phd defense, oldenburg, germany, june, 2014

Q-Balls and Boson Stars in asymptotically flat andAnti-de-Sitter space-time

Jürgen Riedel

PhD Supervisor: Betti Hartmann

Faculty of Physics, University Oldenburg, GermanyModels of Gravity

Oldenburg, June 5th, 2014

Riedel (University Oldenburg) Q-Balls and Boson Stars Oldenburg, June 5th, 2014 1 / 38

Publications:

B. Hartmann(1)(2) and J. Riedel, Phys. Rev. D 86, 104008(2012), arXiv:1204.6239.B. Hartmann and J. Riedel, Phys. Rev. D 87 044003 (2013),arXiv:1210.0096.B. Hartmann, J. Riedel, and R. Suciu(1), Physics Letters B(2013), arXiv:1308.3391,Y. Brihaye(3) and J. Riedel, (2013), arXiv:1310.7223 (acceptedfor publication in Phys. Rev. D 89).Y. Brihaye, B. Hartmann, and J. Riedel, (2014),arXiv:1404.1874.

(1)School of Engineering and Science, Jacobs University Bremen, Germany(2)Universidade Federal do Espirito Santo (UFES), Departamento de Fisica,

Vitoria (ES), Brazil(3)Physique-Mathématique, Université de Mons, 7000 Mons, BelgiumRiedel (University Oldenburg) Q-Balls and Boson Stars Oldenburg, June 5th, 2014 2 / 38

Outline

Brief introduction to general concepts of Q-balls and BosonStars

non-topological SolitonsProperties of Q-balls and Boson Stars

Model to construct Gauss-Bonnet Boson Stars inasymptotically AdS space-time (aAdS)Numerical results of Gauss-Bonnet Boson Stars

Stability aspects of Gauss-Bonnet Boson StarsClassical or absolute StabilityErgoregions and Superradiant Instability

Outlook


Solitons in non-linear field theories

General properties of soliton solutionsLocalized, finite energy, stable, regular solutions of non-linearfield equationsCan be viewed as models of elementary particles

ExamplesTopological solitons: Skyrme model of hadrons in high energyphysics one of first models and magnetic monopoles, domainwalls etc.Non-topological solitons: Q-balls (named after Noether chargeQ) (flat space-time) and boson stars (generalisation in curvedspace-time)


Non-topolocial solitons

Properties of non-topological solitonsSolutions possess the same boundary conditions at infinity asthe physical vacuum stateDegenerate vacuum states do not necessarily existRequire an additive conservation law, e.g. gauge invarianceunder an arbitrary global phase transformation

S. R. Coleman, Nucl. Phys. B 262 (1985), 263, R. Friedberg, T. D. Lee and A. Sirlin, Phys. Rev. D 13 (1976) 2739), D. J. Kaup,

Phys. Rev. 172 (1968), 1331, R. Friedberg, T. D. Lee and Y. Pang, Phys. Rev. D 35 (1987), 3658, P. Jetzer, Phys. Rept. 220

(1992), 163, F. E. Schunck and E. Mielke, Class. Quant. Grav. 20 (2003) R31, F. E. Schunck and E. Mielke, Phys. Lett. A 249

(1998), 389.


Non-topolocial solitons

Model for Q-ballsWith complex scalar field Φ(x, t) : L = ∂µΦ∂µΦ∗ −U(|Φ|), U(|Φ|)minimum at Φ = 0Lagrangian is invariant under transformation φ(x)→ eiαφ(x)

Give rise to Noether charge Q = 1i

∫dx3φ∗φ̇− φφ̇∗)

Solution that minimizes the energy for fixed Q: Φ(x, t) = φ(x)eiωt

Solutions have been constructed in (3 + 1)-dimensionalmodels with non-normalizable Φ6-potential

M.S. Volkov and E. Wöhnert, Phys. Rev. D 66 (2002), 085003, B. Kleihaus, J. Kunz and M. List, Phys. Rev. D 72 (2005), 064002,

B. Kleihaus, J. Kunz, M. List and I. Schaffer, Phys. Rev. D 77 (2008), 064025.


Existence conditions of Q-balls

Condition 1V

′′(0) < 0; Φ ≡ 0 local maximum⇒ ω2 < ω2

max ≡ U′′

(0)

