gauss bonnet boson stars

Gauss–Bonnet Boson Stars

Jürgen Riedelin collaboration with

Betti Hartmann and Raluca Suciu1

Physics Letters B (2013) [arXiv:1308.3391 [hep-th]]

Faculty of Physics, University Oldenburg, GermanyModels of Gravity

ASTROPHYSICS AND COSMOLOGY WORKSHOP, MODELS OF

GRAVITY

Bremen, October 8th 2013

1School of Engineering and Science, Jacobs University Bremen, GermanyJürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars

Motivation

Set up a model to support boson star solutions inGauss-Bonnet gravity in 4 + 1 dimensionsGauss-Bonnet theory which appears naturally in the lowenergy effective action of quantum gravity modelsWe 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.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 do not extend to higher dimensions.

Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars

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 (flat space-time) andboson 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.


Model for Gauss–Bonnet Boson Stars

Action

S =

∫d5x√−g (R + αLGB + 16πG5Lmatter)

LGB =(

RMNKLRMNKL − 4RMNRMN + R2)

(1)

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

Gauge mediated potential

USUSY(|ψ|) = m2η2susy

(1− exp

(− |ψ|

2

η2susy

))(2)

USUSY(|ψ|) = m2|ψ|2 − m2|ψ|4

2η2susy

+m2|ψ|6

6η4susy

+ O(|ψ|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 Q-balls and boson stars in d + 1 dimensions

Einstein Equations are derived from the variation of theaction with respect to the metric fields

GMN +α

2HMN = 8πG5TMN (4)

where HMN is given by

HMN = 2(

RMABCRABCN − 2RMANBRAB − 2RMARA

N + RRMN

)− 1

2gMN

(R2 − 4RABRAB + RABCDRABCD

)(5)

Energy-momentum tensor

TMN = −gMN

[12

gKL (∂Kψ∗∂Lψ + ∂Lψ

∗∂Kψ) + U(ψ)

]+ ∂Mψ

∗∂Nψ + ∂Nψ∗∂Mψ (6)


Ansatz Q-balls and boson stars for d + 1 dimensions

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

∂|ψ|2

)ψ = 0 (7)

Lmatter is invariant under the global U(1) transformation

ψ → ψeiχ . (8)

Locally conserved Noether current jM

jM = − i2

(ψ∗∂Mψ − ψ∂Mψ∗

); jM;M = 0 (9)

The globally conserved Noether charge Q reads

Q = −∫

d4x√−gj0 . (10)


Ansatz Q-balls and boson stars for d + 1 dimensions

Metric in spherical Schwarzschild-like coordinates

ds2 = −N(r)A2(r)dt2 +1

N(r)dr2

+ r2(

dθ2 + sin2 θdϕ2 + sin2 θ sin2 ϕdχ2)

(11)

whereN(r) = 1− 2n(r)

r2 (12)

Stationary Ansatz for complex scalar field

ψ(r , t) = f (r)eiωt (13)

Rescaling using dimensionless quantities

r → rm

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

m(14)


Einstein equations in (d + 1) dimensions

Equations for the metric functions:

N ′(r) = 2r(

1− Nr2 + 2α(1− N)

)− 2

3κr3

NA2U N A2 + ω2f 2 + N2A2f ′2

r2 + 2α(1− N)(15)

A′ =2κr3 (A2N2f ′2 + ω2f 2)3AN2

(2α− 2αN + r2

) (16)

κ = 8πG5η2susy (17)

Matter field equation:(r3ANf ′

)′= r3A

(12∂U∂f− ω2f

NA2

)(18)


Einstein equations in (d + 1) dimensions

Appropriate boundary conditions:

f ′(0) = 0 , n(0) = 0 . (19)

The scalar field function falls of exponentially with

f (r >> 1) ∼ 1

r32

exp(−√

1− ω2r)

+ ... (20)

Therefore we require f (∞) = 0, while we chooseA(∞) = 1 (rescaling of the time coordinate)


Expressions for Charge Q and Radius R

The explicit expression for the Noether charge

Q = 2π2

∞∫0

dr r3ωf 2

AN(21)

Define the radius of the boson star as an averaged radialcoordinate2

R =2π2

Q

∞∫0

dr r4ωf 2

AN(22)

2D. Astefanesei and E. Radu, Nucl. Phys. B 665 (2003) 594[gr-qc/0309131].


