Gauss Bonnet Boson Stars
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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
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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
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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)
Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars
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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.
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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
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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)
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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)
Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars
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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)
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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)
Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars
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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)
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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].
Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars
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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
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Finding ω for solutions
Figure : Effective potential V (φ) = ω2φ2 − U(|Φ|).
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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
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Gauss–Bonnet boson stars with α >= 0
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
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Gauss–Bonnet boson stars with α >= 0
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
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Gauss–Bonnet boson stars with α >= 0
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.
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Gauss–Bonnet boson stars with α >= 0
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.
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Gauss–Bonnet boson stars with α >= 0
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.
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Gauss–Bonnet boson stars with α >= 0
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.
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Gauss–Bonnet boson stars with α >= 0
Figure : Ricci scalar R(0) different values of α.
4
4Yves Brihaye, Physique-Matheematique, Universite de Mons-Hainaut,7000 Mons, Belgium
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Gauss–Bonnet boson stars with α >= 05
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
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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
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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
Jürgen Riedel & Betti Hartmann Gauss–Bonnet Boson Stars