Condition 2ω2 > ω2

min ≡ minφ[2U(φ)/φ2] minimum over all φ

Consequences

Restricted interval ω2min < ω2 < ω2

max ; U′′

(0) > minφ[2U(φ)/φ2]

Q-balls are rotating in inner space with ω stabilized by having alower energy to charge ratio as the free particles


Finding solutions

Analogy to dynamics of a point particle:

r → time, V (φ)→ potential + damping

φ

V(φ

)

−4 −2 0 2 4

−0.0

3−

0.0

10.0

00.0

10.0

2

φ(0)

k=0

k=1

k=2

ω = 0.80

ω = 0.85

ω = 0.87

ω = 0.90

V = 0.0

Figure : Effective potential V (φ) = ω2φ2 − U(φ).


Finding solutions (continued)

r

φ

0 5 10 15 20

−0.1

0.1

0.2

0.3

0.4

0.5

k

= 0

= 1

= 2

φ = 0.0

rφ

0 5 10 15 20 25 30

−0.2

0.2

0.4

0.6

0.8

1.0

k

= 0

= 1

= 2

φ = 0.0

Figure : Profile functions φ(r) fundamental and radially excited for static (left) and rotating(right) solutions.


Thin wall and Thick wall approximation of Q-balls

Thin wall limit: ω ' ωmin; φ(0)� 0

Step-like ansatz [Coleman 1985]: no wall thickness

Modified step-like (egg-shell-like) ansatz: includes wall thickness[Correia et.al. 2001; M.I.T., Copeland, Saffin 2008]

Minimum of total energy ωmin = Emin = 2U(φ0)φ2 , for φ0 > 0

The energy and charge is proportional to the volume which issimilarly found in ordinary matter→ Q = ωφ2VTherefore Q-balls in this limit are called Q-matter and have verylarge charge, i.e. volume

Thick wall limit: ω ' ωmax ; φ(0) ' 0

As ω increases initial density φ(0) gets close to zeroGaussian ansatz: φ(r) = exp

(r2

R2

)


Q-ball profile in 3D: Thick wall to Thin wall transition

�(r) = 0 �(r) = 13

Figure : Q-ball profile function φ(r) for initial field densitie φ(0). Qball 3D Rot. Boson Star 3D


http://localhost/Qball3d.mp4

http://localhost/BosonStar5d_rot.mp4

Affleck-Dine (AD) Mechanism

Attempt to describe Baryogenesis of early universe in theMinimal Supersymmetric extension of the Standard Model(MSSM) of particle physicsIn Scalar Potential (= sauark, slepton, higgs) of MSSMThere exist Flat Directions = AD fieldFlat directions are supersymmetric minima of the scalarpotential.Directions are lifted by supersymmetry breaking effects (onlyapproximately flat)AD condensates forms along flat direction. Fragmentationcreates Q-balls.


Affleck-Dine Condensate and Q-balls


SUSY breaking potentials

Two possible (soft) SUSY breaking potentials: Gravity mediatedpotential and Gauge mediated potentialGravity mediated potential : SUSY breaking throughgravitational interactions.Gauge mediated potential: SUSY breaking through theStandard Model’s gauge interactions.

USUSY(|Φ|) =

{m2|Φ|2 if |Φ| ≤ ηsusy

m2η2susy = const . if |Φ| > ηsusy

(1)

U(|Φ|) = m2η2susy

(1− exp

(− |Φ|

2

η2susy

))(2)

A. Kusenko, Phys. Lett. B 404 (1997), 285; Phys. Lett. B 405 (1997), 108, L. Campanelli and M. Ruggieri, Phys. Rev. D

77 (2008), 043504


Boson star models

Gravitating cousins of Q-ballsSimplest model U = m2|Φ|2 (by Kemp, 1986)

Proper boson stars U = m2|Φ|2 − λ|Φ|4/2(by Colpi, Sharpio and Wasserman, 1986)

Sine-Gordon boson star U = αm2[sin(π/2

[β√|Φ|2 − 1

]+ 1]

Cosh-Gordon boson star U = αm2[cosh(β

√|Φ|2 − 1

]Liouville boson star U = αm2 [exp(β2|Φ|2)− 1

](Schunk and Torres, 2000)


Self-interacting boson star models

Model U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a and b are constants(Mielke and Scherzer, 1981)

Soliton stars U = m2|Φ|2(1− |Φ|2/Φ2

0)2

(Friedberg, Lee and Pang, 1986)

Represented in the limit of flat space− time, by Q -balls asnon-topological solitonsHowever, terms of |Φ|6 or higher-order terms implies that thescalar part of the theory is not re-normalizable


Why study boson stars or Q-balls?

