infrared divergences in the inflationary brane world
DESCRIPTION
Infrared divergences in the inflationary brane world. Oriol Pujol à s Yukawa Institute for Theoretical Physics , Kyoto University. gr-qc/0407085 to appear in JCAP. In collaboration with Takahiro Tanaka & Misao Sasaki. Trobades de Nadal 2004 Universitat de Barcelona, 21/12/04. - PowerPoint PPT PresentationTRANSCRIPT
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