Cosmology in a brane-induced gravity model

with trace-anomaly terms

Shuntaro Mizuno

Kei-ichi Maeda, S.M. , Takashi Torii

Phys. Rev. D 68, 024033 (2003)

Mini Workshop on Brane World @ Tokyo Institute of Technology

2003. 10. 30

Waseda University, Japan

1. Introduction　

2. DGP model　

3. Brane induced gravity in 5D-AdS 　

4. Brane induced gravity with trace anomaly

terms 　5. Summary and discussion

1. Introduction 　Cosmological scenario based on brane models

R-SII model (simple and concrete)

homogeneous part inhomogeneous part

quadratic term

dark radiation

(necessary for comparison with observations)

modified by curvature corrections

many discussions

1, induced gravity term on brane

2, trace anomaly term on brane

4d gravity in infinite volume extradimension

quantum fluctuation of matter fields on brane 3, Gauss-Bonnet term in bulk

non-singular warped compactification

new interesting phenomenology?

2. D-G-P model (Dvali, Gabadadze, Porrati, `00)model

Sbulk =R

d5xp

Äg[m35

2 R]

Smatter =R

d5xp

Äg[Lmé(y)]a flat brane in 5D Minkowski bulk

S = Sbulk + Smatter

gravity matter

graviton

massive scalars/fermions

Sbrane =R

d4xp

Ä (4)g[ñ2

2(4)R]

+Sbrane

quantum interaction between bulk gravity and brane matter

( Sakharov `75, Akama `78 Adler `80 )

Cosmological solutions in DGP model ( Deffayet `01)

ds2 = Än2(ú;y)dú2 + a2(ú;y)çi j dxi dxj + b2(ú;y)dy2

Tñó = é(y)

b diag(Äö;p;p;p;0)[a0]a0b0

= è( 1m3

5öÄ ñ2

m35n2

0f _a0

2

a20

+ kn20

a20g)( matching condition)

H 2 + ka2 = 1

3ñ2 ö4D cosmology

H = 2m35

ñ2

Inflationary solution

è= +1 branch

m5 ò 100MeV present cosmic acceleration even for

è= Ä1 branch H 2 + ka2 = 1

36m45ö2

5D cosmology

övac = 0

è= Ü1

high energy low energy( H ù m35

ñ2 ) ( H ú m35

ñ2 )

Gravitational property in DGP model

Scalar property (gravitational potential)

V(r) =R

GR (t;x;y = 0;0;0;0)dt r ëp

x21 + x2

2 + x23

V(r) / Ä 1r (4D gravity) for r ú ñ2

m35

(small scale)

V(r) / Ä 1r 2 (5D gravity) for r ù ñ2

m35

(large scale)

Tensor property (tensor structure of graviton propagator)g2TñóP ñóãåT0

ãå

(Dvali, Gabadadze, Porrati, `00)

P ñóãå = 12(ëñã ëóå + ëñåëóã ) Ä 1

3ëñóëãå + O(p) cf. 4D massless graviton

P ñóãå = 12(ëñã ëóå + ëñåëóã ) Ä 1

2ëñóëãå + O(p)

excluded for relativistic sources van Dam-Veltman discontinuity

3. Brane induced gravity in 5D-AdS 　

confinement of the massless mode on the brane as R-S model cf. Tanaka (`03) tensor structure

Maeda, S.M. , Torii (`03)

S = Sbulk + Sbrane

model

Sbulk =R

d5xp

Ä (5)g[m35

2(5)R Ä (5)É]

Sbrane =R

d4xp

Äg[ñ2

2 R + Lm Ä ï ](4)É = 1

2[(5)É + ï 2

m45]

field equation

Gñó = Ä (4)Égñó + î 4

6 ï úñó + î 4ôñó Ä Eñó

úñó = Tñó Ä 1ñ2 Gñó

ôñó quadratic with respect to úñó

( effective energy momentum tensor )

cf. Shiromizu, Maeda, Sasaki (`00)

with

5D Einstein eq., Gauss eq. and Cadazzi eq.,

Israel’s junction condition on the brane

cosmological solutions 　case 1: è= 1 branch

(4)É = 0(inflationary solution is the attractor in D-G-P model even for )

