6D Brane Cosmological Solutions
Masato Minamitsuji
(ASC, LMU, Munich)
T. Kobayashi & M. Minamitsuji, JCAP0707.016 (2007) [arXiv:0705.3500]
M. Minamitsuji, CQG 075019(2008) [arXiv:0801.3080]
CENTRA, Lisbon, June 2008
1
Contents
~ Introduction
~ 6D braneworld
~ 6D brane cosmological solutions
~ Tensor perturbations
~ Stability
2
Introduction
Braneworld
One of the most popular and mostly studied higherdimensional cosmological scenarios in the last decade
Matter (SM particles) are confined on the brane
while Gravity can propagate into the bulk
Brane
(SM)
bulk
(Gravity)
Motivated from string / M-theory
Gauge hierarchy problem, Inflation, Dark energy , …
3
5D braneworld
Randall-Sundrum (II) model (RS 1999)
2
2
ds  2 (dz 2  dt 2   ij dx i dx j )
z
Localization of gravity by strong warping
2

8  G4
   
2

H  4 
 O  
   
3


G4  G5 
6
5   2

z
1
4   2   2

  
Standard Cosmology
3-brane 
Vanishing cosmological constant  4  0 cannot be obtained
unless one fine-tunes the value of the brane tension.
4
6D braneworld
The property of a codimension 2 brane is quite different
from that of the codimension 1 brane . Codimension 2 brane
y
   M 64
Codimension 1
Codimension 2
~Conical singularity
x

The tension of the brane  is absorbed into the bulk deficit
angle and does not curve the brane geometry
Self-tuning of cosmological constant ?
5
Models with the compact bulk


Rugby-ball shaped bulk
The compact bulk is supported by the magnetic flux F

Self-tuning of the cosmological constant ?
Caroll & Guica (03), Navarro (03), Aghababaie, et.al (03)
H  H ( B,  6 )
We assume that for a given B0 H ( B0 ,  6 )  0
After the sudden phase transition on the brane  0  1, it seems to be
plausible that the brane keep the initial flat geometry.
however, because of the flux conservation
2  1 B0 R0

2   0 B1 R12
2
B1  B0
Vinet & Cline (04), Garriga & Poratti (03)
H ( B1 ,  6 )  0
6
Nevertheless, as a toy braneworld model with two essential features
Flux stabilized extra dimensions F
Higher codimensions
Stabilization of extra dimensions
In comactifying extra dimensions, d.o.f.s associated with
the shape and size appear in the 4D effective theory.
Flux stabilization of extra dimensions would be useful
6D model (2D bulk) gives the simplest example
C.f. in 5D
d
d is not fixed originally
additional mechanism
quantum corrections,…
7
Northern pole (+-brane)
generalization


Static warped solutions
Southern pole (--brane)
Mukohyama et.al (05) Aghababaie, et. al (03), Gibbons, Gueven and Pope (04)
We derive the cosmological version of these solutions
8
Branes in higher co-dimensional bulk
Codim-2 Brane tension develops the deficit angle
but one cannot put ordinary matter on the brane
Codim >2 = black holes or curvature singularities
One cannot put any kind of matter on the brane
need of regularizations of the brane
Cap region
4-brane
Codimension-1
Codimension-2
Peloso, Sorbo & Tasinato (06), Kobayashi & Minamitsuji (07)
4 
1
1

 

R  8 G4  T  g  T           2 
2
2

 

4D GR
 ~ R 2  8G4( I )T 
I
Scalar mode associated with the compact dimension
Large distances scales L2  R 2
1
~L
Recovery of 4D GR
9
6D brane cosmological solutions
Our purpose is to find brane cosmological solutions in
the following 6D Einstein-Maxwell-dilaton theory
1 
1
2

ab 
S   d x  g  R  ( )  2e    e Fab F 
4
2

0    1   0 pure Einstein-Maxwell model
  1 gauged supergravity
6 
6
Instead of solving coupled Einstein-Maxwell-dilaton system,
we start from (D+2)-dimensional Einstein-Maxwell theory
S
( D  2)
 d
D2
X
D4
1
1  
G 
R  2  
D2
2
4
( D  2)
FMN
( D  2)
F
MN



First, we consider seed solutions in higher dimensions
10
Northern pole (+-brane)


Southern pole (--brane)
11
For a seed (D+2)-dim solution, we consider the
dimensional reduction:
Compactified
Dimensional reduction
dsD2  2  e ( D 4)  x  2 g ab ( x)dx a dx b  e 2  x  mn dy m dy n
6D
with some field identifications
( D  2)
Fab  Fab ( x)

( D  4) dim
( D  2)
FmM  0
D( D  4)
D4
,  
2
D
The effective 6D theory is the same as the one we are interested in
S
6 


1 
1
2

ab 
  d x  g  R     2e  e Fab F 
4
2
12
6
(D+2)-dimensional seed solutions
2 (1 )
2
ds 2  
( D  2)
1
F 
2
2 2

2


d


  dX  dX  
  2 h( )d 2 
2  h( )


