Gauss–Bonnet Boson Stars in AdS - Ibericos, Portugal, 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

IBERICOS 2014Aveiro, Portugal, April 28th, 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 GaussBonnet Boson Stars in AdS

Agenda

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


Solitons in non-linear field theories

General properties of soliton solutionsLocalized, finite energy, stable, regular solutions ofnon-linear field equationsCan be viewed as models of elementary particles

ExamplesTopological solitons: Skyrme model of hadrons in highenergy physics one of first models and magneticmonopoles, domain walls etc.Non-topological solitons: Q-balls (named after Noethercharge Q) (flat space-time) and boson stars(generalisation in curved space-time)


Non-topolocial solitons

Properties of non-topological solitonsSolutions possess the same boundary conditions atinfinity as the physical vacuum stateDegenerate vacuum states do not necessarily existRequire an additive conservation law, e.g. gaugeinvariance under an arbitrary global phasetransformation

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.


Why study Q-balls and boson stars?

Boson starsSimple toy models for a wide range of objects such asparticles, compact stars, e.g. neutron stars and evencentres of galaxiesWe are interested in the effect of Gauss-Bonnet gravityand will study these objects in the minimal number ofdimensions in which the term does not become a totalderivative.Toy models for studying properties of AdS space-timeToy models for AdS/CFT correspondence. Planar bosonstars in AdS have been interpreted as Bose-Einsteincondensates of glueballs


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


Model for GaussBonnet Boson Stars

Einstein Equations are derived from the variation of theaction with respect 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)


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)


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) = f (r)eit (13)


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)


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)


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

r0 2 4 6 8 10

1 .0

0 .

50 .

00 .

51 .

0

Functionsphi(r) & (0) = 0.2phi(r) & (0) = 4.0phi(r) & (0) = 9.0b(r) & (0) = 0.2b(r) & (0) = 4.0b(r) & (0) = 9.0f(r) & (0) = 0.2f(r) & (0) = 4.0f(r) & (0) = 9.0w(r) & (0) = 0.2w(r) & (0) = 4.0w(r) & (0) = 9.0

Figure : Effective potential V (f ) = 2f 2 U(f ).Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in AdS

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 anddifferent values of . max shift: max = Leff


GaussBonnet Boson Stars with < 0 Animation

(Loading Video...)


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, anddifferent values of . max shift: max = +2kLeff


Summary Effect of Gauss-Bonnet Correction

Boson starsSmall coupling to GB term, i.e. small , we find similarspiral like characteristic as for boson stars in pureEinstein gravity.When the Gauss-Bonnet parameter is large enoughthe spiral 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 thefrequency .


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 GaussBonnet Boson Stars in AdS

Stability of AdS

It has been shown that AdS is linearly stable (Ishibashiand Wald 2004).The metric conformally approaches the static cylindricalboundary, waves bounce off at infinity will return infinite time.general result on the non-linear stability does not yetexist for AdS.It is speculated that AdS is nonlinearly unstable since inAdS one would expect small perturbations to bounce ofthe boundary, to interact with themselves and lead toinstabilities.Conjecture: small finite Q-balls of AdS in(3 + 1)-dimensional AdS eventually lead to theformation of black holes (Bizon and Rostworowski 2011).Related to the fact that energy is transferred to smallerand smaller scales, e.g. pure gravity in AdS (Dias et al).


Erogoregions and Instability

If the boson star mass M is smaller than mBQ onewould expect the boson star to be classically stable withrespect to decay into Q individual scalar bosons.Solutions to Einsteins field equations with sufficientlylarge angular momentum can suffer from superradiantinstability (Hawking and Reall 2000).Instablity occurs at ergoregion (Friedman 1978) where therelativistic frame-dragging is so strong that stationaryorbits are no longer possible.At ergoregion infalling bosonic waves are amplifiedwhen reflected.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 hasbeen done (Cardoso et al 2008, Kleihaus et al 2008)


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 andY = 6.


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 andY = 6.


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 GaussBonnet Boson Stars in AdS

Summary Stability Analysis

Strong coupling to gravity: self-interacting rotatingboson stars are destabilized.Sufficiently small AdS radius: self-interacting rotatingboson stars are destabilized.Sufficiently strong rotation stabilizes self-interactingrotating boson stars.Onset of ergoregions can occur on the main branch ofboson star solutions, supposed to be classically stable.Radially excited self-interacting rotating boson starscan be classically stable in aAdS for sufficiently largeAdS radius and sufficiently small backreaction.


Outlook

Analysis the effect of the Gauss-Bonnet term onstability of boson stars.To do a stabiliy analysis similar to the one done fornon-rotating minimal boson stars by Bucher et al 2013([arXiv:1304.4166 [gr-qc])See whether our arguments related to the classicalstability of our solutions agrees with a full perturbationanalysisGauss-Bonnet Boson Stars and AdS/CFTcorrespondanceBoson Stars in general Lovelock theory


Thank You

Thank You!
