Dilute and dense axion stars

Dilute and dense axion stars

The MIT Faculty has made this article openly available.

Please share

how this access benefits you. Your story matters.

Citation

Visinelli, Luca et al “Dilute and Dense Axion Stars.” Physics Letters

B 777 (February 2018): 64–72 © 2017 The Author(s)

As Published

http://dx.doi.org/10.1016/j.physletb.2017.12.010

Publisher

Elsevier

Version

Final published version

Citable link

http://hdl.handle.net/1721.1/115347

Terms of Use

Creative Commons Attribution 4.0 International License

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Dilute

and

dense

axion

stars

Luca Visinelli

a

,

b

,

∗

,

Sebastian Baum

a

,

b

,

∗

,

Javier Redondo

c

,

Katherine Freese

a

,

b

,

d

,

Frank Wilczek

a

,

e

,

f

,

g

a_{TheOskarKleinCentreforCosmoparticlePhysics,DepartmentofPhysics,StockholmUniversity,AlbaNova,10691Stockholm,Sweden} b_{Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,Roslagstullsbacken23,10691Stockholm,Sweden}

c_{UniversityofZaragoza,P.Cerbuna12,50009Zaragoza,Spain}

d_{DepartmentofPhysics,UniversityofMichigan,AnnArbor,MI48109,USA} e_{CenterforTheoreticalPhysics,MIT,Cambridge,MA02139,USA}

f_{DepartmentofPhysicsandOriginsProject,ArizonaStateUniversity,Tempe,AZ25287,USA}

g_{T.D.LeeInstituteandWilczekQuantumCenter,ShanghaiJiaoTongUniversity,Shanghai200240,China}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received1November2017 Accepted5December2017 Availableonline6December2017 Editor:A.Ringwald

Axion stars are hypothetical objects formed of axions, obtained as localized and coherently oscillating solutions to their classical equation of motion. Depending on the value of the field amplitude at the core

|θ0|≡ |θ(r=0)|, the equilibrium of the system arises from the balance of the kinetic pressure and either

self-gravity or axion self-interactions. Starting from a general relativistic framework, we obtain the set of equations describing the configuration of the axion star, which we solve as a function of |θ0|. For small

|θ0|1, we reproduce results previously obtained in the literature, and we provide arguments for the

stability of such configurations in terms of first principles. We compare qualitative analytical results with a numerical calculation. For large amplitudes |θ0|1, the axion field probes the full non-harmonic QCD

chiral potential and the axion star enters the dense branch.

Our numerical

solutions show that in this latter regime the axions are relativistic, and that one should not use a single frequency approximation, as previously applied in the literature. We employ a multi-harmonic expansion to solve the relativistic equation for the axion field in the star, and demonstrate that higher modes cannot be neglected in the dense regime. We interpret the solutions in the dense regime as pseudo-breathers, and show that the life-time of such configurations is much smaller than any cosmological time scale.

©2017 The Author(s). 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

TheQCD axion [1–9]arising within thePeccei–Quinn solution ofthestrongCP-problem[10,11]isoneofthebestmotivateddark matter candidates. Other bosonic darkmatter candidates include axion-likeparticles[12]emerginginmanyextensionsofthe Stan-dardModel,especiallyinstringtheorycompactifications[13–16].

IfbosonscomprisethedarkmatterofourUniverse,theycould form dense (with respect to the average dark matter density) clumpscalledbosonstars[17,18],oraxionstars inthespecificcase ofaxiondarkmatter.(Here“star”isusedtodenoteanobject sus-tainedby hydrostaticequilibrium, whetheror not itemits light.) Such objects have beenlong studied [17–26],and recentlythere hasbeenrevivedinterest[27–36].

*

Correspondingauthors.

E-mailaddresses:[email protected](L. Visinelli),[email protected]

(S. Baum),[email protected](J. Redondo),[email protected](K. Freese),

[email protected](F. Wilczek).

Inthisarticle,westudythestabilityofaxionstarsasafunction of theamplitudeof theaxion field atthe coreof thestar

|θ

0

|

≡

|θ(

r

=

0

)

|

.Ourresultsapply tothefull rangeofaxionmassesfor whichQCDaxionscancompriseallofthedarkmatter.Weidentify three distinct branches of axion stars, distinguished by the field amplitudeatthecore,whichinturndeterminesthedensityofthe star. We should keep in mind, that the axion is a periodic field withamplitudeeffectivelyrestrictedtothedomain0

≤ |θ

0

| ≤

π

.

Forsmallfieldvalues

|θ

0

|



10−6



10−5 _eV

_/

_m



_{withm the}

ax-ion mass, the axion field only probes the harmonic part of the potential, and it can be treated as a free field. In this regime, self-gravity is balanced by the kinetic pressure arising from the uncertainty principle. We call this the dilute axion star branch. Wereproducethepreviousfindingsintheliteratureforthe mass-radius relationship, R

∝

M−1_{,where R andM are theradiusand}

mass ofthe star,respectively. Inthisregime, the configurationis stableagainst perturbation: Fora givenmass M, stars are pulled back to the equilibrium radius if they expand because then the (attractive)self-gravityisstrongerthanthe(repulsive)kinetic pres-https://doi.org/10.1016/j.physletb.2017.12.010

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

sure;conversely,iftheyareperturbedtosmallerradii,theyexpand becausekineticpressurebecomesstrongerthanself-gravity.

ForconfigurationswithM

∼

10−11_M





10−5 _eV

_/

_m



2_,

self-inter-actions cannot be neglected anymore, although the amplitude is still comparatively small,

|θ

0

| ∼

10−6

(

10−5eV

/

m

)

. For QCD

ax-ions,thelowestorderself-interactionisanattractivequarticterm. Foramplitudes

|θ

0

| 

10−6

(

10−5 eV

/

m

)

,theattractivequartic

self-interaction is stronger than gravity, which is negligible in this regime. In thiscritical branch, we find solutions when the quar-ticself-interaction balance the kinetic pressure withmass-radius relation R

∝

M. Note, that this relation implies that axion stars becomelighter withgrowingdensity, suchthat they alwayshave massesM



10−11M_



10−5eV

/

m



2 inthisbranch.However,the solutionsare unstableagainst perturbations:foragiven mass M, starsexpand whenperturbed toradiilarger thanthe equilibrium value since the quartic self-interactions are weaker than the re-pulsivepressure.Eventuallytheconfigurationrelaxestothedilute, interaction-freeregime described intheprevious paragraph. Con-versely, if configurations are perturbed to radii smaller than the equilibriumvalue,the quarticinteraction istoo strongto be bal-ancedbythepressureandthestarcollapsestoevenhigher densi-ties.

It has recently been pointed out, that new stable configura-tions, calleddense axion stars, are obtained when the amplitude ofthe axion field in the corereaches

|θ

0

|

=

O(

1

)

[27]. For such

amplitudes,the axion field scans the full non-perturbative axion potential, andself-interactionsmust be taken into account to all orders.Usingtheassumptionthattheaxionfieldinthestaris co-herentlyoscillatingatasinglefrequency,ascommonlyusedinthe literature, we obtain the mass-radius relation M

∝

R3_{, in}

agree-ment with Ref. [27]. However, we find that the single-harmonic approximation, which holds in the branches described above, is notaccurateforthedensebranch.Usingamulti-harmonic expan-sion,wefindthathigherharmonicsaregeneratedwithamplitudes comparable to the fundamental mode’s amplitude. Heuristically, thepresenceofhigherharmonicscorrespondstothegenerationof (relativistic)axionsbycoalescenceprocessesna

→

a.Wefindthat configurationsonthedensebranchdecayviaemissionof relativis-ticaxions,withlifetimes oforder

τ

life

∼

103

/

m, whichare much

shorterthananycosmologicaltimescale.

