;itwas shownthat differ-enttypesofsolutionsmayexistinthismodel,butforthepurpose of exhibiting our new configuration, we shall only consider the regime of parameters for which the corresponding ANO vortices (without currents) underconsideration are of type II [7] and re-strictattentiontothecasen

=

1 forsimplicity.

2. Smallcondensates

Tomotivate theexistenceofexcited solutions,letusfirst con-sidertheregimewherethecondensateissufficientlysmallthatwe canneglecttheterm

∝

f3in(9)aswellastheback-reactiononh, i.e.theterm

∝

f2 in(8).AssumingthecurrentlessANOvortexto bestable(hencechoosingn

=

1),wefollowtheanalysisofWitten

[20] andperturb thescalarfield

σ

around

σ

(

x

)

≡

0 in the back-groundofthisvortex.Thiswilltelluswhetherinagivensetting thevortexcansustaintheadditionalstructure.

Taking the boundary conditions into account, we assume the followingsimpleansatzfortheHiggsfieldfunction1

h

(

x

)

=



κ

x for 0

≤

x

<

1

/

κ

,

1 for 1

/

κ

≤

x

,

(14)

where

κ

is a freely adjustable constant giving the characteristic size of the Higgs field variations. For x

≥

1

/

κ

, equation (9) then becomes 1 x



x f





=



w

−

γ

2q2

+

γ

3



f

.

(15)

This is a modified Bessel equation whose only solution decaying strictlyfasterthat x−1/2 atinfinityis

f

(

x

)

=

C1K0



w

−

γ

2q2

+

γ

3x



,

(16)

whereC1

∈ R

isanintegrationconstantandK0isthezerothorder modifiedBesselfunctionofthesecondkind.Intheinteriorregion x

<

1

/

κ

,equation(9)becomes 1 x



x f





=



w

−

γ

2q2

+

γ

3

κ

x



f

.

(17)

1 _{Wehaverepeatedthecalculationswithh}

(x)=tanh(κx)withqualitatively sim-ilarresults[33].

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

Tofinditssolutions,itisusefultodefinethevariableY

(

x

)

andthe functionF by Y

(

x

)

≡

√

γ

3

κ

x2

,

F [Y

(

x

)

]

=

exp



Y

(

x

)

2



f

(

x

).

(18)

Equation(17)isthen rewrittenasthelinear,second order differ-entialequation Yd 2_F dY2

+ (

1

−

Y

)

dF dY

+

m F

=

0

,

(19) where m

≡ −



w

−

γ

2q2 4

√

γ

3

κ

+

1 2



.

(20)

Eq.(19)istheconfluenthypergeometricequation[34].Ithasonly oneregularsolutionattheorigin,namely F

(

Y

)

=

C2Lm

(

Y

)

,where

Lm denotes the Laguerre function withparameter m and C2

∈ R

isanother integration constant. So, forx

<

1

/

κ

, the only regular solutionof(17)is

f

(

x

)

=

C2e−√γ3κx

2_/2

Lm

(

√

γ

3

κ

x2

),

C2

∈ R.

(21)

Therequirement that f and f should be continuous at x

=

1

/

κ

givestwomatchingconditions.Thefirstoneisarelationbetween C1 and C2. The other one gives a constraint on m with a dis-crete setof solutions (see [33] for more details). Thatis, regular solutionsdecaying sufficiently fastatinfinity existonly forthese discretevaluesofm,whichwastobeexpectedsincethelinearized version of Eq. (9) solved here is nothing buta two dimensional Schrödingerequationinaconfiningpotentialwithafinitenumber ofboundstates.

Inthe limit

√

γ

3

/

κ

→ ∞

, thesesolutions are localized inthe domainx

1

/

κ

,so that (21)isvalidup to a regionwhere f is exponentiallysmall. Regularsolutions are then givenin termsof theLaguerrepolynomials [34],restrictingm to bea natural inte-ger.Asthemth _{Laguerre polynomialhasm strictly positive roots,} thecorrespondingfunction f hasm nodes(seeFig. 1,dashedlines form

=

1 andm

=

2).Inpractice,thefinitevalueof

κ

givesa cut-offon m, of the orderof

√

γ

3

/(

4

κ

)

. Forthe numerical solutions showninthefigure,weestimatethat

√

γ

3

/(

4

κ

)

≈

4

.

90 form

=

1 and

√

γ

3

/(

4

κ

)

≈

4

.