Expressions for Mass M

Mass for κ = 0

M = 2π2

∞∫0

dr r3(

Nφ′2 +ω2φ2

N+ U(φ)

)(23)

Mass for κ > 0 we define the gravitational mass3 by theasymptotic behaviour:

MG ∼ n(r →∞)/κ (24)

3Y. Brihaye and B. Hartmann, Nonlinearity 21 (2008), 1937, D. Astefaneseiand E. Radu, Phys. Lett. B 587 (2004) 7


Finding ω for solutions

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


Gauss–Bonnet boson stars with α >= 0

ω

Q

0.6 0.7 0.8 0.9 1.0

50

200

1000

5000

20000

α

= 0.0

= 0.025

= 0.05

= 0.075

= 0.1

= 0.2

= 0.3

= 0.4

= 0.45

= 0.5

= 1.0

= 2.0

= 5.0

ω = 1.0

Figure : Charge Q in dependence on the frequency ω for κ = 0.05 and differentvalues of α



ω

M

0.6 0.7 0.8 0.9 1.0

50

200

500

2000

10000

α

= 0.0

= 0.025

= 0.05

= 0.075

= 0.1

= 0.2

= 0.3

= 0.4

= 0.45

= 0.5

= 1.0

= 2.0

= 5.0

ω = 1.0

ω

M

0.6 0.7 0.8 0.9 1.0

200

500

2000

5000

α

= 0.0

= 0.025

= 0.05

= 0.075

= 0.1

= 0.15

= 0.2

= 0.3

= 0.4

= 0.45

= 0.5

= 1.0

= 2.0

= 5.0

ω = 1.0

Figure : Mass M in dependence on the frequency ω for κ = 0.05 (left) and κ = 0.02(right) and different values of α



ω

f(0)

0.6 0.7 0.8 0.9 1.0

01

23

45

α

= 0.0

= 0.025

= 0.05

= 0.075

= 0.1

= 0.2

= 0.3

= 0.4

= 0.45

= 0.5

= 1.0

= 2.0

= 5.0

ω = 1.0

Figure : Field f (0) in dependence on the frequency ω for κ = 0.05 and differentvalues of α



α

ω

0.0 0.2 0.4 0.6 0.8 1.0

0.6

0.7

0.8

0.9

1.0

κ

= 0.02 ωmin

= 0.02 ωcr

= 0.05 ωmin

= 0.05 ωcr

Figure : Frequency ωmin as well as ωcr as function of α for twodifferent values of κ.



r

A

1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

q(0)

= 0.1= 0.5= 1.0= 1.5= 2.2= 3.0= 3.6

rA

1 2 3 4 50.

00.

20.

40.

60.

81.

0

q(0)

= 0.1= 0.5= 1.0= 1.5= 2.2= 3.0= 3.6

t

Q

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.02e

+0

22

e+

03

2e

+0

42

e+

05

_

= 0.025= 0.1= 0.3= 5.0= 20.0= 100.0t = 1.0

Figure : Metric function A(r) for different values of α and f (0),κ = 0.05.



ω

A(0

)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

κ = 0.05 & α

= 0.0= 0.5= 1.0= 2.0= 5.0= 10.0= 25.0= 30.0= 35.0= 40.0= 50.0= 100.0= 1000.0= 5000.0

r

A

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

α

= 0.0

= 0.5

= 2.0

= 5.0

= 10.0

= 25.0

= 30.0

= 35.0

= 40.0

= 50.0

= 100.0

= 500.0

= 5000.0

Figure : A(0) over ω for fixed f (0), different values of α (large alpharegime) and κ = 0.05.



r

N

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

α

= 0.0

= 0.5

= 2.0

= 5.0

= 10.0

= 25.0

= 30.0

= 35.0

= 40.0

= 50.0

= 100.0

= 500.0

= 5000.0

r

f(r)

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

α

= 0.0

= 0.5

= 2.0

= 5.0

= 10.0

= 25.0

= 30.0

= 35.0

= 40.0

= 50.0

= 100.0

= 500.0

= 5000.0

Figure : f (r) and N(r) for fixed f (0), different values of α (large alpharegime) and κ = 0.05.



Figure : Ricci scalar R(0) different values of α.

4

4Yves Brihaye, Physique-Matheematique, Universite de Mons-Hainaut,7000 Mons, Belgium



Figure : Mass-radius relation for the equation of state of neutron stars(left) in comparison to the mass-radius relation of Gauss-Bonnetboson stars (right).

5S. GANDOLFI et al, PHYSICAL REVIEW C 85, 032801(R) (2012)Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars

Outlook

Rotating Gauss-Bonnet Boson Stars (currently underconstruction with Vyes Brihaye)Gauss-Bonnet Boson Stars in AdS space-time (AdS/CFT)Stability analysis of Gauss-Bonnet Boson StarsExcited solutions for Gauss-Bonnet Boson Stars


Existence conditions for solutions

Condition 1:V

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

max ≡ 12U

′′(0)

Condition 2:ω2 > ω2

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

Restricted interval ω2min < ω2 < ω2

max ;U

′′(0) > minφ[2U(φ)/φ2]

Q-balls are rotating in inner space with ω stabilized byhaving a lower energy to charge ratio as the freeparticlesFor USUSY: ω2

max = 1 and ω2min = 0
