PhD Defense, Oldenburg, Germany, June, 2014

Q-Balls and Boson Stars in asymptotically ﬂat and
Anti-de-Sitter space-time
Jürgen Riedel
PhD Supervisor: Betti Hartmann
Faculty of Physics, University Oldenburg, Germany
Models of Gravity
Oldenburg, June 5th, 2014
Riedel (University Oldenburg) Q-Balls and Boson Stars Oldenburg, June 5th, 2014 1 / 38

Publications:
B. Hartmann(1)(2) and J. Riedel, Phys. Rev. D 86, 104008
(2012), arXiv:1204.6239.
B. Hartmann and J. Riedel, Phys. Rev. D 87 044003 (2013),
arXiv:1210.0096.
B. Hartmann, J. Riedel, and R. Suciu(1), Physics Letters B
(2013), arXiv:1308.3391,
Y. Brihaye(3) and J. Riedel, (2013), arXiv:1310.7223 (accepted
for publication in Phys. Rev. D 89).
Y. Brihaye, B. Hartmann, and J. Riedel, (2014),
arXiv:1404.1874.
(1)
School of Engineering and Science, Jacobs University Bremen, Germany
(2)
Universidade Federal do Espirito Santo (UFES), Departamento de Fisica,
Vitoria (ES), Brazil
(3)
Physique-Mathématique, Université de Mons, 7000 Mons, Belgium

Outline
Brief introduction to general concepts of Q-balls and Boson
Stars
non-topological Solitons
Properties of Q-balls and Boson Stars
Model to construct Gauss-Bonnet Boson Stars in
asymptotically AdS space-time (aAdS)
Numerical results of Gauss-Bonnet Boson Stars
Stability aspects of Gauss-Bonnet Boson Stars
Classical or absolute Stability
Ergoregions and Superradiant Instability
Outlook

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)

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.

Non-topolocial solitons
Model for Q-balls
With complex scalar ﬁeld Φ(x, t) : L = ∂µΦ∂µΦ∗ − U(|Φ|), U(|Φ|)
minimum at Φ = 0
Lagrangian is invariant under transformation φ(x) → eiαφ(x)
Give rise to Noether charge Q = 1
i dx3φ∗ ˙φ − φ ˙φ∗)
Solution that minimizes the energy for ﬁxed Q: Φ(x, t) = φ(x)eiωt
Solutions have been constructed in (3 + 1)-dimensional
models with non-normalizable Φ6-potential
M.S. Volkov and E. Wöhnert, Phys. Rev. D 66 (2002), 085003, B. Kleihaus, J. Kunz and M. List, Phys. Rev. D 72 (2005), 064002,
B. Kleihaus, J. Kunz, M. List and I. Schaffer, Phys. Rev. D 77 (2008), 064025.

Existence conditions of Q-balls
Condition 1
V (0) < 0; Φ ≡ 0 local maximum ⇒ ω2 < ω2
max ≡ U (0)
Condition 2
ω2 > ω2
min ≡ minφ[2U(φ)/φ2] minimum over all φ
Consequences
Restricted interval ω2
min < ω2 < ω2
max ; U (0) > minφ[2U(φ)/φ2]
Q-balls are rotating in inner space with ω stabilized by having a
lower energy to charge ratio as the free particles

Finding solutions
Analogy to dynamics of a point particle:
r → time, V(φ) → potential + damping
φ
V(φ)
−4 −2 0 2 4
−0.03−0.010.000.010.02
φ(0)
k=0
k=1
k=2
ω = 0.80
ω = 0.85
ω = 0.87
ω = 0.90
V = 0.0
Figure : Effective potential V(φ) = ω2φ2 − U(φ).

Finding solutions (continued)
r
φ
0 5 10 15 20
−0.10.10.20.30.40.5
k
= 0
= 1
= 2
φ = 0.0
rφ
0 5 10 15 20 25 30
−0.20.20.40.60.81.0
k
= 0
= 1
= 2
φ = 0.0
Figure : Proﬁle functions φ(r) fundamental and radially excited for static (left) and rotating
(right) solutions.

