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

Gauss–Bonnet Boson Stars in AdS
Jürgen Riedel
in collaboration with
Yves Brihaye(1)
, Betti Hartmann(2)(3)
, and Raluca Suciu(2)
Physics Letters B (2013) (arXiv:1308.3391), arXiv:1310.7223, and
arXiv:1404.1874
Faculty of Physics, University Oldenburg, Germany
Models of Gravity
SPRING WORKSHOP 2014, MODELS OF GRAVITY
Bielefeld, Germany
Mai 7th, 2014
(1)
Physique-Mathématique, Université 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)Gauss–Bonnet Boson Stars in AdS
SPRING WORKSHOP 2014, MODELS OF GRAV
/ 26

Agenda
Describe the model to construct non-rotating Gauss-Bonnet
boson stars in AdS
Describe effect of Gauss-Bonnet term to boson star solutions
Stability analysis of rotating Gauss-Bonnet boson star
solutions (ignoring the Gauss-Bonnet coupling for now)
Outlook
/ 26

Solitons in non-linear field theories
General properties of soliton solutions
Localized, finite energy, stable, regular solutions of non-linear
field equations
Can be viewed as models of elementary particles
Examples
Topological solitons: Skyrme model of hadrons in high energy
physics one of first models and magnetic monopoles, domain
walls etc.
Non-topological solitons: Q-balls (named after Noether charge
Q) (flat space-time) and boson stars (generalisation in curved
space-time)
/ 26

Non-topolocial solitons
Properties of non-topological solitons
Solutions possess the same boundary conditions at inﬁnity as
the physical vacuum state
Degenerate vacuum states do not necessarily exist
Require an additive conservation law, e.g. gauge invariance
under 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.
/ 26

Why study boson stars?
Boson stars
Simple toy models for a wide range of objects such as
particles, compact stars, e.g. neutron stars and even centres of
galaxies
We are interested in the effect of Gauss-Bonnet gravity and will
study these objects in the minimal number of dimensions in
which the term does not become a total derivative.
Toy models for studying properties of AdS space-time
Toy models for AdS/CFT correspondence. Planar boson stars
in AdS have been interpreted as Bose-Einstein condensates of
glueballs
/ 26

Model for Gauss–Bonnet Boson Stars
Action
S =
1
16πG5
d5
x
√
−g (R − 2Λ + αLGB + 16πG5Lmatter)
LGB = RMNKL
RMNKL − 4RMN
RMN + R2
(1)
Matter Lagrangian Lmatter = − (∂µψ)∗
∂µψ − U(ψ)
Gauge mediated potential
USUSY(|ψ|) = m2
η2
susy 1 − exp −
|ψ|2
η2
susy
(2)
USUSY(|ψ|) = m2
|ψ|2
−
m2|ψ|4
2η2
susy
+
m2|ψ|6
6η4
susy
+ 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
/ 26

Model for Gauss–Bonnet Boson Stars
Einstein Equations are derived from the variation of the action with
respect to the metric ﬁelds
GMN + ΛgMN +
α
2
HMN = 8πG5TMN (4)
where HMN is given by
HMN = 2 RMABCRABC
N − 2RMANBRAB
− 2RMARA
N + RRMN
−
1
2
gMN R2
− 4RABRAB
+ RABCDRABCD
(5)
Energy-momentum tensor
TMN = −gMN
1
2
gKL
(∂K ψ∗
∂Lψ + ∂Lψ∗
∂K ψ) + U(ψ)
+ ∂Mψ∗
∂Nψ + ∂Nψ∗
∂Mψ (6)
/ 26

Model continued
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
= −
i
2
ψ∗
∂M
ψ − ψ∂M
ψ∗
; jM
;M = 0 (9)
The globally conserved Noether charge Q reads
Q = − d4
x
√
−gj0
. (10)
/ 26

Ansatz non-rotating
Metric Ansatz
ds2
= −N(r)A2
(r)dt2
+
1
N(r)
dr2
+ r2
dθ2
+ sin2
θdϕ2
+ sin2
θ sin2
ϕdχ2
(11)
where
N(r) = 1 −
2n(r)
r2
(12)
Stationary Ansatz for complex scalar ﬁeld
ψ(r, t) = φ(r)eiωt
(13)
/ 26

Ansatz rotating
Metric Ansatz
ds2
= −A(r)dt2
+
1
N(r)
dr2
+ G(r)dθ2
+ H(r) sin2
θ (dϕ1 − W(r)dt)2
+ H(r) cos2
θ (dϕ2 − W(r)dt)2
+ (G(r) − H(r)) sin2
θ cos2
θ(dϕ1 − dϕ2)2
, (14)
with θ = [0,π/2] and ϕ1,ϕ2 = [0, 2π].
Cohomogeneity-1 Ansatz for Complex Scalar Field
Φ = φ(r)eiωt ˆΦ, (15)
with
ˆΦ = (sin θeiϕ1
, cos θeiϕ2
)t
(16)
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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 . (17)
Where Leff is the effective AdS-radius:
L2
eff =
2α
1 − 1 − 4α
L2
; L2
=
−6
Λ
(18)
Chern-Simons limit:
α =
L2
4
(19)
Mass for κ > 0 we deﬁne the gravitational mass at AdS
boundary
MG ∼ n(r → ∞)/κ (20)
/ 26

