gauss-bonnet boson stars ads and flat space-time, golm, germany, 2014-talk
DESCRIPTION
Simple toy models for a wide range of objects such as particles, compact stars, e.g. neutron stars and even centres of galaxies Gauss-Bonnet gravity: its spectrum does not include new propagating degrees of freedom besides gravitation Toy models for AdS/CFT correspondence. Planar boson stars in AdS have been interpreted as Bose-Einstein condensates of glueballsTRANSCRIPT
Gauss–Bonnet Boson Stars in AdS
Jürgen Riedelin collaboration with
Yves Brihaye1, Betti Hartmann, and Raluca Suciu2
Physics Letters B (2013) (arXiv:1308.3391) and arXiv:1310.7223
Faculty of Physics, University Oldenburg, GermanyModels of Gravity
NORDIC STRING MEETING 2014Golm, February 3rd, 2014
1Physique-Mathématique, Université de Mons, 7000 Mons, Belgium
2School of Engineering and Science, Jacobs University Bremen, Germany
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
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.Discovery of the Higgs Boson in 2012: fundamentalscalar fields do exist in nature.
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
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)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
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.
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Why study Q-balls and boson stars?
Q-ballsScalar fields prevented from collapse by Heisenberg’suncertainty principle and repulsive self-interactionSupersymmetric Q-balls have been considered aspossible candidates for baryonic dark matter
Boson starsDescribed by relatively simple equationsSimple toy models for a wide range of objects such asparticles, compact stars, e.g. neutron stars and evencentres of galaxiesGauss-Bonnet gravity: its spectrum does not includenew propagating degrees of freedom besidesgravitationToy models for AdS/CFT correspondence. Planar bosonstars in AdS have been interpreted as Bose-Einsteincondensates of glueballs
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Model for Gauss–Bonnet Boson Stars
Action
S =1
16πG5
∫d5x√−g (R − 2Λ + αLGB + 16πG5Lmatter)
LGB =(
RMNKLRMNKL − 4RMNRMN + R2)
(1)
Matter Lagrangian Lmatter = − (∂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
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Model for Gauss–Bonnet Boson Stars
Einstein Equations are derived from the variation of theaction with respect to the metric fields
GMN + ΛgMN +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)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Ansatz
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)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Ansatz
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)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Einstein equations
Equations for the metric functions:
N ′(r) = 2r
(1− N − r2
3 Λ
r2 + 2α(1− N)
)
− 23κr3
NA2
(UNA2 + ω2f 2 + N2A2f ′2
r2 + 2α(1− N)
)(15)
A′(r) =2κr3 (A2N2f ′2 + ω2f 2)3AN2
(r2 + 2α(1− N)
) (16)
κ = 8πG5η2susy (17)
Matter field equation:(r3ANf ′
)′= r3A
(12∂U∂f− ω2f
NA2
)(18)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Boundary Conditions flat space-time
Appropriate boundary conditions:
f ′(0) = 0 , n(0) = 0 . (19)
We need the scalar filed to vanish at infinity and thereforerequire f (∞) = 0, while we choose A(∞) = 1 (rescaling ofthe time coordinate)If Λ = 0 the scalar field function falls of exponentially with
f (r >> 1) ∼ 1
r32
exp(−√
1− ω2r)
+ ... (20)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
AdS space-time
3
Figure : (a) Penrose diagram of AdS space-time, (b) massive (solid) and massless(dotted) geodesic.
3J. Maldacena, The gauge/gravity duality, arXiv:1106.6073v1Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Boundary Conditions for asymptotic AdS space time
If Λ < 0 the scalar field function falls of with
φ(r >> 1) =φ∆
r∆, ∆ = 2 +
√4 + L2
eff . (21)
Where Leff is the effective AdS-radius:
L2eff =
2α
1−√
1− 4αL2
(22)
With L being the AdS-radius which is related to Λ as:
Λ =−6L2 (23)
Chern-Simons limit:
α =L2
4(24)
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Expressions for Charge Q and Radius R
The explicit expression for the Noether charge
Q = 2π2
∞∫0
dr r3ωf 2
AN(25)
Define the radius of the boson star as an averaged radialcoordinate4
R =2π2
Q
∞∫0
dr r4ωf 2
AN(26)
4D. Astefanesei and E. Radu, Nucl. Phys. B 665 (2003) 594[gr-qc/0309131].
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Expressions for Mass M
Mass for κ = 0
M = 2π2
∞∫0
dr r3(
Nf ′2 +ω2f 2
N+ U(f )
)(27)
Mass for κ > 0 we define the gravitational mass5 by theasymptotic behaviour:
MG ∼ n(r →∞)/κ (28)
5Y. Brihaye and B. Hartmann, Nonlinearity 21 (2008), 1937, D. Astefaneseiand E. Radu, Phys. Lett. B 587 (2004) 7
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Finding solutions: fixing ω
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
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
Figure : Effective potential V (f ) = ω2f 2 − U(f ).
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ = 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
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ = 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
Figure : Mass M in dependence on the frequency ω for κ = 0.05 (left) and κ = 0.02(right) and different values of α
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ = 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.
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ = 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.
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ = 06
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).
6S. GANDOLFI et al, PHYSICAL REVIEW C 85, 032801(R) (2012)Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
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 anddifferent values of α. ωmax shift: ωmax = ∆
Leff
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Gauss–Bonnet boson stars with Λ < 0
ω
M
0.8 1.0 1.2 1.41e+
00
1e+
02
1e+
04
1e+
06
1e+
08
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= 13.0= 14.0= 14.5= 14.95= 15.0 (CS limit)
ω = 1.0
Figure : Mass M in dependence on the frequency ω for Λ = −0.01, κ = 0.02 anddifferent values of α
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Excited Gauss–Bonnet boson stars with Λ = 0
ω
Q
0.5 0.6 0.7 0.8 0.9 1.01e+
03
5e+
03
2e+
04
1e+
05
k=0
k=1
k=2
k=3
α
= 0.0
= 0.1
= 1.0
= 5.0
= 10.0
ω = 1.0
Figure : Charge Q in dependence on the frequency ω for Λ = 0, κ = 0.0, anddifferent values of α
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
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, anddifferent values of α. ωmax shift: ωmax = ∆+2k
Leff
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Outlook
Rotating Gauss-Bonnet Boson Stars paper with YvesBrihaye online: arXiv:1310.7223Gauss-Bonnet Boson Stars and AdS/CFTcorrespondanceStability analysis of Gauss-Bonnet Boson StarsBoson Stars in general Lovelock theory
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS
Thank You
Thank You!
Brihaye, Hartmann, Riedel, and Suciu Gauss–Bonnet Boson Stars in AdS