1 2
Q
2 
2 ( 2  2 ) (1 2 )
  D-dimensional Einstein space ( D) R [ ]   
h( ) has two positive root at   1,  0    1
We compactify (D-4) dimensions in

Magnetic charge


 
12

2
1 2

2 
2 (1 2 ) 3   1  


Q 

2
2
3  2
1   
 5  
1 2

1


3
2
5
2
5  2
(1   2 )(3   2 ) 1   1
  max   :

2
3  2
2(5   )
1 2
1
2
Upper bound
13
Northern pole (+-brane)
 1

Warped
generalization

Southern pole (--brane)
 
14
From the (D+2)-dimensional de Sitter brane solutions
 
D-dimensional de Sitter spacetime
6D cosmological solutions
2
2


b

d

ds 2   1 (d 2  a 2   ij dx i dx j ) 
(
  2 h( )d 2 )
2 h( )

Q
2
2
 t ,    ln( b( )) 
ln

F 
2
2 ( 2  2 ) (1 2 )

1 
2
2
2
a( )  
1/  2
b( )  
Power-law inflationary solutions since 0    1
15
From the Kasner-de Sitter solutions


  dX dX  dt  e  ij dx dx  e  mndy dy
e
3 A ( D  4 ) B
2
 D 1

 sinh 
(2 ) (t  t0 )
 D2

2A
i
e
A B
2B
m

 D 1 
 
 tanh 
(t  t0 ) 

 D2 2
 
The early time cosmology a( )   p 
3(2   2 )  2 6 1   2
p 
3(6  5 2 )
j
n

D 2
3( D  4 )
b    q
 2  6 1   2
q 
6  5 2
(0.33  p  0.48, 0.33  p  0.063, 0  q  0.22, 0  q  0.41)
generalizations of solutions found in KK cosmology
Late time cosmology
1/  2
Power-law a( )  
b( )  
late-time attractors
Maeda & Nishino (85)
solutions are always the
16
Tensor perturbations
Tensor perturbations in 6-dim dS solutions
= Tensor perturbations in (D+2)-dim dS solutions
g MN  g MN dX
2 (1 2 )

1 2
 dt
2
M
dX
e
2 Ht
N

ij
KK decomposition hij 
j
2 Ht
m
n
(TT )

(
t
)

(

)
e
 m m ij
m
 12 2 2
d  1 2 d
 h m    mm  0

d 
d

4

 hij dx dx  e  mn dy dy  (2 D)
i
 d2
d
k2
2
 2  ( D  1)  2 H t   m  m (t )  0
dt e
 dt

TT polarization tensor
17
4D observers on the brane measure the KK masses
M m2 ( )  e ( D4) Ht 2 m2      ~ a (3 2 ) 2(1 2 )
The critical mass
3
 

2
4(1   )
2
2
c
c
m
  c   m ~ a
( 3 2 )  2 2 (1 2 ) c2
Light KK modes may decay slowly
First few KK modes


 0.156
Dashed line= critical KK masses

 0.743

18
  0.33 ( D  4.5)
  0.96 ( D  54)
Red= The first KK mass
Dashed = The critical KK mass
For the increasing brane expansion rate, the first KK mass
tends to be lighter than the critical one.
But one must be careful for the stability of the solutions
19
Summary 1
The 6D brane cosmological solutions are derived via
the dimensional reduction from the higherdimensional de Sitter brane solutions
The 6D brane cosmological solutions are stable
against the tensor perturbations.
For the larger value of the brane expansion rate, the
first KK mass of tensor perturbations becomes
lighter than the critical one, below which the mode
does not decay during inflation
20
Stability
Minkowski branes
stable
Yoshiguchi, et. al (06), Sendouda, et.al (07)
Lee & Papazoglou (06), Burgess, et.al (06)
de Sitter branes
unstable for relatively higher expansion rates
Kinoishita, Sendouda & Mukohyama (07)
Stability of 6-dim dS solutions
= Stability of (D+2)-dim dS solutions
21
Scalar perturbations
 1
ds
2
D2
1  2
 1  2 2 ( w, x)   dx dx 
2(1   2 )  (3   2 )






 2
3 2
2
2
2




 1  21 ( w, x)   2 ( w, x) dw  1  2  1 ( w, x) 
 2 ( w, x)    cos wd 
2
1







(1)
a


2((1   2 )  2 )  
4
1


  21 
 2  
1, w 
2
2
2

2(1   )  (3   )  
1 
 tan w

KK decomposition
i    n ( x  )i ,n ( w)
n
22
The lowest mass eigenvalue is given by
2

(
3


) 
2
2

0  (1   )1 
2 2
 (1   )  
A tachyonic mode appears for the expansion rates

inst
(1   )




3 2
2 2
inst
max
1  2




2
An instability against the scalar perturbations appears in the
de Sitter brane solutions with relatively higher expansion rates.
23
Dynamical v.s. “thermodynamical” instabilities
 1
Kinoshita, et. al showed the equivalence of
dynamical and “thermodynamical” instabilities
See the next slide
in the 6D warped dS brane solutions with flux
compactified bulk
Dynamically unstable solutions
= Thermodynamically unstable solutions
The arguments can be extended to the cases of higher
dimensional dS brane solutions.
24
Thermodynamical relations
Area of de Sitter horizon A 
Magnetic flux
1
2
0
0
H  ( D 2)  ( D  2) 
1  
D  1 
2
  2  d  d F 
 ( D  2)
D 1
Q

