曲がる余剰次元と宇宙項

曲がる余剰次元と宇宙項
- braneworld cosmology -
G  8 GT ?
YITP, Kyoto U
Misao Sasaki
Plan of this talk
1. Introduction
--- relativist/cosmologist’s view of braneworld
--- Randall-Sundrum’s (single) braneworld
2. Brane cosmology from the bulk point of view
3. Alternative models
--- DGP model, Bulk inflaton model, …
4. Einstein Gauss-Bonnet braneworld
5. Summary
1. Introduction



Braneworld  domain wall with matter fields
3-brane = 4D time like (singular) hypersurface in
5D spacetime (bulk).
Brane tension (s)
 vacuum energy 0
→ self-gravitating
domain wall
Causality on the brane
 causality in the bulk
Ishihara ‘01
A new picture of the Universe!
Two different views on gravity in the bulk
string (~particle) picture
geometrical (~wave) picture
emission of closed loops
spacetime fluctuations
This talk is based on a view
from this standpoint
Need to UNIFY the two pictures!
One fact worth keeping in mind ….
In General Relativity or any theory with
1/rn-2 force law in n-dim spacetime,
embedding a self-gravitating brane with co-dim > 2
ie, object with T~ dp(x) where p>2 in n-dim
will generically form a black hole (brane)!
Example: n=4
thin (singular) limit
• domain wall (p=1) →
OK
• string
(p=2) → marginally OK
• monopole (p=3) →
BH!
need to consider a thick brane for co-dim>2
(future issue)

Randall-Sundrum’s (single-) braneworld
RS ‘99
··· 5D Einstein theory with 5<0
1
S  2  d 5 x g  R  25 )   d 4 x q s
4
25 M5
ds 2  dy 2  e
2 y
5D Anti-de Sitter
(AdS5)
  dx  dx for  52s  6 5
• Warp factor b(y)e-|y|/ decreases exponentially
as |y|→.
b y)  e
y
• Extra dim is
effectively compact
with size ~
• gravity is confined
within |y| 
Z2-symmetry
;
2
6
5

4 on the brane  vacuum energy s

s
1
 4  5 
2
12

2
5
)   0 for RS flat brane )
2
On scales > , gravity reduces to 4D Einstein.
On scales < , 5D Einstein gravity is recovered:
G5 M
G5 M
G4 M
 Newton   2  

r r
r
r
for a point mass M on the brane.

8 G4   42 
 52
Randall & Sundrum ’99
Garriga & Tanaka ’99
Only gravity propagates in the bulk
Experimental bound:  <0.1 mm (‘large’ extra-dimension)
cosmology of the early universe is modified significantly
( <0.1 mm  t <10-13 sec)
2. Brane cosmology from the bulk point of view
Binetruy et al. (’99), Krauss (’99), Ida (’99), ...
5D AdS-Schwarzschild as a bulk
2
dR
ds 2   A  R ) dT 2 
 R2d2K ) ;
A R)
T
n

A R)  K 
R2


;
R
K  1, 0,   2G5 M  BH mass
2
2
Induced metric on the brane
2
dsbrane
"bulk"
2

R
 )  2
2
   A  R ) T  ) 
 d

A R) 

 R2 d2K )
 d 2  R2d2K )
R
Junction condition
A  R)
R
T
52
6
   s RS ) 
2
52
6
1
  ; s RS 
2





K
R
K
2
4
H  2    2 
 1 
R R R
3 
12
2
4
6

2
5
  RS flat brane's value

2
 4
 R
 42  8 G4 
 52
• Cosmological evolution deviates from 4D at H  1
• /R4 term due to BH in AdS bulk (5D Weyl tensor ‘E’)
~ ‘radiation’ on the brane
Shiromizu, Maeda & MS ‘99
dark radiation / Weyl fluid ~ conformal field
AdS/CFT
3. Alternative models
RS braneworld  5D Einstein gravity
•
Simplest non-trivial realization of braneworlds.
• There are many other possibilities.
• RS two-brane model → existence of ‘radion’ 
= distance between 2 branes
radion is tachyonic
in cosmology!
m2~ -4H2
Gen & MS (’01)
Frolov & Kofman (’03)
may have interesting
implications
e.g., Kanno, Soda & Wands (’05)
• DGP braneworld
Dvali, Gabadadze & Porrati (’00)
1
1
 5)
5
S  2  d x g R  2  d 4 x q
25 M5
2 4 4
 4)
R
induced gravity on the brane
5D gravity on large scales: r > r0
4D gravity on small scales: r < r0
52
r0  2
4
IR modification of gravity: our universe will NOT
be described by 4D Friedmann if r0~H0-1
2
 
1
1
4
H 
 2    2
 3
r0 r0 
r0


2
2
4
‘Dark energy’ due to 5D gravity, leading to H=2/r0
• 5D theory may not be pure Einstein:
scalar fields f (moduli, dilaton, …) ··· bulk inflaton
model
Kobayashi & Soda, Himemoto & MS, Maeda Wands (’00),…
•
 4,eff
1
 5,eff 
2
 s )
2
5
12
2
  s 
 