Thin wall and Thick wall approximation of Q-balls
Thin wall limit: ω ωmin; φ(0) 0
Step-like ansatz [Coleman 1985]: no wall thickness
Modiﬁed step-like (egg-shell-like) ansatz: includes wall thickness
[Correia et.al. 2001; M.I.T., Copeland, Safﬁn 2008]
Minimum of total energy ωmin = Emin = 2U(φ0)
φ2 , for φ0 > 0
The energy and charge is proportional to the volume which is
similarly found in ordinary matter → Q = ωφ2
V
Therefore Q-balls in this limit are called Q-matter and have very
large charge, i.e. volume
Thick wall limit: ω ωmax ; φ(0) 0
As ω increases initial density φ(0) gets close to zero
Gaussian ansatz: φ(r) = exp r2
R2

Q-ball profile in 3D: Thick wall to Thin wall transition
(r) = 0 (r) = 13
Figure : Q-ball profile function φ(r) for initial field densitie φ(0). Qball 3D Rot. Boson Star 3D

Affleck-Dine (AD) Mechanism
Attempt to describe Baryogenesis of early universe in the
Minimal Supersymmetric extension of the Standard Model
(MSSM) of particle physics
In Scalar Potential (= sauark, slepton, higgs) of MSSM
There exist Flat Directions = AD field
Flat directions are supersymmetric minima of the scalar
potential.
Directions are lifted by supersymmetry breaking effects (only
approximately flat)
AD condensates forms along flat direction. Fragmentation
creates Q-balls.

Afﬂeck-Dine Condensate and Q-balls

SUSY breaking potentials
Two possible (soft) SUSY breaking potentials: Gravity mediated
potential and Gauge mediated potential
Gravity mediated potential : SUSY breaking through
gravitational interactions.
Gauge mediated potential: SUSY breaking through the
Standard Model’s gauge interactions.
USUSY(|Φ|) =
m2|Φ|2 if |Φ| ≤ ηsusy
m2η2
susy = const. if |Φ| > ηsusy
(1)
U(|Φ|) = m2
η2
susy 1 − exp −
|Φ|2
η2
susy
(2)
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

Boson star models
Gravitating cousins of Q-balls
Simplest model U = m2|Φ|2 (by Kemp, 1986)
Proper boson stars U = m2|Φ|2 − λ|Φ|4/2
(by Colpi, Sharpio and Wasserman, 1986)
Sine-Gordon boson star U = αm2 sin(π/2 β |Φ|2 − 1 + 1
Cosh-Gordon boson star U = αm2 cosh(β |Φ|2 − 1
Liouville boson star U = αm2 exp(β2|Φ|2) − 1
(Schunk and Torres, 2000)

Self-interacting boson star models
Model U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a and b are constants
(Mielke and Scherzer, 1981)
Soliton stars U = m2|Φ|2 1 − |Φ|2/Φ2
0
2
(Friedberg, Lee and Pang, 1986)
Represented in the limit of ﬂat space − time, by Q -balls as
non-topological solitons
However, terms of |Φ|6 or higher-order terms implies that the
scalar part of the theory is not re-normalizable

Why study boson stars or Q-balls?
Q-balls
Supersymmetric Q-balls have been considered as possible
candidates for baryonic dark matter
Boson stars and Q-balls
Simple toy models for a wide range of objects such as
particles, compact stars, e.g. neutron stars and even centres of
galaxies
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

Why study higher dimensions?
Set up a model to support boson star solutions in
Gauss-Bonnet gravity in 4 + 1 dimensions
Gauss-Bonnet theory which appears naturally in the low energy
effective action of quantum gravity models
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.
Higher dimensions appear in attempts to find a quantum
description 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: fundamental scalar
fields do exist in nature.

Model for Gauss–Bonnet Boson Stars
Action
S =
1
16πG5
d5
x
√
−g (R − 2Λ + αLGB + 16πG5Lmatter)
LGB = RMNKL
RMNKL − 4RMN
RMN + R2
(3)
Matter Lagrangian Lmatter = − (∂µψ)∗
∂µψ − U(ψ)
Gauge mediated potential
USUSY(|ψ|) = m2
η2
susy 1 − exp −
|ψ|2
η2
susy
(4)
USUSY(|ψ|) = m2
|ψ|2
−
m2|ψ|4
2η2
susy
+
m2|ψ|6
6η4
susy
+ O |ψ|8
(5)
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 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 (6)
where HMN is given by
HMN = 2 RMABCRABC
N − 2RMANBRAB
− 2RMARA
N + RRMN
−
1
2
gMN R2
− 4RABRAB
+ RABCDRABCD
(7)
Energy-momentum tensor
TMN = −gMN
1
2
gKL
(∂K ψ∗
∂Lψ + ∂Lψ∗
∂K ψ) + U(ψ)
+ ∂Mψ∗
∂Nψ + ∂Nψ∗
∂Mψ (8)