2 ( D 1)
D2
2 ( D 1)
D2




2 ( D 1)


D

2
1  





Deficit angles (=brane tensions)    2 (1    )
 
 h, (  1)
  
2
2D
( D 2)
 
 ,  
h, (   )
2

1
(Q  ( D  1) H D A)
2
12
5 


3 
2
1 2

2 
2 (1 2 ) 3   1  


Q 

2
2
3  2
1   
 5  
1 2

1


2
2
25
2 (1 )
2
ds 2  
1
2
2 2

 1  d 2



  dX dX 
  2 h( )d 2 
2  h( )

2
  D-dimensional de Sitter ( D) R [ ]   
h( ) has two positive root at   1,  0    1
Upper bound
5  2
(1   )(3   ) 1  
  max   :

2
3  2
2(5   )
1 2
1
2
2
1 2
26
Intensive variables
~
~
A  A   ,      , ~    
2D


  Q 

D

2

 H 
  DQ   2D     

  H 



The (+)-brane point of view
“Thermodynamics”
~
d (  A) 

1
~
2d     Qd
D
( D  1) H

d ( ) 
1
2
d 
1
dS  (dE  pdV )
T
Somewhat similar to the BH therodynamics
27
“Thermodynamical stability” conditions
  
 0,

D 
~
 H 
~
  


 QH  D   0


Some Identities
1
  



D
 H ~   





H
~
  ( ,  ) 


  ( H D ,  ) ,


  (QH  D ) 
1

~    2 D
H
 

~ 1

  ( ,  )   D  Q 
  
D  Q    




  ( H D ,  )   H  H D   Q     H     H D  

H
H
H


 
The boundary between unstable and stable solutions is given by
the curve, which is determined by the breakdown of one-to-one
map from ( , H ) plane to conserved quantities ( , ~ ) .
~
 ( ,  ) 


 ( H D ,  )   0


28

D4
D
Special limits
1) 6D limit :   0
The curve is exactly boundary between dynamically stable
and unstable modes
Kinoshita, Sendouda & Mukohyama (07)
2) unwarped limit 
crit 2(1   2 )

max
3 2
1
crit  inst
The same thing happens in the higher dimensional geometry.
29
Cosmological evolutions
Cosmological evolutions from (D+2)-dimensional unstable de
Sitter brane solutions
 1
ds 2  dt 2  a 2 (t ) ij dxi dx j  c 2 t (dw2   2 cos 2 wd 2 )
Evolution of the radion mode
a c  dVeff (c)

c   ( D  1)   
a c
dc



2
(3   )  

2 1
1 
(1   2 ) 2


Veff (c) 

c  log( c)
2
3
4
16( 
1 )
2
2(1   )
2
The potential has one local maximum and one local minimum
30
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
31
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
32
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
33
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
34
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
35
effective potential
Two possibilities: toward a stable solution with a smaller radius
decompactification
Flux conservation relates the initial vacuum to final one.
(3   2 ) 2 1
2 2 1
2 4(1   )
4(1   2 ) 2 

 (3   2 ) 2 1  2 1
1  2 
36
inst
(1   )

2

3
2 2
 AdS
4(1   )

2 2

(3   )
2 2
inst  1  AdS   0 a new dS brane solution
2
The corresponding 6D solution is the stable accelerating, powerlaw cosmological solutions.
Inflation
AdS  1  max
Dark Energy Universe ?
2  0 an AdS brane solution
The corresponding 6D solution is the collapsing Universe.
37
Summary
6D brane cosmological solutions in a class of the
Einstein-Maxwell-dilaton theories are obtained via
dimensional reduction from the known solutions in
higher-dimensional Einstein-Maxwell theory.
Higher-dimensional dS brane solutions (and hence the
equivalent 6D solutions) are unstable against scalar
perturbations for higher expansion rates. This also has an
analogy with the ordinary thermodynamics.
The evolution from the unstable to the stable
cosmological solutions might be seen as the cosmic
evolution from the inflation to the current DE Universe.
38
Equivalent 6D point of view
inst  1  AdS
2  0
4D effective theory for the final stable vacuum
Seff

 ( 4) 1 
  d x  qˆ  R  qˆ      2 e 2
2

4
2 2 (1 2 ) 



characterizes the effective scalar potential
The cosmological evolution may be seen as the evolution
from the initial inflation to the current dark energy
dominated Universe.
39
Quantum corrections
Ghilencea, et.al (05), Elizalde, Minamitsuji & Naylor (07)
Stability
Minkowski branes
Einstein-Maxwell
stable Yoshiguchi, et. al (06),
Supergravity
Sendouda, et.al (07)
Lee & Papazoglou (06), Burgess, et.al (06)
marginally stable (with one flat direction)
de Sitter branes
Einstein-Maxwell
Kinoishita, Sendouda & Mukohyama (07)
dS brane solutions are unstable for relatively
higher expansion rates !
40
41