Q-ballsSupersymmetric Q-balls have been considered as possiblecandidates for baryonic dark matter

Boson stars and Q-ballsSimple toy models for a wide range of objects such asparticles, compact stars, e.g. neutron stars and even centres ofgalaxiesToy models for studying properties of AdS space-timeToy models for AdS/CFT correspondence. Planar boson starsin AdS have been interpreted as Bose-Einstein condensates ofglueballs


Why study higher dimensions?

Set up a model to support boson star solutions inGauss-Bonnet gravity in 4 + 1 dimensionsGauss-Bonnet theory which appears naturally in the low energyeffective action of quantum gravity modelsWe are interested in the effect of Gauss-Bonnet gravity and willstudy these objects in the minimal number of dimensions inwhich the term does not become a total derivative.Higher dimensions appear in attempts to find a quantumdescription of gravity as well as in unified models.For black holes many of their properties in (3 + 1) dimensions donot extend to higher dimensions.Discovery of the Higgs Boson in 2012: fundamental scalarfields do exist in nature.


Model for Gauss–Bonnet Boson Stars

Action

S =1

16πG5

∫d5x√−g (R − 2Λ + αLGB + 16πG5Lmatter)

LGB =(

RMNKLRMNKL − 4RMNRMN + R2)

(3)

Matter Lagrangian Lmatter = − (∂µψ)∗ ∂µψ − U(ψ)

Gauge mediated potential

USUSY(|ψ|) = m2η2susy

(1− exp

(− |ψ|

2

η2susy

))(4)

USUSY(|ψ|) = m2|ψ|2 − m2|ψ|4

2η2susy

+m2|ψ|6

6η4susy

+ O(|ψ|8

)(5)

A. Kusenko, Phys. Lett. B 404 (1997), 285; Phys. Lett. B 405 (1997), 108, L. Campanelli and M. Ruggieri, Phys. Rev. D

77 (2008), 043504


Model for Gauss–Bonnet Boson Stars

Einstein Equations are derived from the variation of the action withrespect to the metric fields

GMN + ΛgMN +α

2HMN = 8πG5TMN (6)

where HMN is given by

HMN = 2(

RMABCRABCN − 2RMANBRAB − 2RMARA

N + RRMN

)− 1

2gMN

(R2 − 4RABRAB + RABCDRABCD

)(7)

Energy-momentum tensor

TMN = −gMN

[12

gKL (∂Kψ∗∂Lψ + ∂Lψ

∗∂Kψ) + U(ψ)

]+ ∂Mψ

∗∂Nψ + ∂Nψ∗∂Mψ (8)


Model continued

The Klein-Gordon equation is given by:(�− ∂U

∂|ψ|2

)ψ = 0 (9)

Lmatter is invariant under the global U(1) transformation

ψ → ψeiχ . (10)

Locally conserved Noether current jM

jM = − i2

(ψ∗∂Mψ − ψ∂Mψ∗

); jM;M = 0 (11)

The globally conserved Noether charge Q reads

Q = −∫

d4x√−gj0 . (12)


Ansatz non-rotating

Metric Ansatz

ds2 = −N(r)A2(r)dt2 +1

N(r)dr2

+ r2(

dθ2 + sin2 θdϕ2 + sin2 θ sin2 ϕdχ2)

(13)

whereN(r) = 1− 2n(r)

r2 (14)

Stationary Ansatz for complex scalar field

ψ(r , t) = φ(r)eiωt (15)


Ansatz rotating

Metric Ansatz

ds2 = −A(r)dt2 +1

N(r)dr2 + G(r)dθ2

+ H(r) sin2 θ (dϕ1 −W (r)dt)2

+ H(r) cos2 θ (dϕ2 −W (r)dt)2

+ (G(r)− H(r)) sin2 θ cos2 θ(dϕ1 − dϕ2)2, (16)

with θ = [0,π/2] and ϕ1,ϕ2 = [0,2π].Cohomogeneity-1 Ansatz for Complex Scalar Field

