I. Introduction II. Collision of Domain Walls in 5D Minkowski Space - - PowerPoint PPT Presentation

SLIDE 1

Kei-ichi Maeda Waseda Univ. with G. Gibbons, H. Kudoh, and Y. Takamizu

• Phys. Rev. D70 (2004) 123514 , D73 (2006) 103508

hep-th/0610286, in preparation

• I. Introduction
• V. Collision of Domain Walls in Asymptotically AdS Space
• II. Collision of Domain Walls in 5D Minkowski Space
• IV. Fermion Localization at Collision
• VI. Summary
• III. Reheating by Collision of Branes

SLIDE 2

• I. Introduction

A brane: an interesting object in string theory D3 brane : could be our universe

Some interesting cosmological senarios

Ekpyrotic (or cyclic) universe Brane inflation (Dvali-Tye , Rolling Tachyon , KKLMMT, ・・・)

A brane is usually treated as an infinitesimally thin object

To discuss “matter” on branes (e.g. reheating, localization), we consider a finite thickness of brane.

collision of branes

a brane = a domain wall

SLIDE 3

• II. collision of domain walls in 5D Minkowski space

5D scalar field Φ potential

V = ï 4 Ä à

2 Äë 2Å2

η −η Φ V domain wall Φ=η Φ=−η

à = ëtanh ê y D ë

domain wall solution

• Y. Takamizu & KM: Phys.Rev. D70 (2004) 123514
• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

Φ η −η y

2à = @V @à

SLIDE 4

Collision of two domain walls Two boosted domain walls with velocities v and −v initial condition

à(y; 0) = àv(y + y0; 0) Ä àÄv(y Ä y0; 0) Ä 1

v −v y Φ −y0

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

y y0

unit : ë= 1

SLIDE 5

collide once v=0.4

ö

à = 1

collide twice v=0.2

2 ê _ à2 + à02ë + ï 4 Ä à2 Ä ë

2Å2

à(t; y)

The number of bounces highly depends on the initial velocity. N-bounce sols. : a fractal structure

Anninos, Oliveira and Matzner, PRD 44 (1991) 1147

SLIDE 6

• III. Reheating by Collision of Branes

a scalar field σ confined on a brane, which is coupled to Φ

Lint = 1 2g2à2

W (ú

)õ

2

àW (ú ) : evaluated on a domain wall (a brane)

τ : a proper time on a domain wall |in > |out > |in > |out > particle creation quantization of a scalar field σ with a “time-dependent mass”

SLIDE 7

spectrum: gaussian enough reheating !

më ò 1015[GeV] N Ä1=4

b

ê ñ g 10Ä5 ëÄ1 í TR 1010GeV ì

:mass scale of domain wall

quantum creation of particles

ö= 20g4Nb; n = 25Dg4Nb

ñ g ù ï D Nb n ö 0.4 1.0 1.414 1 3.69Ç10Ä7 2.05Ç10Ä7 0:01 10 0.447 1.16Ç10Ä7 2.05Ç10Ä7 0.2 1.0 1.414 2 7.19Ç10Ä7 3.90Ç10Ä7 10 0.447 2.26Ç10Ä7 3.91Ç10Ä7 0.4 1.0 1.414 1 3.57Ç10Ä3 2.01Ç10Ä3 0.1 10 0.447 1.16Ç10Ä3 2.05Ç10Ä3 0.2 1.0 1.414 2 6.65Ç10Ä3 3.81Ç10Ä3 10 0.447 2.24Ç10Ä3 3.88Ç10Ä3

SLIDE 8

• III. Fermion Localization at Collision
• G. Gibbons, KM &Y. Takamizu : hep-th/0610286

5D four-component fermion Ψ coupled to the scalar field Φ

Ä

MDMâ + gàâ = 0

DM = @M + 1 4! ^

A ^ BMÄ ^ A ^ B

Ä

^ A ^ B = Ä [ ^ AÄ ^ B]

Ä

M = eM ^ AÄ ^ A

fÄ

^ A; Ä ^ Bg = 2ë ^ A ^ B Two chiral states

âÄ = 1 2 ê 1 Ä Ä

^ 5ë

â

â+ = 1 2 ê 1 + Ä

^ 5ë

â

âÄ = † †

Ä

Ć

Ä

!

â+ = † †

+

†

+

!