When

|θ

0

|



O(

1

)

, axions stars are short lived solutions of

the relativistic equation, elsewhere known as oscillons [37–44]. In the literature, similar objects have also been called pseudo-breathers [45], axitons [26], or oscillatons when driven by grav-ity[23,46,47].Since gravityis negligibleinthe densebranch,the axion field is described by the Klein–Gordon equation with the QCDchiralpotential(the

χ

-Gordonequation).Thereisalargebut scatteredliterature on finding solutions to related equations.For example,in one dimension, assuming a cosine potential leads to the Sine–Gordon equation, which admits localized breather solu-tionsthatarenotharmonic[48],i.e.whichfeatureaninfinite col-lectionofhigherharmonics.Inthreedimensionsoscillonsclosely resemble the breather solutions of the one dimensional Sine– Gordonequation, but they differin that they radiate energy and thusdecayinafinitelifetime,thoughslowlyrelativetothe “natu-ral”timescalesetbytheinversemassoftheparticles.

Justifying and expandingupon this concise summary, the re-mainderofthispaperisasfollows.InSec.2wesetout thebasic equations.InSec.3we findnumericallystablesolutionsand pro-vide quantitative resultsfor thedilute andthe criticalaxion star branches.InSec. 4we discussthe densebranch,andanalyzethe equilibriumandmetastabilityofdenseconfigurationsina relativis-ticframework.InSec.5weusethemass-radiusdiagramtosketch aqualitativestorylineforaxionstars,andinSec.6wesummarize andconclude.

2. Axionstars

2.1. AxionLagrangian

The axionresults frompromotingthe flavor-neutralCP violat-ing angle of the standard model,

θ

, to a dynamical field [2,1] in thePeccei–Quinnmechanism[10,11].Thecanonicalnormalization ofthedynamicalangle

θ (

x

)

requiresanewenergyscale f ,the ax-iondecay constant,to define theaxion field a

(

x

)

= θ(

x

)

f . In the following we will refer to both

θ

and a as the axion field. The dynamicsoftheaxion fieldundertheinfluenceofgravityare de-scribedbytheaction

S

=



d4x

√

−

g

L

=

=



d4x

√

−

g



1 2



∂

μa

 

∂

μa



−

V

(

a

/

f

)



,

(1)

wherethemetric gμν isdetermined by theEinsteinequation for theenergymomentumtensoroftheaxionfieldT μν

(

a

)

.Weadopt theaxionpotential[49,50],

V

(θ )

=



4 cz



1

−



1

−

4czsin2

(θ/

2

)



,

(2)

where



4

≈ (

75

.

5 MeV

)

4 isthe topologicalsusceptibility[50–52] andcz

≈

z

/(

1

+

z

)

2

≈

0

.

22 withtheratiooftheupanddownquark massesz

=

mu

/

md

≈

0

.

48.Note,thattheminimumofthepotential isatV

(

0

)

=

0 andthemaximumatV

(

π

)

= 

4



₁

₋

√

₁

₋

_4c

z



/

cz. Theaxionmassm andthequarticcouplingconstant

λ

aredefined through m2

=

1 f2 d2V d

θ

2





θ=0

=



4 f2

=



57

μ

eV10 11_GeV f



2

,

(3)

λ

=

1 f4 d4_V d

θ

4





θ=0

= −(

1

−

3cz

)

m2 f2

.

(4)

Assumingsphericalsymmetry andexpandingthemetric tolinear orderaboutflatspaceyieldsthelineelement

ds2

=

g_μνdxμdxν

= (

1

+

2

φ)

dt2

− (

1

−

2

φ)

dr2

−

r2d



2

,

(5)

where

φ

isthegravitationalpotential,which satisfiesthePoisson equationwithenergydensity

ρ

=

T00

(

a

)

,andd



isthedifferential solidangle.Inthefollowing,werescaletimeandradiusast

→

mt andr

→

mr,respectively,sothattheLagrangianinEq.(1)reads

L

= 

4

˙

θ

2 2

−

|θ



_|

2 2

− ˜

V

(θ )

,

(6)

where a dot indicates a derivative with respect to the rescaled time, a prime indicates a derivative withrespect to the rescaled radius, andV

˜

(θ )

≡

V

(θ )/

4. Couplingthe Poissonequation with the equation of motion obtained fromthe Lagrangian density

L

gives

¨θ = (

1

+

4

φ)



2

θ

 r

+ θ





₊

₄

_{˙φ ˙θ − (}

₁

₊

₂

_φ)

dV

˜

(θ )

d

θ

,

(7)

φ



+

2

φ

 r

=

4

π

β

ρ

˜

,

(8)

˜

ρ

=

ρ

kin

+

ρ

grad

+

ρ

pot

=

˙θ

2

2

+

|θ



_|

2

2

+ ˜

V

(θ ),

(9)

where

β

≡

G f2

= (

f

/

mPl

)

2 with the Planck mass mPl

=

1

.

221

×

reaches

ρ

˜

∼

1 when

|θ|

∼

π

andtheaxion potentialsaturates.In Eq.(9),wedenotethecontributionstotheenergydensityfromthe kinetic, gradient,and potential componentsseparately. Note,that thegradientenergyisduetothemomentumoftheaxionarising fromtheuncertaintyprinciple.Sofar,theonlyapproximationused isthatgravityisweak,

φ



1.

We anticipate one of the results of this paper, namely that thesystemcan bestudiedintwodifferentregimesdependingon whethertheaxion field is

|θ|



1 (the “dilute” andthe“critical” axion starregimes)or

|θ|



1 (the “dense”axion starregime).In thediluteandcriticalregimes,theaxionscomprisingthestarare non-relativistic and the tools described in Sec. 2.2 below apply. When

|θ|

∼

1,afullrelativisticdescriptionisneeded,aswesketch inSec.3.3.

2.2. Non-relativistic(singleharmonic)limit

When the non-relativisticlimit applies, the axion mass isthe largest energyscale inthe problem, so that axion stars oscillate atafrequencyveryclosetotheaxionmassm.Despitenon-linear interactions arisingfroma cosineor achiralpotential precluding axion stars solutions fromhavingone single frequency,for small fieldconfigurations

|θ|



1,theone-frequencyapproximation

θ

= (

r

)

cos

(

ω

t

) ,

(10)

suffices.Here,

ω

isthetotalenergyofaconstituentaxion,inunits of the axion mass. We write

ω

=

1

+



, where



accounts for thecontributionfromthebinding,kineticandself-interaction en-ergies, while the one accounts for the rest mass energy. In the non-relativisticapproximation,wehave

|



|



1 and

ω

≈

1.

We further assume that gravity is a weak effect, so that we candropalltermscontaining

φ

inEq.7,exceptfortheterm2

φθ

whichisofthesameorderas

¨θ + θ =



1

−

ω

2



_θ

_{≈ −}

₂

_

_θ

_.Wesplit

thepotentialintoamasstermandtheselfinteractionas

˜

V

(θ )

=

1



4 m2 2 a 2

₊

Vself

(θ )



4

=

θ

2 2

+ ˜

Vself

(θ ) .

(11)

InsertingtherepresentationinEq.(10)intoEqs.(7)–(9)and aver-agingovertheperiod2

π

/

ω

,weobtain





+

2



 r

2



W1

()

+ φ +

ω

2

₋

₁ 2



,

(12)

φ



+

2

φ

 r

4

π

β

ρ

˜

,

(13)

˜

ρ

˜

ρ

kin

+ ˜

ρ

grad

+ ˜

ρ

pot

.

(14)

In the last expressions, we have introduced the energy density terms

˜

ρ

kin

=

ω

2 4



2

_,

_ρ

_˜

grad

=

|



_|

2 4

,

ρ

˜

pot

=



2 4

+

W

(),

(15)

andwehavedefinedtheeffectiveself-interactionpotentialandits firstderivativethrough W

()

=

1 2

π

2π



0

˜

Vself

(θ )

d

(

ω

t

),

(16) W1

()

=

2 dW

()

d



2

.