32 form

=

2,respectively.2 Sincenonlinear ef-fectstend to reduce the numberof solutions, we thus expect to haveafewofthem whenworkingwiththefullsetofequations. Thisiscompatiblewiththefactthatfortheseparameters,only so-lutionswith 0,1, and2 nodes exist,seethe next section. Inthe following,forsimplicitywe denotebym thenumberofnodesof thefunction f ,althoughitdoesnotexactlyfollow(21).

TheexistenceofboundstatesaroundanotherwisestableANO vortexisthereasonwhyanordinarycosmicstringcanbecome su-perconducting[20].Our discussionheregoesone stepfurther by showingthatthere isa finitenumberofsuch bound states,each ofwhichleadingtoaninstabilityinthecurrentcondensate,an in-stabilitythatwillbetamedbythenonlinearterms:onetherefore expectsthat for each value of w, the full nonlinearset of equa-tions should lead to a series ofsolutions with finite energy per unitlengthandtensionintheformofagroundstateandexcited modes.

Indeed,sofar, wehaveworkedto linearorderin f .When in-cludingthenonlinearterm

∝

f3 anditsbackreactiononh and P ,

2 _{Thedifferencebetweenthesetwovaluesisduetotheback-reactionofthe}

con-densateontheHiggsfield,whichisnottakenintoaccountinthelinearanalysis. Whilethisisalimitationontherelevanceoftheboundonm,the latterisstill expectedtogivethecorrectorderofmagnitude.

Fig. 1. Solutionswithm=1 and 2 inthechirallimit w=0.Theparametersare γ2=5000,γ3=100,q=0.1 andα=10−2.Dashedlinesshowthecorresponding

solutionsinthesmallcondensateapproximationfor√γ3/(4κ)1,seeEq.(21).

Underthisassumption, f/f(0)is,foreachvalueofm,auniversalfunctionofthe rescaledcoordinateγ31/4κ1/2x hereused.Althoughthequalitativebehaviorofthe

solutioniswellcapturedbythe perturbativeapproximation,the actualbehavior requiresthefullnumericalsolution.Onecanalsocheckthatthebehaviorcloseto thestringcore(r→0)correspondstotheexpected1−f/f(0)∝r2_.

Fig. 2. Value f(0)ofthecondensatefunctionalongthestringaxisasafunction ofthestateparameter w forexcitedstatenumbersm=0,1,2,3 andthe same parametersasinFig. 1.

theamplitudeoftheregular,asymptoticallydecayingsolutions de-pends on the values of w. Since the boundary conditions and asymptotic behaviorsofbounded solutions remain similar tothe linear case, one can conjecture they remain discrete, andcan be continuously deformedinto thelinear solutions,so itseems rea-sonabletoassumethatthequalitativefeaturesofthesolutionsdo notchange.Thisresultsinadiscretesetofseriesofsolutions,each ofthemextendingoverafiniteintervalofw.Wepresentthese so-lutions numericallyin thenext section. In Fig. 2, we presentthe dependenceof w on f

(

0

)

duetothe nonlinearandbackreaction terms.

3. Numericalresults

Wehavesolved numericallythesetofcoupled non-linear dif-ferential equations (7), (8), and (9), subject to the appropriate boundaryconditions(10)usingan iterative Newton–Raphson/col-location method with automatic grid selection [35]. The rela-tive errors of our solutions are typically of the order of 10−7 to 10−10_{. We have studied a large number of parameter values,}

α

∈ [

10−3

,

10−1

]

,

γ

i

∈ [

1

,

103

]

, i

=

2

,

3 and q

∈ ]

0

,

1

]

, butreport

onlyon onespecific casethat isgeneric enoughanddisplays the main qualitativefeatures. Thechoice was madeinorder tofulfill thenecessary constraints(seee.g. [26])by alarge margin andto allowtheexistenceofsolutionsinthechirallimit w

=

0.The chi-ralsolutionswithm

=

1

,

2 nodesforourchoiceofparameters are shown in Fig. 1, together with the approximate perturbative so-lutions. The constant

κ

:=

h

|

x=0 is extractedfromthe numerical

JID:PLB AID:32618 /SCO Doctopic: Theory [m5Gv1.3; v1.199; Prn:14/02/2017; 10:25] P.4 (1-6) 4 B. Hartmann et al. / Physics Letters B•••(••••)•••–•••

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Fig. 3. CurrentI asafunctionofthestateparameterw andthesameparameters asinFig. 1.