Φ = φ(r)eiωt Φ̂, (17)

withΦ̂ = (sin θeiϕ1 , cos θeiϕ2)t (18)


Boundary Conditions

Rescaling using dimensionless quantities

r → rm

, ω → mω , ψ → ηsusyψ , n→ n/m2 , α→ α/√

m (19)

Appropriate boundary conditions non-rotating:

φ′(0) = 0 , N(0) = 1. (20)

Appropriate boundary conditions rotating:

B′(0) = 0H ′(r = 0)

r2 = 0 W ′(0) = 0 φ(0) = 0. (21)

We need the scalar filed to vanish at infinity and thereforerequire φ(∞) = 0, while we choose A(∞) = 1 (rescaling of thetime coordinate)Field equations depend only on the dimensionless couplingconstants α and κ = 8πG5η

2SUSY


Asymptotic Behaviour

If Λ < 0 the scalar field function falls off with

φ(r >> 1) =φ∆

r∆, ∆ = 2 +

√4 + L2

eff . (22)

Where Leff is the effective AdS-radius:

L2eff =

2α

1−√

1− 4αL2

; L2 =−6Λ

(23)

Chern-Simons limit:

α =L2

4(24)

Mass for κ > 0 we define the gravitational mass at AdSboundary

MG ∼ n(r →∞)/κ (25)


Gauss–Bonnet Boson Stars with Λ < 0

ω

Q

0.8 0.9 1.0 1.1 1.2 1.3 1.4

110

100

1000

10000

α

= 0.0= 0.01= 0.02= 0.03= 0.04= 0.05= 0.06= 0.075= 0.1= 0.2= 0.3= 0.5= 1.0= 5.0= 10.0= 12.0= 15.0 (CS limit)

1st excited modeω = 1.0

Figure : Charge Q in dependence on the frequency ω for Λ = −0.01, κ = 0.02 and differentvalues of α. ωmax shift: ωmax = ∆

LeffUnfolding DB GB0 DB GB15 DB GB0 ex DB GB15 ex


http://localhost/UnfoldGBAdSBosonStar.mp4

http://localhost/QoverOmegaGB_L_0.mp4

http://localhost/QoverOmegaGB_L_15.mp4

http://localhost/QoverOmegaGB_L_0_node5.mp4

http://localhost/QoverOmegaGB_L_15_node5.mp4

Excited Gauss–Bonnet Boson Stars with Λ < 0

ω

Q

0.6 0.8 1.0 1.2 1.4 1.6

1100

10000

k=0 k=1 k=2 k=3 k=4

α

= 0.0= 0.01= 0.05= 0.1= 0.5= 1.0= 5.0= 10.0= 50.0= 100.0= 150.0 (CS limit)

ω = 1.0

Figure : Charge Q in dependence on the frequency ω for Λ = −0.01, κ = 0.02, and differentvalues of α. ωmax shift: ωmax = ∆+2k

Leff


Summary Effect of Gauss-Bonnet Correction

Boson starsSmall coupling to GB term, i.e. small α, we find the samespiral-like characteristic as for boson stars in pure Einsteingravity.When the Gauss-Bonnet parameter α is large enough the spiral’unwinds’.When α and the coupling to gravity κ are of the samemagnitude, only one branch of solutions survives.The single branch extends to the small values of the frequencyω.


AdS Space-time

(4)

Figure : (a) Penrose diagram of AdS space-time, (b) massive (solid) and massless (dotted)geodesic.

(4)J. Maldacena, The gauge/gravity duality, arXiv:1106.6073v1Riedel (University Oldenburg) Q-Balls and Boson Stars Oldenburg, June 5th, 2014 29 / 38

Stability of AdS

It has been shown that AdS is linearly stable (Ishibashi and Wald2004).The metric conformally approaches the static cylindricalboundary, waves bounce off at infinity and will return in finitetime.general result on the non-linear stability does not yet exist forAdS.It is speculated that AdS is nonlinearly unstable, because onewould expect small perturbations to bounce off the boundary,interact with themselves and lead to instabilities.Conjecture: small finite Q-balls of AdS in (3 + 1)-dimensionalAdS eventually lead to the formation of black holes (Bizon andRostworowski 2011).Related to the fact that energy is transferred to smaller andsmaller scales, e.g. pure gravity in AdS (Dias et al).