(17)

For

|



|



1,Eq.(12)isaSchrödingerequationfortheradial eigen-function



witheigen-energy



,whiletheenergydensityreduces

Table 1

ThecoefficientsinthetruncatedseriesexpansionofthechiralpotentialinEq.(18), afterthecorrectionsdescribedbelowEq.(20)andforz=0.48.

v0=1.30264 v1= −1.4403 v2=0.1692 v3= −0.0404 v4=0.0105 v5=0.001636

to

ρ

˜

= 

2

_/

_{2 since thecontributions from thegradient termand}

self-interactionsarenegligible.

We stressthatourprocedure,whichinvolves theaverageover 2

π

/

ω

of the equation of motion leads to the same results as what was obtained in Ref. [53], where the authors neglect the rapidly oscillating terms proportional to powers of exp

(

i

ω

t

)

. As long as gravity is negligibleand the single-harmonic approxima-tioninEq.(10)holds,Eqs.(12)–(14)arevalidevenforrelativistic axions. We anticipate,that for (mostof) the dense branch, grav-ityisindeednegligiblebutthesingleharmonicapproximationno longerholds.

2.3. Axionpotential

WeexpandtheexpressioninEq.(2)as

˜

V

(θ )

=

∞



h=0 vhcos

(

h

θ ),

(18) v0

=

1 2

π

2π



0

˜

V

(θ )

d

θ,

(19) vh>0

=

1

π

2π



0

˜

V

(θ )

cos h

θ

d

θ.

(20)

In our numericalcalculation, we truncate thesum inEq.(18) to thefirstfivetermsh

≤

5.Thisattainsaprecisionbelow1%with re-specttothechiralpotentialinEq.(2);thisprecisionisbetterthan the accuracy of the chiral perturbation theory itself. We slightly modify thecoefficients vh sothat thetruncatedpotential shows: I)thesameminimumV

˜

(

0

)

=

0,II)thesamemassV

˜

θ θ

=

1,andIII) thesamequarticcouplingV

˜

θ θ θ θ

= λ

φasthefullchiralpotentialin Eq. (2), wherethe (negative) quantity

λ

φ

= −(

1

−

3cz

)

isrelated totheaxionquarticself-interactionconstantas

λ

= λ

φ

(

m

/

f

)

2.The numerical values of the corresponding corrected coefficients are giveninTable 1forz

=

0

.

48.

Theeffectivenon-relativisticpotentialinEq.(16)is

W

()

=





h vhJ0

(

h

)

−



2 4

,

(21)

where J0

(

x

)

istheBesselfunctionofthefirstkindoforderzerofor

theargumentx.Noticethatthecosinepotentialisrecoveredinthe limit cz

→

0,equivalent tosetting v0

=

1, v1

= −

1,andallother

vh equaltozeroinEq.(18).ThesetofEqs.(12)–(14)hasbeen ex-tensively applied to self-gravitating systems madeof bosons. For the caseofaxions,the free case W1

()

=

0 has beenstudied in

Refs. [23,54,55] following the seminal work in Refs. [17,18]. The potential expanded to the quartic interactions has been studied inRefs. [56,57,53,58].Ref. [27]considers thesetofEqs.(12)–(14) withthecosinepotential,usingtheexpressionfortheenergy den-sity (in ournotation)

ρ

˜

= 

2

_/

_{2, insteadof ourEq.(15)obtained}

contributions fromself-interaction andkinetic energyto the en-ergydensity,whichsourcesthegravitationalpotential.Asweshow below,thosecontributionstotheenergydensityaffecttheresults forthe“dense”branch.

3. Numericalresultsinthesingleharmonicapproximation

3.1.Axionstarbranches

We numericallysolve for the radial profile

(

r

)

appearing in thesetofEqs(12)–(14),asafunctionofthefrequency

ω

.We im-posetheboundaryconditions

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

˜

ρ

0

= (

1

+

ω

2

)

|

0

|

2

/

4

+

W

(

0

),

(

r

→ ∞) =

0

,







_r=0

=

0

,

φ (

r

→ ∞) =

0

,

(22)

where

ρ

˜

0 istherescaledenergydensityatr

=

0 andthecore

am-plitude



0istheamplitudeoftheaxionfieldatr

=

0.Weobtain

aradialprofile

(

r

)

viaashootingmethod,thatisbyvaryingthe value of the core amplitude



0 until we find a profile that

de-caysasexp

(

−

kr

)/

r atasufficiently larger. The solutionwe seek showsno nodes, andcorresponds to the lowest energy state for a given value of



. See Ref. [59] for excited states of an axion star witha quartic potential. We find solutions for all values of

ω

within the range (0,1), although the numerics are particularly trickyas we approach

ω

=

0. For each value of

ω

, we obtain a uniquevalueofthecoreamplitudeandauniqueprofile.Giventhe radial profile, we obtain the total mass M

=



d3r

ρ

and the ra-diusR oftheaxionstar,thelatterdefinedastheradiuscontaining 90 %of theenergy[18].InFig. 1,we show themass-radius rela-tionforthreevaluesof f

= {

1011

,

1013

,

1015

}

GeV.1 Eachpointon theline ischaracterized by a fixed value of

ω

andthe core am-plitude



0.Forincreasingvalueof



0,weidentifythreedifferent

regimes:thedilutebranch(

|

0

|

 β

1/2),theunstablecritical

con-figurations (

β

1/2

 

0



1), and the dense branch (



0



1). For

thecriticalline(reddashed line)and(mostof)the densebranch (dashedblackline),gravityisnegligible.Thenwefinduniversal so-lutionswhenexpressedintermsofthenaturalunitsofstarmass, f2

/

m, andradius, 1

/

m. However, gravity isrelevantin thedilute branch,wheresolutionsdependonthevalueof f through

β

. 3.2.Non-relativisticsolutions

Inthissection, we presentheuristicarguments explainingthe numericalresultsobtained inthe previous Sec. 3.1forthe dilute andcritical brancheswhere



0



1; see also [56,57] for a

simi-larapproach. These branchescan be understood interms of the different contributions to the axion star energy U : the gravita-tionalbindingenergy, thegradientenergy, andthe(quartic) self-interactioncontribution, U

∝ −

G M 2 R

+



d3r



f2 2

(θ



₎

2

₊

λ

φ 4

!



4

_θ

4



= −

G M2 R

+

α

k f2

|

0

|

2 2R2 R 3

₊

_α

4

λ

φ 4

!



4

_|

0

|

4R3

.

(23)

Here,

α

kand

α

4 aredimensionlessparameterswhichweinsertto

matchtheanalyticalresultsderivedfromEq.(23)withthe numer-icalsolution.Estimatingthemassoftheaxionstaras

1 _{Note,thatfor f=10}15_{GeV somefine-tuningofthemisalignmentangleis}

re-quiredtoavoidoverclosureoftheUniverse[60,61].

Fig. 1. Lineofequilibriumsolutionsofthenon-relativisticaxion-starequationsalong thedilutebranchfor f=1011_{GeV (blue),f=10}13_{GeV (green), f=10}15_GeV

(or-ange),connectingtotheunstablebranchalongthecriticalline(reddashed).Central densityincreases withthearrows.Alsoshownisthemeta-stabledensesolution (dashedblack).Notethattheseresultsareobtainedinthesingle-harmonic approx-imationandthustheblackdashedcurvedescribingthedenseregimeshouldnot betrusted.(Forinterpretationofthereferencestocolorinthisfigure,thereaderis referredtothewebversionofthisarticle.)