Finding solutions: ﬁxing ω
f
V(f)
−4 −2 0 2 4
−0.03−0.010.000.010.02
f(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
−0.10.10.20.30.40.5
k
= 0
= 1
= 2
φ = 0.0
r
Φ
0 5 10 15 20 25 30
−0.20.20.40.60.81.0
Figure : Effective potential V(f) = ω2φ2 − U(f).
/ 26

Gauss–Bonnet Boson Stars with Λ < 0
ω
Q
0.8 0.9 1.0 1.1 1.2 1.3 1.4
110100100010000
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 mode
ω = 1.0
Figure : Charge Q in dependence on the frequency ω for Λ = −0.01, κ = 0.02 and different
values of α. ωmax shift: ωmax = ∆
Leff
/ 26

Gauss–Bonnet Boson Stars with Λ < 0 Animation
(Loading Video...)
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Excited Gauss–Bonnet Boson Stars with Λ < 0
ω
Q
0.6 0.8 1.0 1.2 1.4 1.6
110010000
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, and different
values of α. ωmax shift: ωmax = ∆+2k
Leff
/ 26

Summary Effect of Gauss-Bonnet Correction
Boson stars
Small coupling to GB term, i.e. small α, we ﬁnd similar spiral
like characteristic as for boson stars in pure Einstein gravity.
When the Gauss-Bonnet parameter α is large enough the
spiral ’unwinds’.
When α and the coupling to gravity (κ) are of the same
magnitude, only one branch of solutions survives.
The single branch extends to the small values of the frequency
ω.
/ 26

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:1106.6073v1
/ 26

Stability of AdS
It has been shown that AdS is linearly stable (Ishibashi and Wald
2004).
The metric conformally approaches the static cylindrical
boundary, waves bounce off at infinity will return in finite time.
general result on the non-linear stability does not yet exist for
AdS.
It is speculated that AdS is nonlinearly unstable since in AdS
one would expect small perturbations to bounce of the
boundary, to interact with themselves and lead to instabilities.
Conjecture: small finite Q-balls of AdS in (3 + 1)-dimensional
AdS eventually lead to the formation of black holes (Bizon and
Rostworowski 2011).
Related to the fact that energy is transferred to smaller and
smaller scales, e.g. pure gravity in AdS (Dias et al).
/ 26

Erogoregions and Instability
If the boson star mass M is smaller than mBQ one would
expect the boson star to be classically stable with respect to
decay into Q individual scalar bosons.
Solutions to Einsteins field equations with sufficiently large
angular momentum can suffer from superradiant instability
(Hawking and Reall 2000).
Instablity occurs at ergoregion (Friedman 1978) where the
relativistic frame-dragging is so strong that stationary orbits
are no longer possible.
At ergoregion infalling bosonic waves are amplified when
reflected.
The boundary of the ergoregion is defined as where the
covariant tt-component becomes zero, i.e. gtt = 0
Analysis for boson stars in (3 + 1) dimensions has been done
(Cardoso et al 2008, Kleihaus et al 2008)
/ 26

Stability of Boson Stars with AdS Radius very large
ω
Q
0.2 0.4 0.6 0.8 1.0 1.2
1e+021e+041e+061e+08
Y= 0.00005Y= 0.00001
κ & Y & k
0.1 & 0.00005 & 0
0.01 & 0.00005 & 0
0.001 & 0.00005 & 0
0.01 & 0.00001 & 2
0.001 & 0.000007 & 0
0.001 & 0.000007 & 4
Figure : Charge Q in dependence on the frequency ω for different values of κ and Y = −6Λ.
/ 26

Ergoregions Animation
(Loading Video...)
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Stability for Boson Stars in AdS
ω
Q
0.5 1.0 1.5 2.0
1e+021e+041e+061e+08
Y= 0.1Y= 0.01Y= 0.001
κ
0.1,0.01,0.001,0.0001
0.1,0.01,0.001,0.0001
0.1,0.05,0.01,0.005,0.001
Figure : Charge Q in dependence on the frequency ω for different values of κ and Y = −6Λ.
/ 26

Stability Analysis for Radial Excited Boson Stars in
AdS
ω
Q
0.4 0.6 0.8 1.0 1.2 1.4
1e+021e+031e+041e+05
k=0 k=1 k=2 k=3 k=5
κ & Y
0.01 & 0.001
Figure : Charge Q in dependence on the frequency ω for different excited modes k.
/ 26

Summary Stability Analysis
Strong coupling to gravity: self-interacting rotating boson
stars are destabilized.
Sufficiently small AdS radius: self-interacting rotating boson
stars are destabilized.
Sufficiently strong rotation stabilizes self-interacting rotating
boson stars.
Onset of ergoregions can occur on the main branch of boson
star solutions, supposed to be classically stable.
Radially excited self-interacting rotating boson stars can be
classically stable in aAdS for sufficiently large AdS radius and
sufficiently small backreaction.
/ 26

Outlook
Analysis the effect of the Gauss-Bonnet term on stability of
boson stars.
To do a stabiliy analysis similar to the one done for non-rotating
minimal boson stars by Bucher et al 2013 ([arXiv:1304.4166
[gr-qc])
See whether our arguments related to the classical stability of
our solutions agrees with a full perturbation analysis
Gauss-Bonnet Boson Stars and AdS/CFT correspondance
Boson Stars in general Lovelock theory
/ 26

Thank You
Thank You!
/ 26

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

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