Ä@5âÄ + Ä

ñ@ñâ+ + gàâÄ = 0

@5â+ + Ä

ñ@ñâÄ + gàâ+ = 0

In Minkowski background

SLIDE 9

Localization on a brane

Assume massless fermion on the brane

Ä

ñ@ñ† + = 0

Ä

ñ@ñ† Ä = 0

âÄ(x; z) = †

Ä(x)fÄ(z)

â+(x; z) = †

+(x)f+(z)

Static domain wall

à = è ëtanh z D è= 1 : kink

è= Ä1 : antikink

Ä@5fÄ + gàfÄ = 0 @5f+ + gàf+ = 0 è= 1

f+ / 1 (cosh(z=D))gD

è= Ä1

fÄ / 1 (cosh(z=D))gD

positive-chirality: localized negative-chirality: localized

• 10
• 5

5 10

• 1
• 0.5

0.5 1

fÄ

• 10
• 5

5 10

• 1
• 0.5

0.5 1

f+

Jackiw-Rebbi (76), Rubakov-Shaposhnikov(83) Randjbar-Daemi-Shaposhnikov(00) Bajc-Gabadadze(00), Kehagias-Tamvakis (01)

SLIDE 10

â(K)(x; z) = @

(4)

†+(x)f+(z)

(4)

†+(x)f+(z) 1 A fÜ(z) = î Ä (gD + 1

2)

2pô DÄ (gD) ï1=2 h cosh ê z D ëiÄgD â(A)(x; z) = @

(4)

†Ä(x)fÄ(z) Ä

(4)

†Ä(x)fÄ(z) 1 A aK = hâ(K); âi and aA = hâ(A); âi

annihilation operators

normalized wave function wave function of localized fermions on a kink and on an antikink

SLIDE 11

Time-dependent Background Ansatz 1: 3-space is flat

†

Ä =

1 (2ô )3=2 ei~

k~ x† Ä(t; z : ~

k) †

+ =

1 (2ô )3=2 ei~

k~ x† +(t; z : ~

k)

(@5 + gà)†

+ Ä (i@0 + ~

k~ õ )†

Ä = 0

(@5 Ä gà)†

Ä + (i@0 Ä ~

k~ õ )†

+ = 0 Ansatz 2: Low energy state

~ k ô 0

†

+ =

† †

+u

†

+d

!

†

Ä =

† †

Äu

†

Äd

!

u: up d: down

i@0†

Äu = (@5 + gà)† +u

i@0†

+u = (Ä@5 + gà)† Äu

u!d

and k : u-d mixing time-dependence : chirality mixing

SLIDE 12

â = B B B B @ 1 1 1 C C C C A †

+(z; t) +

B B B B @ 1 Ä1 1 C C C C A †

Ä(z; t)

†(K)

+ (z; t; ù

) = r ç+ 1 2 ~ †

(K) (ç

(z Ä ù t)) †

(K) Ä (z; t; ù

) = i ç ù ç+ 1 r ç+ 1 2 ~ †

(K) (ç

(z Ä ù t)) †(A)

Ä (z; t; ù

) = r ç+ 1 2 ~ †

(A) (ç

(z Ä ù t)) †(A)

+ (z; t; ù

) = Äi ç ù ç+ 1 r ç+ 1 2 ~ †

(A) (ç

(z Ä ù t))

wave function on a moving kink wave function on a moving antikink up state

SLIDE 13

^ â = â(K)

in (x; z; ù

)aK + â(A)

in (x; z; Äù

)aA + â(B)

in (x; z)aB

Fermion Localization on Colliding Branes

bK = ã

KaK + å AaA

bA = ã

AaA + å KaK

Bogoliubov transformation â(K)

in (x; z; ù

) ò ã

Kâ(K)

• ut(x; z; Äù

) + å

Kâ(A)

• ut(x; z; ù

) + ç

Kâ(B)

• ut(x; z)

â(A)

in (x; z; Äù

) ò ã

Aâ(A)

• ut(x; z; ù

) + å

Aâ(K)

• ut(x; z; Äù

) + ç

Aâ(B)

• ut(x; z)

Mode mixing by domain wall collision

à(t; z) : colliding two domain walls (Sec. II)

Fermion wave functions

Before collision

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

v −v ^ â = â(K)

• ut(x; z; Äù

)bK + â(A)

• ut(x; z; ù

)bA + â(B)

• ut(x; z)bB

After collision

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

v −v

SLIDE 14

hNKi ë hKAjby

KbKjKAi = jã Kj2 + jå Aj2

hNAi ë hKAjby

AbAjKAi = jã Aj2 + jå Kj2

hNKi ë hK0jby

KbKjK0i = jã Kj2

hNAi ë hK0jby

AbAjK0i = jå Kj2

Two cases :

(1) same amount of fermion on each brane (2) one brane is empty

• 10
• 5

5 10

• 1
• 0.5

0.5 1

~ âÄ

• 10
• 5

5 10

• 1
• 0.5

0.5 1

~ â+

• 10
• 5

5 10

• 1
• 0.5

0.5 1

~ â+

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

jKAi = ay

Kay Aj0i

Initial state

jK0i = ay

Kj0i

Initial state after collision after collision

SLIDE 15

(2)v=0.4 (1) v=0.8

Fermions transfer to the vacuum brane

0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 n z 0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 n z 0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 n z 0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 z n+ n- 0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 z n+ n- 0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 z n+ n-

Bogoliubov coefficients by solving the Dirac eqs.