M

=



d3r

ρ

∼ 

4

|

0

|

2R3

,

(24)

we can expressthe centralamplitude as

|

0

|

2

∼

M

/(

4R3

)

,and

thetotalenergyU canberewrittenas

U

∝ −

G M 2 R

+

α

k f2_M 2



4_R2

+

α

4

λ

φ 4

!

M2



4_R3

=

=

f2 m



−

β ˜

M

_˜

2 R

+

α

k

˜

M 2R

˜

2

+

α

4

λ

φ 4

!

˜

M2

˜

R3



.

(25)

In the last equality, we have used the scaling property of the Schrödinger–Poissonequation,writingthemassandtheradiusof the starinterms ofdimensionlessquantities, M

˜

=

M

(

m

/

f2

₎

_and

˜

R

=

mR.Thenaturalscaleforthemassandtheradiusoftheaxion stararethen

f2 m

=

3

×

10 −20_M 



10−5_eV m



3

,

(26) 1 m

=

3

×

10 −11_R 



10−5eV m



,

(27)

whereM_ andR_ arerespectivelythemassandtheradiusofthe Sun.Theequilibriumconfigurationsoftheaxionstarcanbe quali-tativelyobtainedbyminimizingtheenergydensityinEq.(25)with respectto R,

˜

whilefixingtheaxionstarmassor,equivalently,the totalnumberofaxions N

=

M

/

m. Thisgivesaquadraticequation whosesolutionscorrespondtotheradiusofthestarforeitherthe dilutebranch(R

˜

₊)orthecriticalbranch(R

˜

₋),namely

˜

R_±

=

α

k 2

β ˜

M



1

±



1

−

α

4

|λ

φ

|β ˜

M 2 2

α

_k2

.

(28)

The stability of the solution is determined by the sign of

∂

2U

/∂

R2



_R=R

±. Solutions in the dilute branch (

ρ

˜

0

 β

) are

sta-ble,while those inthe criticalbranch (

β

 ˜

ρ

0



1) are unstable.

Matchingontoournumericalresultsfromsection3.1,weobtain

α

k

=

9

.

9

,

α

4

=

1

.

7

,

(29)

independentofthevalueof

β

.

Thedilutebranchoftheaxion starcorresponds tothe equilib-riumbetweenthegradientenergyandgravity. Dependingonthe valueofthedecayconstant,equilibriumconfigurationsofthistype populatethelinewithnegativeslopeinFig. 1with f

=

1011_GeV

(blue), f

=

1013GeV (green),or f

=

1015GeV (orange),withthe mass-radiusrelation

˜

R₊





λφ→0

=

α

k

β ˜

M

.

(30)

For configurations lying above thisequilibrium line, the gravita-tional pull overcomes gradient pressure, so these configurations contract. On the contrary, configurations lying below the mass-radiuslineinEq.(30)arerestoredtotheequilibriumconditionby thegradientpressureterm.Hence,arestoringforce actstovanish anydeviationfromthestableequilibrium.

Thecriticalbranch,thedashedredlineinFig. 1,correspondsto thebalanceofthegradientandthequarticself-interactionenergy contributions,withmass-radiusrelation

˜

R₋





G→0

=

α

4

|λ

φ

| ˜

M 8

α

k

.

(31)

Deviations from this configuration are pushed either further to-wards the dilute branch or to further contraction andare hence unstable.Asolutionfortheradiusoftheaxionstarexistsaslong asthequantitybelowthesquare rootinEq.(28)ispositive,that iswhenthemassofthestarissmallerthanthecriticalvalue

˜

M_∗

=



2

α

2 k

α

4

|λ

φ

|β

=

1

.

3

×

109



−λ

φ



1011GeV f



,

(32)

whichcorrespondstotheradius R

˜

_∗

=

andtothecoreamplitude

˜

R_∗

=

α

k 2

β ˜

M_∗

=



α

4

|λ

φ

|

8

β

,

(33)

|

∗0

| =

√

32

β

α

k

α

4

|λ

φ

|

=

8

.

8

×

10−8

|λ

φ

|



f 1011_GeV



.

(34)

ThevaluesofM

˜

_∗and R

˜

_∗ definetheturningpointinthetopright corner of Fig. 1, corresponding to the transition from the dilute tothecriticalbranch.Inthecriticalbranch,adensersolution cor-responds to moving along the red dashed linein Fig. 1 towards the bottom left of the figure, with the star contracting and be-cominglighter.Since inthis branchthecore amplitude increases as



0

= 

∗0M∗

/

M, non-perturbative dynamics becomes relevant

when



0

≈

1,oratatypicalmass

˜

M

(

0

=

1

)

≈ 

∗0M

˜

∗

=



4

α

k

α

4

|λ

φ

|



3/2

=

110

|λ

φ

|

3/2

,

(35)

˜

R

(

0

=

1

)

≈



_α

k

α

4

|λ

φ

|

=

2

.

4



−λ

φ

.

(36)

ThesevaluesofM

˜

(

0

=

1

)

andR

˜

(

0

=

1

)

markthesecond

turn-ing point inthe bottom-left region ofFig. 1.Forlarger values of thecoreamplitude, theaxion field exploresthe wholechiral po-tentialandadifferenttreatmentisneeded.

Fig. 2. Thefrequencyoftheaxionstarω(blacksolidline)asafunctionofthe coreamplitude0forournumericalsolutionsofthenon-relativisticstability

equa-tions,(12)–(14).Wealsoshowthecontributionstothetotalenergyfromthekinetic (bluedottedline),gradient(orange dashed line),and potentialenergy(red dot-dashedline).Inthedensebranch,i.e.01,thesolutionisnotconsistentwith

thenon-relativisticapproximation.(Forinterpretationofthereferencestocolorin thisfigure,thereaderisreferredtothewebversionofthisarticle.)

3.3. Non-perturbativesolution

The axion star solutions found for



0



1 correspond to a

clumpofaxions whosetotalmass andradiusare largerthan the criticalvaluesinEqs.(35)and(36).Forsuchconfigurations,higher order terms in the attractive self-interacting potential cannot be neglected anda newregime isobtained, oftenreferred to asthe “dense”axionstarregimeintherecentliterature[27,32].Weshow thenumericalresultsforthemass-radiusrelationobtainedinthe dense branchconfigurationwiththesolidblacklineinFig. 1. Fit-ting the curve far from the turning point leads to the relation

˜

R

=

0

.

6M

˜

1/3_{.Thisregime corresponds toclassicallystable}

config-urationwithanalmostconstantdensity

ρ

∼ 

4 _{intheinnercore.}

For themass-radius relation,we have obtainedthe same power-lawexponent (1/3)asin Ref.[27],because suchdependence fol-lowsfromthefactthatthesolutioninthedensebranchsaturates theQCDpotentialandleadstoaconstantdensityofthestar.

However,thestructureofoursolutiondiffersgreatlyfromwhat was obtainedinRef. [27]. We disagreeon their interpretation of the equilibrium of the axion star in the dense branch for three main reasons. I) We have included the self-interactions and the gradientenergytermsthroughEq.(15).Thesetermscannotbe ne-glected,asweshowinFig. 2.II)InRef.[27]thesetofequationsis solved intheThomas–Fermiapproximation,that isneglecting the Laplacianof



appearingontheleft-handsideofEq.(12).III)Most importantly,thesingle-harmonicapproximationinEq.10doesnot holdinthenon-perturbativeregime.

InFig. 2,weshowthedifferentcontributionstothemassofthe axion star, M

=



d3_r

_ρ

_{, fromthe various components in Eq.(9),}

namely uα

=



d3_r

_ρ

α

/

M, where

α

∈

[kin

,

grad

,

pot],asa function ofthecoreamplitude.Inthe



0



1 (



0



1) regimeshown, the

starisinthecritical(dense)branch.Inthecriticalbranch,the ki-neticandpotentialenergiesbothcontributeafactorequalto1

/

2.

