Gauss–Bonnet Boson Stars in AdS, Bielefeld, Germany, 2014

GaussBonnet Boson Stars in AdS

Jrgen Riedelin collaboration with

Yves Brihaye(1), Betti Hartmann(2)(3) , and Raluca Suciu(2)

Physics Letters B (2013) (arXiv:1308.3391), arXiv:1310.7223, andarXiv:1404.1874

Faculty of Physics, University Oldenburg, GermanyModels of Gravity

SPRING WORKSHOP 2014, MODELS OF GRAVITYBielefeld, Germany

Mai 7th, 2014

(1)Physique-Mathmatique, Universit de Mons, 7000 Mons, Belgium

(2)School of Engineering and Science, Jacobs University Bremen, Germany

(3)Universidade Federal do Espirito Santo (UFES), Departamento de Fisica, Vitoria (ES), Brazil

Brihaye, Hartmann, Riedel, and Suciu (Jacobs University Bremen)GaussBonnet Boson Stars in AdSSPRING WORKSHOP 2014, MODELS OF GRAVITY Bielefeld, Germany Mai 7th, 2014 1

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Agenda

Describe the model to construct non-rotating Gauss-Bonnetboson stars in AdSDescribe effect of Gauss-Bonnet term to boson star solutionsStability analysis of rotating Gauss-Bonnet boson starsolutions (ignoring the Gauss-Bonnet coupling for now)Outlook


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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)


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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.


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Why study boson stars?

Boson starsSimple toy models for a wide range of objects such asparticles, compact stars, e.g. neutron stars and even centres ofgalaxiesWe 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.Toy 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


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Model for GaussBonnet Boson Stars

Action

S =1

16piG5

d5xg (R 2 + LGB + 16piG5Lmatter)

LGB =(RMNKLRMNKL 4RMNRMN + R2

)(1)

Matter Lagrangian Lmatter = () U()Gauge mediated potential

USUSY(||) = m22susy(

1 exp( ||

2

2susy

))(2)

USUSY(||) = m2||2 m2||4

22susy+m2||664susy

+ O(||8

)(3)

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


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Model for GaussBonnet Boson Stars

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

GMN + gMN +

2HMN = 8piG5TMN (4)

where HMN is given by

HMN = 2(RMABCRABCN 2RMANBRAB 2RMARAN + RRMN

) 1

2gMN

(R2 4RABRAB + RABCDRABCD

)(5)

Energy-momentum tensor

TMN = gMN[

12gKL (KL + LK) + U()

]+ M

N + NM (6)


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Model continued

The Klein-Gordon equation is given by:( U

||2) = 0 (7)

Lmatter is invariant under the global U(1) transformation ei . (8)

Locally conserved Noether current jM

jM = i2

(M M

); jM;M = 0 (9)

The globally conserved Noether charge Q reads

Q =

d4xgj0 . (10)


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Ansatz non-rotating

Metric Ansatz

ds2 = N(r)A2(r)dt2 + 1N(r)

dr2

+ r2(d2 + sin2 d2 + sin2 sin2 d2

)(11)

whereN(r) = 1 2n(r)

r2(12)

Stationary Ansatz for complex scalar field

(r , t) = (r)eit (13)


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Ansatz rotating

Metric Ansatz

ds2 = A(r)dt2 + 1N(r)

dr2 + G(r)d2

+ H(r) sin2 (d1 W (r)dt)2+ H(r) cos2 (d2 W (r)dt)2+ (G(r) H(r)) sin2 cos2 (d1 d2)2, (14)

with = [0,pi/2] and 1,2 = [0,2pi].Cohomogeneity-1 Ansatz for Complex Scalar Field

= (r)eit , (15)

with = (sin ei1 , cos ei2)t (16)


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Boundary Conditions for asymptotic AdS space-time

If < 0 the scalar field function falls of with

(r >> 1) =r

, = 2 +

4 + L2eff . (17)

Where Leff is the effective AdS-radius:

L2eff =2

1

1 4L2;L2 =

6

(18)

Chern-Simons limit:

=L2

4(19)

Mass for > 0 we define the gravitational mass at AdSboundary

MG n(r )/ (20)


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Finding solutions: fixing

f

V (f )

4 2 0 2 40 .