16  f 
2
5
2
5,eff determined by
5D dynamics of f
H2=4,eff/3 ··· Hubble parameter on the brane
1/2=-5,eff/6 ··· Effective AdS curvature in the bulk
Low energy (H  1) effective theory on the brane
= 4D Einstein-scalar system
Minamitsuji, Himemoto & MS, Kanno & Soda (’03)
Brane can inflate without inflaton on the brane,
while the bulk is collapsing!
Signatures of 5D bulk appear only if H  1
• bulk degrees of freedom = Kaluza-Klein (KK) modes

  y,x  )  0  x  )   dp f p  y ) p  x  )

‘zero’ mode
 2 9 2
m   p  H
KK modes
4

~ massive fields on the brane
2
KK modes are non-negligible when H  1
• important difference from KK cosmology:
KK effects on the brane  KK modes in KK cosmology
(classical) KK modes act as
negative energy dust on the brane!
Minamitsuji, MS
& Langlois ‘05
• 5D action with higher derivatives
Gauss-Bonnet term ··· ghost-free; inspired by string theory
LGB  R2  4Rab Rab  Rabcd Rabcd
e.g., Mavromatos &
Papantonopoulos ‘05
Einstein Gauss-Bonnet (EGB) gravity:
1
S  2  d 5 x  g  R  25   LGB 
2 5 M5
•
•
GB term is topological in 4D. EGB is non-trivial only for D5.
Natural extension of Einstein theory in the sense that
no derivatives higher than second appear in the field eqns.
Lovelock ’71
•
A brane can be consistently embedded in the EGB bulk
despite the presence of R2 terms.
Deruelle & Delezel ‘00, Charmousis & Dufaux ‘02
Davis ’02, Gravanis & Willison ’02, ……..
4. Einstein Gauss-Bonnet (EGB) braneworld
1
S  2  d 5 x g  R  25   LGB    d 4 x q Lbrane
4
2 5 M5
• EGB also admits AdS bulk, but with curvature radius given by
4 5 
4 5 
1
1 
1
1 

1- 1 
 or 2 
1 + 1 

2
4 
3 
4 
3 
4
convenient parameter   2 (We assume >0 & 5<0)
0    1 for -, 1     for +
‘-’ branch reduces to AdS in Einstein in the limit →0:
1/2 → 1/02 ≡-5/6
‘+’ branch is known to be unstable
(but may have interesting cosmological implications)
• (Z2-symmetric) de Sitter brane in AdS bulk

ds 2  dy2  b2  y ) dt 2  H 2cosh 2  Ht ) d23)
b y)  H sinh  y /
tension:
s  s RS 1  H
2 2
)
 
2
1

1

2
H
 3


)
de Sitter with radius H-1
2
)
  
1  2 


-1/2
; s RS 
6
 52
t =const. hypersurface in 5D
(two identical H4 cut and pasted at radius y=y0 , b(y0)=1)
0
• Flat
(Minkowski) brane if s is tuned:
ds 2  dy 2  e
2 y
  dx  dx
    
for s  s RS 1  1  
 3  2 
1 / 2
• Gravity
on the brane reduces to 4D Einstein on
scales > , as in the case of RS braneworld.
(because the extra-dimension is essentially compact)
However, ……..
Gravity never becomes 5D even at short distances!
Deruelle & MS ‘03
 0

4
2
  0.3
 Newton  
Geff  x ) M
r
; x
r
scalar-tensor
(~Brans-Dicke)
theory at r < 
Davis ’04
  0.6
  0.9
Experimental bound on  is relaxed substantially:
1km ~
<  ~< 100km is also possible if  ~ 1
Thus, 5D EGB leads to intrigueing possibilities that
• Size (curvature radius) of the extra-dim may be
macroscopic.
 ~10 km is an interesting scale for SN/NS/GW physics
• Scalar-tensor gravity on the brane at short distances,
while Einstein (tensor) gravity at long distances.
Emergence of an (effectively massive) 4D scalar field
Impact on the early universe cosmology?
 ~10 km  t ~10-4 sec
Linearized gravity on de Sitter brane
Minamitsuji & MS ‘04
2


T
 □4  4H 2 )   5  ;    2H 2 2  1)
6 1 
 d G  h   3H 2 h 
1
 52
2  H
2 2
1
T

fluctuation of
brane position
 1 
)
pure Einstein?
 D D  □4  3H 2    

 H 2 2 1 
1
1
negligible at r  

 y h  y0 )
5D (KK) effect
2 2
→
BD gravity
2 H  1


1
 y h  y0 ) 
 52



dpGp  y0 ; ) T
p ,2 )
‘KK propagator’
Integro-differential equations
spin 2 part of matter
stress tensor
5. Summary

Braneworld gives a new picture of the universe

Different models can lead to a variety of predictions
difference can appear at short or long distances
In
DGP braneworld, 4D at r  r0 , 5D at r  r0
Infinitely large extra dimension
“dark energy” without dark energy
In
EGB braneworld, BD at r   , Einstein at r  
KK modes summed up to give 4D-like behavior
Extra dim can be as large as  ~ 10km
Effect of KK modes (5D gravity)
in the braneworlds is highly non-trivial:
brane geometry

effective 4D geometry in conventional
KK cosmology
KK modes lead to
• cosmological constant in DGP
• massive scalar in EGB
• negative energy dust in RS
• ….
There are surely much more to be done
in braneworld cosmology.
string-inspired models, cosmological perturbations,
‘thick’ braneworld, higher co-dimensions,
black holes, quantum effects, ….
and
observational predictions!
any hint of deviation from Einstein?
Models with instabilities should not be rejected at once.
After all, instabilities are needed in cosmology!

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