Model continued
The Klein-Gordon equation is given by:
−
∂U
∂|ψ|2
ψ = 0 (9)
Lmatter is invariant under the global U(1) transformation
ψ → ψeiχ
. (10)
Locally conserved Noether current jM
jM
= −
i
2
ψ∗
∂M
ψ − ψ∂M
ψ∗
; jM
;M = 0 (11)
The globally conserved Noether charge Q reads
Q = − d4
x
√
−gj0
. (12)

Ansatz non-rotating
Metric Ansatz
ds2
= −N(r)A2
(r)dt2
+
1
N(r)
dr2
+ r2
dθ2
+ sin2
θdϕ2
+ sin2
θ sin2
ϕdχ2
(13)
where
N(r) = 1 −
2n(r)
r2
(14)
Stationary Ansatz for complex scalar ﬁeld
ψ(r, t) = φ(r)eiωt
(15)

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
, (16)
with θ = [0,π/2] and ϕ1,ϕ2 = [0, 2π].
Cohomogeneity-1 Ansatz for Complex Scalar Field
Φ = φ(r)eiωt ˆΦ, (17)
with
ˆΦ = (sin θeiϕ1
, cos θeiϕ2
)t
(18)

Boundary Conditions
Rescaling using dimensionless quantities
r →
r
m
, ω → mω , ψ → ηsusyψ , n → n/m2
, α → α/
√
m (19)
Appropriate boundary conditions non-rotating:
φ (0) = 0 , N(0) = 1. (20)
Appropriate boundary conditions rotating:
B (0) = 0
H (r = 0)
r2
= 0 W (0) = 0 φ(0) = 0. (21)
We need the scalar ﬁled to vanish at inﬁnity and therefore
require φ(∞) = 0, while we choose A(∞) = 1 (rescaling of the
time coordinate)
Field equations depend only on the dimensionless coupling
constants α and κ = 8πG5η2
SUSY

Asymptotic Behaviour
If Λ < 0 the scalar ﬁeld function falls off with
φ(r >> 1) =
φ∆
r∆
, ∆ = 2 + 4 + L2
eff . (22)
Where Leff is the effective AdS-radius:
L2
eff =
2α
1 − 1 − 4α
L2
; L2
=
−6
Λ
(23)
Chern-Simons limit:
α =
L2
4
(24)
Mass for κ > 0 we deﬁne the gravitational mass at AdS
boundary
MG ∼ n(r → ∞)/κ (25)

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
Unfolding DB GB0 DB GB15 DB GB0 ex DB GB15 ex

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

Summary Effect of Gauss-Bonnet Correction
Boson stars
Small coupling to GB term, i.e. small α, we ﬁnd the same
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.

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

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 and 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, because one
would expect small perturbations to bounce off the boundary,
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).

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)

Superradiant Instability
Penrose process (Penrose 1969, Christodoulou 1970) scattering of particles off
Kerr BH =⇒ reduction of BH mass.
Superradiant scattering of wave packet off Kerr BH (Misner 1972,
Zeldovitch 1971)
"Black Hole Bomb" (Press & Teukolsky 1972, Zeldovich 1971, Cardoso et al 2004). Black
hole surrounded by a mirror =⇒ exponential growth of
modes and instability due to superradiant scattering
Natural mirror provided by anti-de Sitter spacetimes (AdS)

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.
DB GB0 DB GB15

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.

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.

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, which are classically stable.
Radially excited self-interacting rotating boson stars can be
classically stable in aAdS for sufficiently large AdS radius and
sufficiently small backreaction.

Outlook
Analyse the effect of the Gauss-Bonnet term on stability of
boson stars.
To do a stability analysis of Gauss-Bonnet boson stars based on
the work on 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

Thank You
Thank You!

PhD Defense, Oldenburg, Germany, June, 2014

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PhD Defense, Oldenburg, Germany, June, 2014