Infrared divergences in the inflationary brane world

Oriol PujolàsYukawa Institute for Theoretical Physics, Kyoto University

In collaboration withTakahiro Tanaka& Misao Sasaki

gr-qc/0407085to appear in JCAP

Trobades de Nadal 2004 Universitat de Barcelona, 21/12/04

??T

Motivation

2 = ? ?T

in BW cosmology

Motivation

2 = ? ?T

How do IR divergences look like in the BW??

Is the backreaction from quantum effects important?

Bulk inflaton model: Bulk scalar with light mode drives inflation on the brane

in BW cosmology

Describe bulk inflaton: modific of RS to include period of infl: inflaton in brane or in bulkIt’s well known that in dS the BD vac suffers from IR divergencesDo the kk modes modify the fluctuations?

• IR divergences in de Sitter

• IR divergences in de Sitter Brane World

• Application: Bulk inflaton model

• Conclusions

PLAN

IR divergence in de Sitter

Light scalars in de Sitter in Bunch Davies vacuum

4

22

effBD

H

m

2 2effm m R

26

effBD

H gm

T

Broadening of the homogeneous mode

for 0effm

2 2 2 23( )ds dt a t dS Massless scalar in de Sitter

3 32

( 2)0 0t ta a

a

e.o.m.

0(1) (1)(0) C (2) (2) 3(0) C a dt

2 2 2 23( )ds dt a t dS Massless scalar in de Sitter

e.o.m.

0dS invariant

dS

(1) (1)(0) C (2) (2) 3(0) C a dt

3 32

( 2)0 0t ta a

a

2 2 2 23( )ds dt a t dS Massless scalar in de Sitter

e.o.m.

0dS invariant

dS

But KG norm* * 3

(0) (0) (0) (0) ia (1)C

dS invariant vacuum2

(1) (1)(0) C (2) (2) 3(0) C a dt

3 32

( 2)0 0t ta a

a

22

32

2

1 ( 0)

AF

dt

a

Allen Follaci vacuum

is a free parameter0• breaks dS inv.

Allen Follaci vacuum

• breaks dS inv.

22 221

tanh ( 0)AF

t

(in 3 dimensions)

Allen Follaci vacuum

Vilenkin Ford ’82Linde ’82

• breaks dS inv.

22 2 321

tanh AF

t H t

(in 3 dimensions)

22 2 321

tanh AF

t H t

Allen Follaci vacuum

Vilenkin Ford ’82Linde ’82

• breaks dS inv.

• is finiteAF

T

4 2 4 2 2t tvAF AF

T H g H H g

21

2S Massless minimally coupled

Special case:

21

2S Massless minimally coupled

Garriga Kirsten vacuum0

0 lim 0GK AF

Special case:

21

2S Massless minimally coupled

Garriga Kirsten vacuum0

0 lim 0GK AF

is finite and dS-invariantGK

T

Special case:

22 1

AF

but

const Shift symmetry 22T g

why?

2 x y x y

In summary, in de Sitter space:

large and

(massless minimal coupling)

some regular dS invariant vacuum exists

(effectively massive but not minimal c.)

but is regular

2 3

AFH t

AFT

effm H 2 T

0 , 0effm

0 , 0effm

does it mean that in the brane worldthere arelight cone divergences?…

2 3H t but … if ,

even in the massive case, the wave function of the bound state diverges on the light cone … ??

IR divergences in the Brane World

0bsm

0bsm MinimalNon-minimal

Model:

one de Sitter brane in a flat bulk n+2 dimensions

(Vilenkin-Ipser-Sikivie ’83)

Model:

one de Sitter brane in a flat bulk n+2 dimensions

(Vilenkin-Ipser-Sikivie ’83)

2 2 2 2

1nds dr r dS

De Sitter

in Rindler coords:

Model:

one de Sitter brane in a flat bulk n+2 dimensions

(Vilenkin-Ipser-Sikivie ’83)

2 2 2 2

1nds dr r dS

De Sitter

in Rindler coords:

0r ‘light cone’

Generic scalar field

2 22 2

12

2

eff

MS R K

bulk brane

Flat bulk2 20( )eff effM M r r

Spectrum

Continuum of KK modes

m

2

nH One bound state, with mass

/ 2KKm n H

22

2bs effn

Hm n Mn

( )

/ 2

( )( (, ' ( ')) ) ( ') dS KK Kbs b K Ss d

nH

U r UG x x G dm GU r U r r

/ 2( )