Fig. 3. TherescaledaxionstarradiusR times˜ theaxionfrequencyω,asafunction ofthecoreamplitude0.

Thisresultcanbeinterpretedbythefactthatthewavefunctionof thecoherentaxionfieldundergoesharmonicoscillations,withthe energydensityequipartitioned betweenthekinetic and potential terms.However, aswe approach thedense regime, the contribu-tionfromthegradienttermincreases,totheextentthatfor





1 all three components contribute with a similar magnitude. Thus fordense axion starsthe energydensity mustincludeall energy contributions.Also,theThomas–Fermiapproximation isnot justi-fiedsincetheLaplaciantermiscrucialforsolvingEq.(12)inthe wholedomainshowninFig. 2 and 3.

Inmoredetail,thestructureofadenseaxionstarlooksas fol-lows. The stellar core is composed of relativistic axions since in thatregion

ω

2

_

_{∼ ∇}

2

_

_{,althoughself-interactionsarenotentirely}

negligible.Aswemoveoutofthecore,thereisanintermediate re-gionwheretheself-interactionsbalancethegradientterm.Finally, intheoutmostpartself-interactionsareagainnegligible.

To further illustrate that the axion field is relativistic in the denseregime, inFig. 2 weshow theaxion energyper particle

ω

(blacksolid line), whichdrops tozero for





1, duetothe fact that self-interactionsincrease with



0. Then, the non-relativistic

condition

ω



π

/ ˜

R,whichexpresses thatthetypical momentum oftheaxionismuchsmallerthanitsenergy,nolongerholds.Fig. 3 alsoshowsthisconclusion,sincethequantity

ω

R decreases

˜

from beingmuchlargerthanonetoaconstantvalue

∼

3 forwhichthe non-relativisticinterpretationdoesno longerhold. The inequality mR



1,or R

˜



1,whichholds eveninthe densebranch, isnot sufficienttojustifyanon-relativisticapproach.

In addition, our solution shows that gravity is negligible ev-erywhere inside the star. The gravitational energy density at a distancer from the center of the star is

ρ

G

=

G

ρ

M

(

r

)/

r, where M

(

r

)

isthemassenclosedwithintheradiusr,sowecanwrite

ρ

G

ρ

=

β ˜

M

˜

R

=

4

.

6

β ˜

R 2

_,

₍₃₇₎

where in the last step we have used the parametrization R

˜

=

0

.

6M

˜

1/3. Hence,gravity can beneglected for R

˜



√

1

/β

. For R

˜

=

O(

1

)

,gravitycanbesafelyneglectedaslongas f



mPlor

β



1,

which is therange of parameters considered in thiswork. How-ever, fordense axion star solutions oflarger mass,gravity could eventuallybecomeimportantagainforR

˜

≈ (

4

.

6

β)

−1/2.Wedonot considerthislatterpossibilityhere.

Aswehavepreviouslydiscussed,thesolutionsobtainedinthe densebranch arenotself-consistent becausethesingle frequency approximationinEq.(10)isnot justifiedonthebasisofthe find-ings in Fig. 3. When the amplitude of the axion field becomes



=

O(

1

)

,theaxion fieldsprobes thefull chiralpotentialandall orders ofself-interactionbecomerelevant.Then, higherharmonic modesoftheaxionfieldwhosefrequencyisamultipleofthe fun-damentalmode

ω

=

m aregeneratedwithamplitude comparable tothatofthefundamentalmode.Inthenextsectionwetherefore startoverfromEq.(7)andperformamulti-harmonicexpansion.

4. Oscillons

4.1. Generalitiesontherelativisticequation

Based on the findings of the previous Section, axions in the dense regime



0



O(

1

)

can be studied using a relativistic

ap-proach andignoring gravity. For simplicity, we derive results for theillustrativecaseofacosinepotential

V

(θ )

= 

4

(

1

−

cos

θ ) ,

(38)

obtainedfromthechiralpotentialEq.(2)forcz

→

0.Inthatcase, therelativisticequationofmotionistheSine–Gordonequation

¨θ − θ



₋

2

r

θ



₊

_sin

_θ

₌

₀

_.

₍₃₉₎

Wewish toidentifytheoscillonsolutionsofEq.(39),namelythe solutions that are spatially-localized and time-periodic. Such so-lutions circumvent Derrick’stheorem [62], which states that the scalar field Lagrangian in Eq. (1) expressed in flat space–time doesnotadmit time-independent,finiteenergysolutions because shrinkinganon-zerofieldconfigurationeffectivelyreducesthe to-tal energy of the system [63–67]. Although the ansatz we used previously,Eq.(10),isnot aproper solutionforthenon-time av-eragedpotential,weexpectittobeareasonableapproximationat the transition fromthe non-relativistic to the relativistic domain when



0

∼

1.

There is a long historyofsearching foroscillons of theSine– Gordonequation, withthemostpositiveoutcome beingsolutions thatlast

O(

100–1000

)

oscillationsinunitsof1

/

m[68,42,69,70,44]. The generalconsensus is that absolutely stablesolutions do not exist,althoughweknowofnodefiniteproof.Inanycase,ismuch thatwecanlearnaboutunstableoscillonsfromtheliterature.

Foraxionsinparticular, KolbandTkachev[26] discovered the so called “axitons” when studying the cosmologicalevolution of theaxionfieldinthedarkmattercontext.Theyfollowedthe evo-lution of the Sine–Gordon equation in an expanding Universe in which theaxion mass stronglydependson thecosmic time, and identifiedaninstabilityconditionthatleadstosmallclumpsofthe axionfield withlargevalues

θ

∼

π

todisappearinburstsof rela-tivisticaxions.Thisinstability,whichoriginatesfromtheattractive quarticself-interactionterm,iswellknowninthecondensed mat-ter community and has been recently revisited in Ref. [30]. In that paper, theauthors followthe collapseof a dilute axion star with a mass slightlyabove the critical value M_∗. The axion star solutionshowsa self-similarcollapsethat endswhen thecentral amplitude saturates theaxion potential. Then,the axion field os-cillatesforafewtimes,radiatingrelativisticaxionsandrelaxingto a small amplitude which is nevertheless larger than the starting

value.Suchinstabilitiesaretriggeredforafewtimesuntilthe cen-tralamplituderelaxestothestability regiondescribedabove.The simulationsincludegravity,sothatthefinalstatecanstillbea di-luteaxionstar,butthedynamicsofthecollapseandtheradiation ofrelativisticaxionshappensatvery smallradii wheregravity is negligiblecomparedwiththeself-interactionsandgradients.

The simulations in Ref. [30] are of considerable phenomeno-logical interest, since in principle the collapse of dilute stars is themostnaturalmechanism toproduce dense axionstars. How-ever, one can address the question of dense axion star stability separatelyfromtheirpossiblecosmologicalorigin.Forsuchatask weneedothermeans.ApromisingapproachemergedinRef.[71], wheretheauthorsconverttheSine–Gordonequationintoaseries ofequationswithdifferentharmonics.

4.2. Beyondthe1stharmonicapproximation

Ageneraltime-periodicsolutioncanbe writtenintermsofan infinitenumerablesetofharmonics.Thus wecanwrite our oscil-lonansatzas

θ

as[72]

θ

=



n



2n+1

(

r

)

cos [

(

2n

+

1

)

ω

t]

,

(40)

which,oncepluggedintotheSine–GordonEq.(39),yieldsasetof coupledequationsforthedifferentharmonics,

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩



₁

+

2_r



₁

+

ω

2



1

=

I0

,



₃

+

2_r



₃

+ (

3

ω

)

2



3

=

I1

,



₅

+

2_r



₅

+ (

5

ω

)

2



5

=

I2

,

..

