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 )