SLIDE 16

g = 2 g = 2:5 ù jã

Kj2

jå

Kj2

jç

Kj2

jã

Kj2

jå

Kj2

jç

Kj2

0.3 0.94 0.056 0.004 0.47 0.53 0.00 0.4 0.87 0.12 0.01 0.57 0.40 0.03 0.6 0.69 0.30 0.01 0.78 0.17 0.05 0.8 0.42 0.55 0.03 0.88 0.02 0.10

Bogoliubov coefficients A few percent of fermions escape to bulk space

jç

Kj2 ú 1

The number of fermions are conserved as a whole

jã

Kj2 +jå Kj2 ô1

Because of the left-right symmetry,

jå

Kj2 =jå Aj2

jã

Kj2 =jã Aj2

SLIDE 17

g-dependence

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 |αK|2, |βK|2 g

hNKi = jã

Kj2 + jå Aj2 ô 1

hNAi = jã

Aj2 + jå Kj2 ô 1

hNKi = jã

Kj2

hNAi = jå

Kj2

(1) collision of two fermion branes (2) collision of fermion-vacuum branes

0.2 0.4 0.6 0.8 1 1.2 1.4

• 40
• 30
• 20
• 10

10 20 30 40 n z 0.2 0.4 0.6 0.8 1 1.2

• 40
• 30
• 20
• 10

10 20 30 40 n z

SLIDE 18

jãK j2 ; jåK j2 = 1 2 h 1 Ü sin (3 p 2 " g = p ï + C ã ;å (ù)) i

The amount of fermions on each wall depends sensitively on

ù a n d g p ï

(2) If we change the incident velocity very little, the number of bounces changes. This causes a drastic change of å nal distribution

• f fermions on each wall.

Some remarks:

(1) g < 2=D : the localization of fermions on a domain wall is not suécient.

jãKj2 + jå

Aj2 = 1:28

g = 1 ; ù = 0 :8 " = Ü 1

SLIDE 19

• IV. collision of domain walls in AdS space

AdS M

• Y. Takamizu & KM: Phys.Rev. D73 (2006) 103508

BPS domain wall solution

V (à) = ê @W @à ë2 Ä 8 3î

2 5 W 2

W ë 1 D í à Ä 1 3à3 Ä 2 3 ì

superpotential

àK(y) = tanh ê y D ë ds2 = e2AK(y)(Ädt2 + dx2) + dy2

AK(y) = Ä4 9î

2 5

n ln h cosh ê y D ëi + tanh2(y=D) 4 Ä y D

• :

Eto-Sakai, PRD68(2003)125001 Arai et al., PLB556 (2003) 192-202

SLIDE 20

Initial setting Two domain walls in asymptotocally AdS background v −v Φ −y0

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

y y0

scalar field metric

Lorentz boost

−y0 y y0 A

SLIDE 21

° à = à00 Ä 3 _ B _ à+ 3B0à0 Ä 1 2e2AV 0(à) ; ° B = B00 Ä 3 _ B2 + 3B02 + 2 3î

2 5e2AV (à)

° A = A00 + 3 _ B2 Ä 3B02 Ä î

2 5( _

à2 Ä à02 + 1 3e2AV (à))

Dynamical equations

_ BB0 Ä A0 _ B Ä _ AB0 + _ B0 = Ä2 3î

2 5 _

àà0 2B02 + B00 Ä A0B0 Ä _ A _ B Ä _ B2 = Ä1 3î

2 5( _

à2 + à02 + e2AV (à))

Constraint equations

ds2 = e2A(t;z)( Ädt2 + dz2 ) + e2B(t;z)dx2

metric form

Dynamics

SLIDE 22

We recover the same results for weak gravity limit (κ5<<1)

Φ becomes unstable Stability

Effect of gravity

î

5 = 0:15

î

5 = 0:25

SLIDE 23

Spacetime evolves into a singularity

metric curvature invariant

eA

Rñóö

õRñóö õ

singularity singularity

Khan-Penrose: Nature 229 (1971) 185

• cf. plane wave collision

F.J. Tipler : PRD 22 (1980) 2929

SLIDE 24

event horizon wall position singularity

Takamizu, Kudoh, KM, in preparation spacelike singularity

Domain walls after collision are moving outside event horizon

“We” do not see a singularity

SLIDE 25

• V. Summary

We discuss collision of domain walls (branes) In 5D Minkowski background We find a bounce (or a few bounces) of domain walls. We study particle production at the collision. reheating of the universe Including gravitational effects We study dynamics of spacetime with asymptotically AdS formation of singularity event horizon appears We analyze localization of fermions on branes. localized after collision transfer to vacuum brane v and g-dependence

SLIDE 26

Remarks

Our analysis is based on field theory (supergravity). It may be more important to study collision of branes based on superstring or M-theory. It may be interesting to see what happens on fermion distribution when gravity is included. One may look for the origin of matter (baryon asymmetry) in a braneworld scenario.