.

(41)

Here,wehaveintroducedthenotation

Im

=

1

π

2π



0 d

φ

cos

((

2m

+

1

)φ)

×

×

sin





n



2n+1

(

r

)

cos [

(

2n

+

1

)φ

]

.

(42)

ThesetofEq.(41)isageneralizationofEq.(10)whenhigher har-monicsotherthanthefundamentalmode

ω

areconsidered;when truncatingthesumatn

=

0 weobtainthesingleharmonic approx-imationEq.(12)with



1

≡ 

.

Asanexample,weconsiderthecasewherewealsoincludethe firstterm beyondthesingle-harmonic approximation besidesthe fundamentalmode

ω

.Thisgives



₁

+

2 r



 1

=

I1

−

ω

2



1

,

(43)



₃

+

2 r



 3

=

I3

−

9

ω

2



3

,

(44) I2n+1

≈

2

(

−

1

)

nJ2n+1

(

1

)

+ 

3D2n+1

(

1

),

(45)

wherewe haveapproximatedthecomputation ofthe coefficients I1 andI3 byexpandingaround



3

=

0,with

D1

(

1

)

≈ −



2₁

8

,

and D3

(

1

)

≈

1

−



2₁

4

.

(46)

Infact, the solutions foundin Sec. 3.3correspond to the zeroth-orderapproximation ofthefull non-linearsolution,whilesolving thesetofEqs.(43)–(44)givesthenext-to-leadingorder contribu-tion.

Fig. 4. Thefirstharmonic1(blue)andthesecondharmonic3(red),satisfying

thesetofEqs.(43)–(44),asafunctionoftheradiusinunitsofthe axionmass. Wefixtheaxionfrequencyω=2π/T ,withT=7.(Forinterpretationofthe refer-encestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthis article.)

At r

→ ∞

,solutionsmust approachzero,with



1

,



3



1.In

thisregime,thehigherharmonic



3 mustsatisfy



₃

+

2 r



 3

+



9

ω

2

−

1





3

= −



3₁ 24

.

(47)

which,intherange1

/

3

<

ω

<

1,isanoscillatorysolutioninspace withwavelength

√

9

ω

2

₋

_1.Thus,if

_ω

_>

₁

_/

_{3,the axionstar}

con-figuration radiatesenergyawaythrough thethirdharmonic,with a contribution that increases withthe total energy of the axion

ω

.Equation(47)caninprinciplebeusedtocomputethelifetime of the axion starin the dilute branch,where the third harmon-icsisaverysmallperturbationoftheexactsolution.Forvaluesof

ω

<

1

/

3,thereisnoradiationsolutionatr

→ ∞

andhigher har-monicshave tobe takenintoaccount through Eq.(41). InFig. 4, wesolvethesetinEqs.(43)–(44)foragivenfrequency

ω

=

2

π

/

T , with T

=

7,using ashooting methodto obtain the initial condi-tionsfor



1 and



3 atr

=

0 thatsatisfy



1

(

+∞)

= 

3

(

+∞)

=

0.

Westress thattheamplitudesofthe1stand3rdharmonicareof thesameorderofmagnitudeeverywhereinthestar, demonstrat-ing thatthesingleharmonicapproximationsufficientforthecase of dilute axions stars doesnot suffice forthe description of the denseregime.

5. Discussion

We haveshown that when



0



O(

1

)

, axions are relativistic

andaxion starsenterthe densebranchregime wherethe config-uration behaves asa meta-stable oscillonofthe

χ

-Gordon equa-tion,witha characteristiclifetime.Forafree field,bosons stream away fromtheoscilloncoreofastarofradius R,withalifetime

τ

lin

=

0

.

836

√

2R2 _{[73] andwitha radiation spectrum peakingat}

ω

lin with width



lin

= (

2

τ

lin

)

−1. Including a more realistic

non-linearself-interactionpotentialmodifiesthespectrumbylowering thepeakfrequencyatalowervalue

ω

nl

<

ω

lin,withanewwidth



nl

<



lin.FollowingRef.[73],anoscillonformsifthetwospectra

donotsignificantlyoverlap,thatiswhen

ω

lin

−

ω

nl

>



lin

+ 

nl

2

≈ 

lin

.

(48)

The computation of the oscillon lifetime for a quartic self-inter-action has been addressed in Refs. [74,73], where the relatively longlifetime (on the scaleof the intrinsictimescale m−1_{) of}

os-cillons is explained by the relatively small overlap between the oscillationfrequencies. Followingthismethod,we estimateofthe lifetimeofanoscillonforacosinepotentialas

τ

life

=

1

α

(

Eosc

−

E∞

)

≈

700 m

=

10 −8_s

_,

₍₄₉₎

wherewe haveused theparameters

α

=

5

×

10−5_{, E}

osc

=

402

.

1,

andE_∞

=

372

.

8,followingRefs.[74,73]withtheaxionLagrangian inEq. (1)anda Gaussian ansatz forthe radial wave function. In short,theenergyofan oscillonisdescribed byitsradiusand am-plitude,anddampedoscillationsintheoscillondevelopalongthe line of constant minimum energy [42,43]. We performed an in-dependent check of these results by using the solutions of the time-independent Eq. (12) in the dense branch as initial condi-tions which we time-evolve with the Sine–Gordon Eq. (39), as prescribedinRef.[71].Althoughthisinitialwavefunctionisnota propersolutiontotheSine–Gordonequation, ournumerical solu-tionsyieldbreathersolutions.ForaperiodTnl

=

7

.

0 asconsidered

above,we findthat thesolutiondecaysafter

τ

life

≈

1200

/

m. This

resultisofthesameorderofmagnitudeaswhatweobtained us-ingEq.(49)

τ

life

=

O



103 m



≈

10−7s



10−5_eV m



.

(50)

Thefactthat pseudo-breathersexisthasbeenshowninRef. [45], where the existence of a finite life-time solution to the Sine– Gordonequation hasbeenrelatedto thesingular behaviorofthe solution atzero, when an oscillating function has been imposed as the boundary condition of the solution at infinity. Pseudo-breathers are ultimately decaying states, as discussed in length in Ref. [26], where it is found numerically that such solutions areunstableandfragmentintosmallerclumps.Thedynamicsand theinitialconditionsconsideredinRef.[26]arehoweverdifferent fromours, sincetheauthorsconsider acosmologicalevolutionof theaxionfield with“whitenoise” initialconditions,andincluded theHubblerateintheequationofmotion.

Recently, Helfer etal. [31] have studied the stability of axion stars including gravity and non-linear effects, finding that stable denseprofilesmaybepossiblewhen f



0

.

1MPl,theexactvalue

depending on the axion starmass. In anycase, the energy scale f involved iswell above the scales we consider here. Forvalues of f below this critical value, the axion star either collapses to a blackhole ordissolves by the emission of relativisticparticles, consistentlywiththepuffingout obtainedinRef. [30]andinthis work.

6. Conclusions

Inthispaper, we havediscussed theproperties ofaxion stars for all allowed values of the core amplitude of the axion field



0. In particular, we have discussed how classically stable

solu-tionscanarisefromtheinterplaybetweenself-gravity,axion self-interactions,thepressureduetotheHeisenberg uncertainty prin-ciple,andthekineticenergy. Usingassumptions commonlymade inthe literature, we haveobtained a set ofequations describing

coherentaxionfieldoscillationsinsidetheaxionstarinthe single-harmonic approximation. For small core amplitudes



0



1, we

confirmedknownresults foraxionstars inthedilute andcritical branches, andprovided a heuristic interpretation ofthose results fromfirstprinciples.