( )nbs N

I MrU

rr 0 / 2n

( )

/ 2

( ) ( ')KK KKdS

m

nH

U rdm Gr U

22 2 / 2p m n

( )dSmG

( ) ( ')KK KKU r U r

0bsm For ,

the KK contribution

Exactly massless bound state 0bsm

Exactly massless bound state 0bsm

0

02 t

AF vacuum

A) Bound state:

A) Bound state:

0

0

B) KK modes:

simple poles: regular

double pole:

2 t

2 log r

Exactly massless bound state

AF vacuum

0bsm

A) Bound state:

0

0

B) KK modes:

simple poles: regular

double pole:

2 t

2 log r

Exactly massless bound state

AF vacuum

light cone div.

light cone div.

0bsm

0 02 2 2 2 2 double simple

bs bs KK KK

2 ( ) logbsU r t r

Regular on the light cone

0 02 2 2 2 2 double simple

bs bs KK KK

=In fact,

Regular on the light conebut its derivatives are NOT

(4 dim)

0 02 2 2 2 2 double simple

bs bs KK KK

=In fact,

Regular on the light conebut its derivatives are NOT

diverges on the LC in 4 and 6 dimensions if

T

0

0 02 2 2 2 2 double simple

bs bs KK KK

22

2 2( ) t h1

anbs tU r

sinh

cosh

T r t

R r t

0 02 2 2 2 2 double simple

bs bs KK KK

22

2

22 ( )

1bs

Tr

RU

Divergence at !!0R

sinh

cosh

T r t

R r t

Continuation of decaying mode grows!!tanh cotht

(even with )

0 02 2 2 2 2 double simple

bs bs KK KK

22

2

22 ( )

1bs

Tr

RU

Divergence at !!0R

0

2

2nT

R

Massless minimally coupledSpecial case:

const

is finite and dS-invariantGK

T

again, because of the shift symmetry

( )bs r is constant

2

22 constbs

AF

so, again

Note:

0M

Garriga Kirsten vacuum ?? 00 lim 0GK AF

Massless minimally coupledSpecial case:0M

2 x y x y ( , ) ( , ) 2 ( , )D x x D y y D x y

( , )D x y

Application: bulk inflaton model

a bulk scalar field in ‘almost’-Randall-Sundrum II model has a light bound state in the spectrum, and a potential that drives inflation

bs

bulk

brane

Scales:

bound state dominates

higher dimensional effects are important

Bound state dominates for

??

Bulk inflaton model

, , bsH m

1H

bsm H

1H

Backreaction?

Light bound state bsm H

2

2 24

( )bs

bs

U rm

H

2

2

( ) ( ')bs bs

bs

D U r U rm

HT

D

Light bound state bsm H

Regular on the light cone (thanks to the KK modes)

(in the bulk)

2

2 24

( )bs

bs

U rm

H

2

2

( ) ( ')bs bs

bs

D U r U rm

HT

D

Bound state wave functioncorresponding to 0bsm

// 2

2 ( )( )s n

nb NU

I Mr

rr

Regular on the light cone (thanks to the KK modes)

Light bound state bsm H

2

22

2 max , ,bs

T Hm

HHM

Light bound state bsm H

( , , ) bsm M H two possibilities for

cancellation (fine tuning)

, , 1M

H H

No fine tuning No large backreaction

on the brane:2

2 TOT

2 BS

2" "BS

no bound statebound state

22

232bsm 2

22 3" " 2m

3for 0 and 16M

Conclusions

0bsm 0bsm (and either or )0M

• The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone

• but it can not avoid an IR divergence within the bulk

• is it possible to avoid this divergence by modifying vacua of KK modes??

0bsm (and either or )0M

Conclusions

• The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone

• but it can not avoid an IR divergence within the bulk

• is it possible to avoid this divergence by modifying vacua of KK modes??

0bsm (and either or )0M

Conclusions

• a regular and dS inv vacuum exists

0M 0bsm

• The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone

• but it can not avoid an IR divergence within the bulk

• is it possible to avoid this divergence by modifying vacua of KK modes??

0bsm (and either or )0M 0bsm

• when the lowest lying mode is light, the dS-invariant vacuum can generate a large if mbs

fine tuned

• no fine tuning of mbs

no large backreaction in the bulk inflaton model

• perturbations on the brane dominated by b.s. if

• can be mimicked by a massive mode ?

Conclusions

T

bsm H2 KK

• a regular and dS inv vacuum exists

0M 0bsm