Fig. 4. Energyperunitlengthε(redsolid)andtensionμ(bluedashed)asfunctions ofthestateparameterw andforthesameparametersasinFig. 1intheelectric regime(top)andinthemagneticregime(bottom),respectively.

data.Ournumerical resultsshow that forthis choiceof parame-ters chiral solutions can only possess up to 2 nodes, i.e., m

≤

2. ThisisshowninFig. 2,wherewegive thevalueof f

(

0

)

as func-tion of the state parameter w. This shows that, while for node number m

=

0

,

1

,

2 electric, magnetic as well as chiral solutions exist,m

=

3 solutionsarealwayselectric.

Fig. 3displaysthecurrent I ofthesuperconductingstring.We observe that the maximal possible current on the string in the magneticregimedecreaseswithincreasingnodenumber.InFig. 4, weshowtheenergyperunitlength

ε

aswellasthetension

μ

of thestrings. Thequalitative behavior ofthesequantities issimilar inallcases.

Carterdevisedamacroscopicstabilitycriterion[36]statingthat a necessary condition for stability of superconducting strings is that the velocities of longitudinal (l), cl,and transverse (t) per-turbations,ct,givenby cl

=



−

d

μ

d

ε

,

ct

=



μ

ε

,

(22)

respectively,shouldbe realforthestringtobestable. Thisoccurs forthe stateparameter lessthana limitingvalue(in general dif-ferentforeachseriesofsolutions)whichwedenoteby wCarter.As canbeseenfromFig. 4,inourexamplewehave wCarter

≈ −

4 for m

=

0, and wCarter

≈ −

9 for m

=

1

,

2. Weexpect that w can be-comepositiveforadifferentchoiceofparameters,asdemonstrated form

=

0 in[23].

Thiscriterionbeinganecessarycondition,itpermitstorestrict the rangeof parameters inwhich stablesolutions mayexist,but doesn’tguaranteestability,eventolinearorder:theymaysupport unstablemodesnot capturedbythismacroscopiccriterion. Deter-miningthepresenceorabsenceofsuchmodesrequireslinearizing the field equations from the Lagrangian (1). A full linear stabil-ityanalysisis beyondthescope ofthepresentletterandwill be presentedin[33].Hereweexemplifythepointbyfocusingon per-turbationsof f only,keepingtheotherfieldsfixed.3

Inserting f

(

x

)

=

f0

+ δ

f ,where f0

(

x

)

is asolution tothe full set ofequations(7),(8),(9)and

δ

f

(

x

)

isrealandsmall, we find that (9) becomes – neglecting terms of order

(δ

f

)

2 and higher – a Schrödinger-type equation for

δ

f with potential Veff

(

x

)

=

w

+

3

γ

2f02

−

γ

2q2

+

γ

3h20, where h0

(

x

)

is a solution to (7), (8),

(9). We find that for the solutions plotted in Fig. 1, this poten-tial isstrictly positive forthefundamental solution,butbecomes negative insome rangeof the coordinate x for the excited solu-tions for w

≤

0. This indicates a possible instability.Solving nu-merically thetime-dependent equation on

δ

f , we found that an unstable modeindeedexists.Thisisillustrated inFig. 5,showing Veff forthefundamental andthetwoexcited solutionscomputed inthesmall-condensatelimit aswell asthecorresponding unsta-blemodes.Sinceaninstabilityalreadyshowsupinthislinearized model, one should envisage that most of the excited solutions may be unstable. The question remains to what extent this is a generic (parameter-dependent) statement. Moreover, in the mag-netic regime for which w

>

0, there must exist a threshold wth abovewhichthepotentialbecomespositivedefiniteagain.The rel-evant solutionsmaythenbestableprovided wth isbelow wCarter definedabove.

4. Discussion

Theabovestringsareofthepurelyspatiallylocaltypeasfaras the currentisconcerned, andthismay,insome GUT-based mod-els forinstance,beconsidered unlikely,asmostsymmetriesthen are expectedtobegauged. Indeed,manysuperconducting cosmic string models consider actual electromagnetic currents,and may even be used to produce large-scale primordial magnetic fields. The excited statesfound hereare based onan uncoupled model, but we haveconstructed thecorresponding solutions in thecase where

σ

is givena gaugecouplingtoo,i.e.foreσ

=

0,andfound that the basic features are left unchanged [33]. Besides, as dis-cussed in Ref. [37], the influence of such an extra coupling is expected to be mostly negligible on the microstructure and we confirmedthisexpectationfortheexcitedsolutions. Ontheother hand,suchacouplingcouldsourceadifferentexternalfield, lead-ing to a more complicated large scale magnetic structure; this shouldbeconsideredinacosmologicalcontext.