Erogoregions and Instability

If the boson star mass M is smaller than mBQ one wouldexpect the boson star to be classically stable with respect todecay into Q individual scalar bosons.Solutions to Einsteins field equations with sufficiently largeangular momentum can suffer from superradiant instability(Hawking and Reall 2000).Instablity occurs at ergoregion (Friedman 1978) where therelativistic frame-dragging is so strong that stationary orbitsare no longer possible.At ergoregion, infalling bosonic waves are amplified whenreflected.The boundary of the ergoregion is defined as where thecovariant tt-component becomes zero, i.e. gtt = 0Analysis for boson stars in (3 + 1) dimensions has been done(Cardoso et al 2008, Kleihaus et al 2008)


Superradiant Instability

Penrose process (Penrose 1969, Christodoulou 1970) scattering of particles offKerr BH =⇒ reduction of BH mass.

Superradiant scattering of wave packet off Kerr BH (Misner 1972,

Zeldovitch 1971)

"Black Hole Bomb" (Press & Teukolsky 1972, Zeldovich 1971, Cardoso et al 2004). Blackhole surrounded by a mirror =⇒ exponential growth ofmodes and instability due to superradiant scatteringNatural mirror provided by anti-de Sitter spacetimes (AdS)


Stability of Boson Stars with AdS Radius very large

ω

Q

0.2 0.4 0.6 0.8 1.0 1.21e+

02

1e+

04

1e+

06

1e+

08

Y= 0.00005Y= 0.00001

κ & Y & k

0.1 & 0.00005 & 0

0.01 & 0.00005 & 0

0.001 & 0.00005 & 0

0.01 & 0.00001 & 2

0.001 & 0.000007 & 0

0.001 & 0.000007 & 4

Figure : Charge Q in dependence on the frequency ω for different values of κ and Y = −6Λ.DB GB0 DB GB15


http://localhost/QoverOmega_rot_0.mp4

http://localhost/QoverOmega_rot_1.mp4

Stability for Boson Stars in AdS

ω

Q

0.5 1.0 1.5 2.01e+

02

1e+

04

1e+

06

1e+

08

Y= 0.1Y= 0.01Y= 0.001

κ

0.1,0.01,0.001,0.0001

0.1,0.01,0.001,0.0001

0.1,0.05,0.01,0.005,0.001

Figure : Charge Q in dependence on the frequency ω for different values of κ and Y = −6Λ.


Stability Analysis for Radial Excited Boson Stars inAdS

ω

Q

0.4 0.6 0.8 1.0 1.2 1.41e+

02

1e+

03

1e+

04

1e+

05

k=0 k=1 k=2 k=3 k=5

κ & Y

0.01 & 0.001

Figure : Charge Q in dependence on the frequency ω for different excited modes k .Riedel (University Oldenburg) Q-Balls and Boson Stars Oldenburg, June 5th, 2014 35 / 38

Summary Stability Analysis

Strong coupling to gravity: self-interacting rotating bosonstars are destabilized.Sufficiently small AdS radius: self-interacting rotating bosonstars are destabilized.Sufficiently strong rotation stabilizes self-interacting rotatingboson stars.Onset of ergoregions can occur on the main branch of bosonstar solutions, which are classically stable.Radially excited self-interacting rotating boson stars can beclassically stable in aAdS for sufficiently large AdS radius andsufficiently small backreaction.


Outlook

Analyse the effect of the Gauss-Bonnet term on stability ofboson stars.To do a stability analysis of Gauss-Bonnet boson stars based onthe work on non-rotating minimal boson stars by Bucher et al2013 ([arXiv:1304.4166 [gr-qc])See whether our arguments related to the classical stability ofour solutions agrees with a full perturbation analysisGauss-Bonnet Boson Stars and AdS/CFT correspondanceBoson Stars in general Lovelock theory


Thank You

Thank You!


phd defense, oldenburg, germany, june, 2014

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