For





1, the “dense” regime, we recover similar results to thoseinRef.[27] whenusingthesingle-harmonic approximation, inparticular,the massradiusrelation R

∝

M1/3_{.However, we}

ar-guethatthesingle-harmonicapproximationdoesnotholdforthe denseregimeandthusadifferentapproachisneeded,takinginto account higher harmonics.In the end, we arrive a very different physicalinterpretationofthedenseregime.Wefindgravitytobe negligiblefor



=

O(

1

)

.Denseaxionstarsshould besolutionsto theSine–Gordon(or

χ

-Gordon)equationdescribingtheaxionfield insidethestar.We computedthelifetimeofdenseconfigurations using both the semi-analytical procedure described in [42,43,74, 73] and by using our single-harmonic solutions as initial condi-tions, whichwe time-evolvednumericallyusing theSine–Gordon equation asprescribedin[71,72].Both methodsyieldcomparable lifetimesoforder

τ

life

∼

103

/

m,muchshorterthanany

cosmologi-caltimescale.

We conclude that if dense axion stars can be formed, they would immediately (on cosmological scales) radiate relativistic axions and decay. Since axion stars in the critical branch are unstable against perturbations and either expand to stable di-lute configurations or contract to the dense branch and subse-quently decay, stable axion stars with mass M

> ˜

M∗

(

f2

_/

_m

₎

_∼

10−11M_



10−5 eV

/

m



2appearimplausible.

Additionalnote

Duringthefinalpreparationofthemanuscriptaftercompletion of this work we received [75,76], partially overlapping with this work.

Acknowledgements

We would like to thank Eric Braaten, Katherine Clough, Mal-colm Fairbairn, Oleg Gnedin, Thomas Helfer, ˘Zelimir Marojevi ´c, David J. E.Marsh, and ScottTremaine for the useful discussions andcommentsthatledtothepresentwork.

SBandLV wouldliketothanktheUniversity ofMichiganand theMassachusettsInstituteofTechnology,wherepartofthiswork wasconducted,forhospitality.

SB, KF, and LV acknowledge support by the Vetenskapsrådet (Swedish Research Council) through contract No. 638-2013-8993 andtheOskarKleinCentreforCosmoparticlePhysics.KF acknowl-edges supportfromDoEgrantDE-SC007859andtheMCTPatthe UniversityofMichigan.JRissupported bytheRamonyCajal Fel-lowship2012-10597,thegrantFPA2015-65745-P(MINECO/FEDER), theEU throughtheITN “Elusives”H2020-MSCA-ITN-2015/674896 and the Deutsche Forschungsgemeinschaft under grant SFB-1258 asaMercatorFellow. FW’sworkissupportedby theU.S. Depart-mentofEnergyundergrantDE-SC0012567,theEuropeanResearch Councilundergrant742104,andtheVetenskapsrådet(Swedish Re-searchCouncil)underContractNo.335-2014-7424.

References

[1] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223–226, https://doi.org/10.1103/ PhysRevLett.40.223.

[2] F. Wilczek, Phys. Rev. Lett. 40 (1978) 279–282, https://doi.org/10.1103/ PhysRevLett.40.279.

[3] M.Shifman,A.Vainshtein, V.Zakharov,Nucl. Phys.B166(1980) 493–506,

https://doi.org/10.1016/0550-3213(80)90209-6, www.sciencedirect.com/ science/article/pii/0550321380902096.

[4] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103–107, https://doi.org/10.1103/ PhysRevLett.43.103,http://link.aps.org/doi/10.1103/PhysRevLett.43.103. [5]A.R.Zhitnitsky,Sov.J.Nucl.Phys.31(1980)260.

[6] M.Dine,W.Fischler,M.Srednicki,Phys.Lett.B104(1981)199,https://doi.org/ 10.1016/0370-2693(81)90590-6.

[7] J.Preskill,M.B.Wise,F.Wilczek,Phys.Lett.B120(1983)127–132,https://doi. org/10.1016/0370-2693(83)90637-8.

[8] L.F.Abbott,P.Sikivie,Phys.Lett.B120(1983)133–136,https://doi.org/10.1016/ 0370-2693(83)90638-X.

[9] M.Dine,W.Fischler,Phys.Lett.B120(1983)137–141,https://doi.org/10.1016/ 0370-2693(83)90639-1.

[10] R.D.Peccei,H.R.Quinn,Phys.Rev.Lett.38(1977)1440–1443,https://doi.org/ 10.1103/PhysRevLett.38.1440.

[11] R.D.Peccei,H.R.Quinn,Phys.Rev.D16(1977)1791–1797,https://doi.org/10. 1103/PhysRevD.16.1791.

[12] P.Arias,D.Cadamuro,M.Goodsell,J.Jaeckel,J.Redondo,A.Ringwald,J. Cos-mol. Astropart. Phys. 1206 (2012) 013, https://doi.org/10.1088/1475-7516/ 2012/06/013,arXiv:1201.5902.

[13] P.Svrcek,E.Witten,J.HighEnergyPhys.06(2006)051,https://doi.org/10.1088/ 1126-6708/2006/06/051,arXiv:hep-th/0605206.

[14]A.Arvanitaki,S.Dimopoulos,S.Dubovsky,N.Kaloper,J.March-Russell,Phys. Rev.D81(2010)123530,arXiv:0905.4720.

[15] A.Ringwald,J.Phys.Conf.Ser.485(2014)012013,https://doi.org/10.1088/1742 -6596/485/1/012013,arXiv:1209.2299.

[16]J.Halverson,C.Long,P.Nath,arXiv:1703.07779,2017.

[17] D.J.Kaup,Phys. Rev.172(1968)1331–1342,https://doi.org/10.1103/PhysRev. 172.1331.

[18] R.Ruffini,S.Bonazzola,Phys.Rev.187(1969)1767–1783,https://doi.org/10. 1103/PhysRev.187.1767.

[19] A.Das,J.Math.Phys.4(1963)45–51,https://doi.org/10.1063/1.1703887. [20] D.A.Feinblum,W.A.McKinley,Phys.Rev.168(1968)1445,https://doi.org/10.

1103/PhysRev.168.1445.

[21] A.F.D.F.Teixeira,I.Wolk,M.M.Som,Phys.Rev.D12(1975)319–322,https:// doi.org/10.1103/PhysRevD.12.319.

[22] M.Colpi,S.L.Shapiro,I. Wasserman,Phys. Rev.Lett. 57(1986) 2485–2488,

https://doi.org/10.1103/PhysRevLett.57.2485.

[23] E.Seidel,W.M.Suen,Phys. Rev.Lett.66 (1991)1659–1662,https://doi.org/ 10.1103/PhysRevLett.66.1659.

[24] I.I.Tkachev,Phys.Lett.B261(1991)289–293, https://doi.org/10.1016/0370-2693(91)90330-S.

[25] E.W.Kolb,I.I.Tkachev,Phys.Rev.Lett.71(1993)3051–3054,https://doi.org/10. 1103/PhysRevLett.71.3051,arXiv:hep-ph/9303313.

[26] E.W.Kolb,I.I.Tkachev,Phys.Rev.D49(1994)5040–5051,https://doi.org/10. 1103/PhysRevD.49.5040,arXiv:astro-ph/9311037.

[27]E.Braaten,A.Mohapatra,H.Zhang,Phys.Rev.Lett.117(2016)121801,arXiv: 1512.00108.

[28] J.Eby,P.Suranyi,L.C.R.Wijewardhana,Mod.Phys.Lett.A31(2016)1650090,

https://doi.org/10.1142/S0217732316500905,arXiv:1512.01709.

[29] J.Eby,M.Leembruggen,P.Suranyi,L.C.R.Wijewardhana,J.HighEnergyPhys. 12(2016)066,https://doi.org/10.1007/JHEP12(2016)066,arXiv:1608.06911. [30] D.G.Levkov,A.G.Panin,I.I.Tkachev,Phys.Rev.Lett.118(2017)011301,https://

doi.org/10.1103/PhysRevLett.118.011301,arXiv:1609.03611.

