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What are universal features of What are universal features of gravitating Q-balls? gravitating Q-balls?
Takashi Tamaki(Nihon University)with Nobuyuki Sakai (Yamagata University)
I. IntroductionQ-balls:a kind of nontopological solitons consist of scalar fields
Self-gravity may not be important?2
2( )2
mV flat ×
with self-gravity ○ (mini-boson star)
Kaup, PR172, 1331 (1968)
gravity-mediation typegravity-mediation typein the Affleck-Dine (AD)in the Affleck-Dine (AD)mechanism mechanism
222
grav.( ) 1 ln2
mV K
M
flat○0K
0K ×
○0K
0K ○ ``Q-balls” surrounded by Q-matter
gravity
TT, N. Sakai, PRD83, 084046 (2011)
Here, gauge-mediation typeHere, gauge-mediation typeseek for universal features!!seek for universal features!!
24
gauge 2( ) ln 1V m
m
TT, N. Sakai, PRD84, 044054 (2011)
II. Model
normalizationm
24ln(1 )
VV
m
m
t mt r mr
24
gauge 2( ) ln 1V m
m
III. flat casegauge22
'' 'dV
r d
2 2 2 22
gauge: ln(1 )2 2
V V
2 2: 2
Particle motion in V
V2
2( 0) 0
d V
d
max( ) 0V trivial20
20 2
r
Thick-wall solution
Thin-wall solution
V
2 0
V
2 2 r
r
IV. gravitating case
2gauge2
2
2 ' ''' '
dVAA
r A d
22
2: 2
2 V
Particle motion in V
Signature of depends on r. depends on ``time”.
Thick-wall
Thin-wall
0.9995
0.9996
0.9997
0.9998
0.9999
1
1.0001
1.0002
1.0003
0 100 200 300 400 500
(a)
r~
A(r)
(r)
0
0.02
0.04
0.06
0.08
0.1
0 20 40 60 80 100 120 140
(b)
(r)
r
=0
=0.0005
~
~
0.2
0.4
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10
(a)
r
A(r)
(r)
~ 0
2
4
6
8
10
12
0 2 4 6 8 10
(b)
=0
=0.0005
r~
(r)
~
2 0.001
2 1.65
2: 0.0005Gm
Thick-wall Thin-wall
0
1 10-5
2 10-5
3 10-5
4 10-5
5 10-5
0 0.02 0.04 0.06 0.08 0.1
infinityr=0
-V
~
~
-2 10-7
-1 10-7
0
1 10-7
2 10-7
3 10-7
4 10-7
0.005 0.01 0.015 0.02 0.025 0.03
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12
infinityr=0
~-V
~
V changes ``quickly”.
2For various ?
2 0.001 2 1.65
0.0001
0.001
0.01
0.1
1
0 200 400 600 800 1000 1200
2
Q
A
C
=0=0.0005
D
B
C
=0.004
strong gravity
weak gravity
WHY ???
0 Q 0 0Q
V. thick-wall case2 1 ( )h r weak gravity 2 1 ( )A f r
take up to first order in h,f.
Let us evaluate
V0 : ( 0)r
0V Evaluate from2 2 20(0) 0h
We used Maclaurin expansion and neglected 50( )O
For 0 0
For 0 We should compare with(0)h 2
2 2 22
28t i
t i
rG G r A V
A
2 20, . ., 2i e and weak gravity
2 2 216r h r
We assume0( ) 1r for
Cr
C=const.
and approximating ( ) ( ) 0C
h h
220 2
8(0)
3
Ch
substitute into
2 2 20(0) 0h
420 2 2
3
8 3C
2 2C
2 2C
0 2 3 20 3
1 1Q r
as in the flat case
2
0
3
2 2C
2 3 20 3
10Q r
V. conclusion
Gravitating Q-balls
24
gauge 2( ) ln 1V m
m
The thick-wall limit is completely different from the flat case.
From the analytic estimation, our results hold for general potentials if a positive mass term is a leading order.
0Q exists !!
420 2 2
3
8 3C
2 2C
2 2C
0 2 3 20 3
1 1Q r
as in the flat case
2
0
3
2 2C
2 3 20 3
10Q r
10-8
10-7
10-6
10-5
0.0001
0.001
10-10 10-9 10-8 10-7 10-6 10-5
=10-4
=10-5
=10-6
0
2
Basic equations
important quantities
Total energy(Hamiltonian)
2
2 2: , :mE m Q
E QM M
サイズの議論mini-BS 1/m の半径 GM~1/m
2pMMm
BS 4 の相互作用項の存在が本質的 e.g.,
2 2 22
2( ) (1 )
p
U mM
GM~1/m_re
Λ 大
2 2pm M
U
1
rem m
2 3
2
p pM MM
m m
m(neutron) 程度 M=M( 太陽 )