3 _{A commentaboutself-consistencyisinorderhere.Strictlyspeaking,the}

stabil-ityanalysisshouldbedonebyconsideringvariationsofthethreefields indepen-dently,astheyareallrelatedbythefieldequations,anditisnotnecessarilycleara priori whatcanbelearnedbyconsideringperturbationsofonefieldonly.However, onecanshowthatinthepresentcasetheexistenceofaninstabilityatfixedh and P impliesthatthefullsystemisunstable(althoughtheconversemaynotbetrue). Theproofwillbegivenin[33],alongwithamoredetailedandcompletestability analysis.

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

Fig. 5. Toppanel:EffectivepotentialVeffforthesolutionswithm=0 (red,

continu-ous),m=1 (blue,dashed),andm=2 (green,dot-dashed)inthesmall-condensate limit.Theprofileofh isahyperbolictangentwithparameterschosentomatch thoseofthenumericalsolutionwithm=0.Middlepanel:profileoftheunstable modeforthesolutionwithm=1.(Thenormalizationisarbitrary.)Bottompanel: profilesofthetwounstablemodesforthesolutionwithm=2.

Wewouldalsoliketoremarkthatthemainqualitativefeatures of these solutions persist in a dynamical space–time, i.e. when couplingthe matter equationsto theEinstein equation. We have checkedthisnumericallyandwillreport onthedetailsofour re-sultselsewhere[33].

Wefindsomeofthesesolutionstofulfillthemacroscopic sta-bilitycriterionofCarter,whichusesintegrated,macroscopic

quan-tities regardless of the details ofthe underlying field theoretical model.Incontrast,themicroscopicstability analysis ofthese solu-tions is very much model-dependent. Since the integrated quan-tities stem from the underlying model though, we expect that a solution that is macroscopicallyunstable will also be unstable when investigatingthemicro-physics. Ourpreliminary results in-dicate that the excited solutions may be classically unstable mi-croscopically,sowe shouldnotexpectthemtobedirectly observ-ablethrough, e.g., gravitationallensing.Besides,itwould bevery difficult to distinguish an excited from a standard cosmic string through the deficitanglealone. Onthe other hand, thefact that thesesolutionsareunstablemayofferapossibilitytodetectthem through particleemission during the collision andrecombination of two strings. For instance, the large fluctuations of the fields during the collision maylocallyexcite the condensate, producing a solution with m

=

0 along a segment of one or both strings, the sharp transition regions between the parts with m

=

0 and m

=

0 producingacurrentkink propagatingalongthestringaxis. Since thesolutionswithm

=

0 areunstable,they willlocally de-cay to the fundamental one, emitting high-energy particles. One can thus expect radiation with highenergy to be emitted along aone-dimensional regionofspace,followingthetrajectory ofthe kink. In return,this emission should reduce the average velocity ofthestringnetwork,transformingitskineticenergyintomassive particleradiation.

Interestingly,theexcitedsolutionsmayhavecloseanaloguesin Bose–Einstein condensates, which would open possible paths to observationsin laboratoryexperiments.It isnow well known[1]

that rotating condensates develop vortex lines akin to cosmic strings. Let usconsider a cold gas madeof two different atoms, orasinglespeciesofatomswithtwointernalstatescoupling dif-ferentlyto someexternal electromagneticfieldscreatingthe con-finingpotential.Inprinciple,thepotentialsexperiencedbythetwo typesofatomscanbe tunedsothat onlyonetypeofatoms con-densesin thelowest-energystate, while theother type can con-denseinsideavortexcore.Thevortexcouldthensupporta super-current,exactlylikethestringsconsideredinthisletter.Moreover, fromthesimilaritybetween(7)andthestationaryGross–Pitaevskii equation, one can expectto findthe samestructure of solutions, with one fundamental configuration minimizing the energy and, depending on the parameters,a set ofexcited ones witha more complicate structure for the condensate of the second type of atoms.(Thiswillbetruegenericallyprovidedthepotential experi-encedbythesecondtypeofatomsisquadraticclosetothevortex line.)Tothebestofourknowledge,thequestionofwhethersuch asetupispossibleornotisstillpending.

Acknowledgements

BH would like to thank Brandon Carter for discussions dur-ing the initial stages of this work. BH would like to thank CNPq forfinancial support underBolsadeprodutividadeGrantNo. 304100/2015-3.BHandPPwouldliketothankFAPESPforfinancial supportunderProjectNo. 2015/02563-8.PPwouldliketothankthe LabexInstitut Lagrange deParis(referenceANR-10-LABX-63) part oftheIdexSUPER,withinwhichthisworkhasbeenpartlydone.

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