0 3

0 .0 1

0 .0 0

0 .0 1

0 .0 2

f(0)

k=0

k=1

k=2

= 0.80 = 0.85 = 0.87 = 0.90V = 0.0

r

0 5 10 15 200 .

10 .

10 .

20 .

30 .

40 .

5

k= 0= 1= 2 = 0.0

r

0 5 10 15 20 25 30

0 .

20 .

20 .

40 .

60 .

81 .

0

Figure : Effective potential V (f ) = 22 U(f ).Brihaye, Hartmann, Riedel, and Suciu (Jacobs University Bremen)GaussBonnet Boson Stars in AdS

SPRING WORKSHOP 2014, MODELS OF GRAVITY Bielefeld, Germany Mai 7th, 2014 12/ 26

GaussBonnet Boson Stars with < 0

Q

0.8 0.9 1.0 1.1 1.2 1.3 1.4

11 0

1 00

1 00 0

1 00 0

0

= 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 mode = 1.0

Figure : Charge Q in dependence on the frequency for = 0.01, = 0.02 and differentvalues of . max shift: max = Leff


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GaussBonnet Boson Stars with < 0 Animation

(Loading Video...)


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HandBrake 0.9.9 2013051800

GBAdSBosonStar.mp4Media File (video/mp4)

Excited GaussBonnet Boson Stars with < 0

Q

0.6 0.8 1.0 1.2 1.4 1.6

11 0

01 0

0 00

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

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

Figure : Charge Q in dependence on the frequency for = 0.01, = 0.02, and differentvalues of . max shift: max = +2kLeff


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Summary Effect of Gauss-Bonnet Correction

Boson starsSmall coupling to GB term, i.e. small , we find similar spirallike characteristic as for boson stars in pure Einstein gravity.When the Gauss-Bonnet parameter is large enough thespiral unwinds.When and the coupling to gravity () are of the samemagnitude, only one branch of solutions survives.The single branch extends to the small values of the frequency.


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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:1106.6073v1Brihaye, Hartmann, Riedel, and Suciu (Jacobs University Bremen)GaussBonnet Boson Stars in AdS


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 will return in finite time.general result on the non-linear stability does not yet exist forAdS.It is speculated that AdS is nonlinearly unstable since in AdSone would expect small perturbations to bounce of theboundary, to 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).


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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)


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Stability of Boson Stars with AdS Radius very large

Q

0.2 0.4 0.6 0.8 1.0 1.21e +

0 21 e

+ 04

1 e+ 0

61 e

+ 08

Y= 0.00005Y= 0.00001

& Y & k0.1 & 0.00005 & 00.01 & 0.00005 & 00.001 & 0.00005 & 00.01 & 0.00001 & 20.001 & 0.000007 & 00.001 & 0.000007 & 4

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


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Ergoregions Animation

(Loading Video...)


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HandBrake 0.9.9 2013051800

Ergo.mp4Media File (video/mp4)

Stability for Boson Stars in AdS

Q

0.5 1.0 1.5 2.01e +

0 21 e

+ 04

1 e+ 0

61 e

+ 08

Y= 0.1Y= 0.01Y= 0.001

0.1,0.01,0.001,0.00010.1,0.01,0.001,0.00010.1,0.05,0.01,0.005,0.001

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


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Stability Analysis for Radial Excited Boson Stars inAdS

Q

0.4 0.6 0.8 1.0 1.2 1.41e +

0 21 e

+ 03

1 e+ 0

41 e

+ 05

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

& Y0.01 & 0.001

Figure : Charge Q in dependence on the frequency for different excited modes k .Brihaye, Hartmann, Riedel, and Suciu (Jacobs University Bremen)GaussBonnet Boson Stars in AdS


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, supposed to be classically stable.Radially excited self-interacting rotating boson stars can beclassically stable in aAdS for sufficiently large AdS radius andsufficiently small backreaction.


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Outlook

Analysis the effect of the Gauss-Bonnet term on stability ofboson stars.To do a stabiliy analysis similar to the one done for non-rotatingminimal boson stars by Bucher et al 2013 ([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


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Thank You

Thank You!


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