[31]T. Helfer,D.J.E. Marsh,K. Clough, M. Fairbairn, E.A. Lim,R. Becerril,arXiv: 1609.04724,2016.

[32] E.Braaten,A.Mohapatra,H.Zhang,Phys.Rev.D96(2017)031901,https://doi. org/10.1103/PhysRevD.96.031901,arXiv:1609.05182.

[33] E.Braaten,A.Mohapatra,H.Zhang,Phys.Rev.D94(2016)076004,https://doi. org/10.1103/PhysRevD.94.076004,arXiv:1604.00669.

[34] Y.Bai,V.Barger,J.Berger,J.HighEnergyPhys.12(2016)127,https://doi.org/ 10.1007/JHEP12(2016)127,arXiv:1612.00438.

[35]J. Eby, M. Leembruggen, J. Leeney, P. Suranyi, L.C.R. Wijewardhana, arXiv: 1701.01476,2017.

[36]V.Desjacques,A.Kehagias,A.Riotto,arXiv:1709.07946,2017.

[37] R.F.Dashen,B.Hasslacher,A.Neveu,Phys.Rev.D11(1975)3424–3450,https:// doi.org/10.1103/PhysRevD.11.3424.

[38]A.E.Kudryavtsev,JETPLett.22(1975)82.

[39] V. Makhankov, Phys. Rep. 35 (1978) 1–128, https://doi.org/10.1016/0370 -1573(78)90074-1, http://www.sciencedirect.com/science/article/pii/ 0370157378900741.

[40] J.Geicke,Phys.Scr.29(1984)431,http://stacks.iop.org/1402-4896/29/i=5/a=003. [41]V.E.Zakharov,E.A.Kuznetsov,Zh.Eksp.Teor.Fiz.91(1986)1310–1324.

[42] M. Gleiser, Phys. Rev. D 49 (1994) 2978–2981, https://doi.org/10.1103/ PhysRevD.49.2978.

[43] E.J. Copeland, M. Gleiser, H.R. Muller, Phys. Rev. D 52 (1995) 1920–1933,

https://doi.org/10.1103/PhysRevD.52.1920,arXiv:hep-ph/9503217.

[44] P.Salmi,M.Hindmarsh,Phys.Rev.D85(2012)085033,https://doi.org/10.1103/ PhysRevD.85.085033,arXiv:1201.1934.

[45]J.N.Hormuzdiar,S.D.H.Hsu,arXiv:hep-th/9906058,1999.

[46] T. Matos, L.A. Ureña-López, Class. Quantum Gravity 17 (2000) L75, http:// stacks.iop.org/0264-9381/17/i=13/a=101.

[47] L.A. Urena-Lopez,Class.Quantum Gravity19(2002) 2617–2632,https://doi. org/10.1088/0264-9381/19/10/307,arXiv:gr-qc/0104093.

[48] M.J. Ablowitz, D.J. Kaup, A.C. Newell,H. Segur, Phys. Rev. Lett. 30 (1973) 1262–1264,https://doi.org/10.1103/PhysRevLett.30.1262.

[49] P.DiVecchia,G.Veneziano,Nucl.Phys.B171(1980)253–272,https://doi.org/ 10.1016/0550-3213(80)90370-3.

[50] G.GrillidiCortona,E.Hardy,J.PardoVega,G.Villadoro,J.HighEnergyPhys. 01(2016)034,https://doi.org/10.1007/JHEP01(2016)034,arXiv:1511.02867. [51] P. Petreczky, H.-P. Schadler, S. Sharma, Phys. Lett. B 762 (2016) 498–505,

https://doi.org/10.1016/j.physletb.2016.09.063,arXiv:1606.03145.

[52] S. Borsanyi, et al., Nature 539 (2016) 69–71, https://doi.org/10.1038/ nature20115,arXiv:1606.07494.

[53] A.H.Guth,M.P.Hertzberg,C.Prescod-Weinstein,Phys.Rev.D92(2015)103513,

https://doi.org/10.1103/PhysRevD.92.103513,arXiv:1412.5930. [54]L.M.Widrow,N.Kaiser,Astrophys.J.416(1993)L71–L74.

[55] D.J.E. Marsh, A.-R. Pop, Mon. Not. R. Astron. Soc. 451 (2015) 2479–2492,

https://doi.org/10.1093/mnras/stv1050,arXiv:1502.03456.

[56] P.-H. Chavanis, Phys. Rev. D 84 (2011) 043531, https://doi.org/10.1103/ PhysRevD.84.043531,arXiv:1103.2050.

[57] P.H.Chavanis,L.Delfini,Phys.Rev.D84(2011)043532,https://doi.org/10.1103/ PhysRevD.84.043532,arXiv:1103.2054.

[58] E.Cotner,Phys.Rev.D94(2016)063503,https://doi.org/10.1103/PhysRevD.94. 063503,arXiv:1608.00547.

[59] S.Davidson,T.Schwetz,Phys.Rev.D93(2016)123509,https://doi.org/10.1103/ PhysRevD.93.123509,arXiv:1603.04249.

[60] M.P.Hertzberg,M.Tegmark,F.Wilczek,Phys.Rev.D78(2008)083507,https:// doi.org/10.1103/PhysRevD.78.083507,arXiv:0807.1726.

[61] L.Visinelli,P.Gondolo,Phys.Rev.D80(2009)035024,https://doi.org/10.1103/ PhysRevD.80.035024,arXiv:0903.4377.

[62] G.H. Derrick, J. Math. Phys. 5(1964) 1252–1254, https://doi.org/10.1063/1. 1704233.

[63] G.Rosen,J.Math.Phys.9(1968)999–1002,https://doi.org/10.1063/1.1664694. [64] T.D.Lee,Phys.Rep.23(1976)254–258,https://doi.org/10.1016/0370-1573(76)

90045-4.

[65] R.Friedberg,T.D.Lee,A.Sirlin,Nucl.Phys.B115(1976)1–31,https://doi.org/ 10.1016/0550-3213(76)90274-1.

[66] R.Friedberg,T.D.Lee,A.Sirlin,Nucl.Phys.B115(1976)32–47,https://doi.org/ 10.1016/0550-3213(76)90275-3.

[67] Nucl.Phys.B269(1986)744,https://doi.org/10.1016/0550-3213(86)90520-1. [68]I.L.Bogolyubskiˇı,V.G.Makhan’kov,Zh.Eksp.Teor.Fiz.PismaRed.24(1976)15.

[69] M.Gleiser,A.Sornborger,Phys. Rev.E62(2000)1368–1374,https://doi.org/ 10.1103/PhysRevE.62.1368,arXiv:patt-sol/9909002.

[70] G.Fodor,P.Forgacs,P.Grandclement,I.Racz,Phys.Rev.D74(2006)124003,

https://doi.org/10.1103/PhysRevD.74.124003,arXiv:hep-th/0609023.

[71] B. Piette,W.J.Zakrzewski,Nonlinearity11(1998)1103,http://stacks.iop.org/ 0951-7715/11/i=4/a=020.

[72] G.L. Alfimov,W.A.B. Evans, L.Vázquez,Nonlinearity 13(2000) 1657,http:// stacks.iop.org/0951-7715/13/i=5/a=313.

[73] M.Gleiser,D.Sicilia,Phys.Rev.D80(2009)125037,https://doi.org/10.1103/ PhysRevD.80.125037,arXiv:0910.5922.

[74] M.Gleiser, D.Sicilia, Phys. Rev.Lett.101(2008)011602, https://doi.org/10. 1103/PhysRevLett.101.011602,arXiv:0804.0791.

[75]E.D.Schiappacasse,M.P.Hertzberg,arXiv:1710.04729,2017.