H 2 + ka2 = î 2

eff3 ö+ 2

3ö0ñ2

ö0 ë ï + 6m65=ñ2

with

ö0 ò öcr

present cosmological acceleration+

confinement of the massless mode

effective Friedmann equation

with ë ë 6m65=(ñ2ö0)

1=ñ2 ! (1+ ë)=ñ2

(4)É = 0

öö0

effective gravitational constant changes

î 2eã =1

ñ2

case 2: è= Ä1

(5D Friedmann solution is the attractor in D-G-P model)

effective Friedmann equation

H 2 + ka2 = î 2

eff3 ö î 2

eã = 8ôGeã = 1ñ2

Ç1+ 2ë

1Äp

1+2ëö=ö0

É

4d-like cosmology

(4)É = 0

case 3: è= Ä1 (4)É > 0

effective cosmological constant(4)Éeã ' 6m6

5ï ñ2

(4)É ú (4)É ï ù m65=ñ2for

suppressed!! for cosmological constant problem Éeã =m2

pl ' 10Ä 120

É=m2pl '

Ä(TeV)=mpl

Å4' 10Ä 60

SUSY

mï =mpl ' 1015 ÇÄm5=mpl

Å3=2

1TeV < m5 < 108GeV

1010GeV < mï < mpl

　Further extension of the model 　 In addition to dark energy ( cosmological constant problem),

other interesting cosmological consequences by other curvature correction terms?

trace anomaly of matter field on brane

gñó < Tñó >= cF Ä aG + dr 2RF = CñóöõCñóöõ , G = RñóöõRñóöõ Ä 4RñóRñó + R2

a = 1360(4ô)2 (NS + 11NF + 62NV ) c = 1

120(4ô)2 (NS + 6NF + 12NV )

d = 1180(4ô)2 (NS + 6NF Ä 18NV )

（ free, massless, conformally invariant case ）

U(N) super Yang-Mills theory a = c = N 2

64ô2 (= k3); d = 0

adding the counter term d = ãN 2=16ô2(= k1)

In 4D theory, inflationary solution is obtained

4. Brane induced gravity with trace anomaly terms　 S.M., Maeda , Torii (2003)Model

S = Sbulk + Sbrane

Sbulk =R

d5xp

Ä (5)g[m35

2(5)R Ä (5)É]

Sbrane =R

d4xp

Äg[m35K

Ü +m2

pl

2 R + Lm Ä ï + L tr]

Effective energy momentum tensor on the brane

úñó = Äï gñó + Tñó Ä m2plGñó + H (1)

ñó + H (3)ñó

H (1)ñó = Äk1(2RRñó Ä 1

2gñóR2 Ä 2r ñr óR + 2gñór ã r ã R)

H (3)ñó = k3(ÄR õ

ñ Róõ + 23RRñó + 1

2gñóRõúRõú Ä 14gñóR2)

Basic equation 　　　(ö = 0;k = 0;E00 = 0; (4)É = 0)

deviation from 4D theory

°H +ê3H Ä _H

2H

ë_H + V0(H ) = 0

V0(H ) =m2

pl

12k1H Ä k3

12k1H 3 Ä ï

36k1H Ä èm35

6k1

q1+ ï 2

36m65H 2

with

Cosmological scenario

max[( m5mp l )

6; ïm4

pl] ò 10Ä 120

(m5 ò 10MeV)Cf. Dvali et al

è= 1 mpl ùq

23ö0=ñ

value of the parameters

k3 ò O(1) number of species

k1 ò 109k3 first inflation

GUT scale

second inflation

present

quintessential inflation driven by induced curvature terms

H

V(H )

mplp3k3

mplp6k1

firstinflation

q23ö0=ñ

secondinflation

creation

5. Summary and discussion

tensor structure becomes 4d massive (problematic)

Next, for the relevant gravitational property on brane, we consider generalized DGP model with bulk cosmological constant and brane tension like RS model.

Last, for the early stage of inflationary solution,we consider other curvature correction term, trace anomaly.

First, I review the cosmology based on DGP model.

alternative to dark energy

creation early inflation

natural explanation for zero vacuum energy

effective gravitational constant on brane changes

effective cosmological constant on brane suppressed