2009.6.5 大阪市立大学セミナー
Horava-Lifshitz重力理論とはなにか？
早田次郎
京都大学理学研究科
Ref. Chiral Primordial Gravitational Waves from a Lifshitz Point
T.Takahashi & J.Soda, arXiv:0904.0554 [hep-th],
to appear in Phys.Rev.Lett.
How to get to Planck scale?
There are two well known the paths to reach the Planck scale.
Hawking radiation
GW
Exponential
red shift
BH
Quantum
fluctuations
inflation Exponential
red shift
Quantum fluctuations
In reality, it would be difficult to observe Hawking radiation.
However, we may be able to observe primordial gravitational waves!
Hence, in this talk, I will mostly discuss an inflation.
The universe is so transparent for GW!
The reaction rate is much smaller than the expansion rate in the cosmic history.

H
 T

 Mp



3
1
reaction rate
T
H expansion rate
M p   8 GN 
reaction rate
1/ 2
 2.4  1018 GeV
Hence, PGW can carry the information of the Planck scale.
Namely, one can see the very early universe!
Indeed, we can indirectly observe PGW through CMB
or directly observe PGW by LISA or DECIGO.
What kind of smoking gun of the Planck scale can be expected?
A brief review of Inflation
M p2
 1 

4
4
d
x

g
R

d
x

g






V
(

)


 2

2 
general relativity
S
FRW universe
ds 2  dt 2  a 2 (t )  dx 2  dy 2  dz 2   a 2 ( )  d 2   ij dx i dx j 
conformal time
dynamics
H2 
1
3M p2
1 2

 2   V ( ) 
We will consider a chaotic inflation.
slow roll
deSitter universe
a(t )  eH t
All of the observations including CMB data
are consistent with an inflationary scenario!
  3H  V '( )  0
H
a
a
PGW must exist if you assume inflation!
ds 2  a 2 ( )  d 2   ij  hij  dxi dx j 
Tensor perturbation
S
action for GW
M p2
8
d
4
hij , j  hi i  0
x hij hij  hij hij 
h
GW propagating in the z direction can be written in the TT gauge as
h
polarization ds  dt  dz  (1  h )dx  (1  h )dy  2hdxdy
2
2
2
2
2
Gravitational waves in FRW background are equivalent
to two scalar fields
 A  M p hA / 2 A  , 
with
 A '' 2
a
 A ' k 2 A  0
a
Bunch-Davis vacuum
Super-horizon
H
A 
2
hA 
Power spectrum
Wavelength of
fluctuations
length scale
  c  decaying mode
a
k
H
Mp
2 H2
Ph  2 2
 Mp
H 1
Sub-horizon

1
eik
a 2k
Quantum fluctuations
t
Is general relativity reliable?
Length Scale
horizon size
1028 cm
1024 cm 1Mpc
k
1021 cm 1kpc
galaxy scale
Initial conditions are set
deep inside the horizon
N  16
For GUT scale inflation
1/ H
H 1
1027 cm
1033 cm
1034 cm
Planckian region
t
We are looking beyond the Planck scale!
We need quantum gravity!!
Quantum Gravity and Renormalizability
A difficulty
UV divergence in general relativity can not be renormalized
GN   2
G
n
N
k 2   2  c2 k 2
k 2n
n
Higher curvature improves the situation
1 1
1
4 1
4 1
4 1
 2 GN k 2  2 GN k 2 GN k 2 
2
k
k
k
k
k
k
1
 2
k  GN k 4
but suffers from ghosts
1
1
1


k 2  GN k 4 k 2 k 2  1
GN
That is why many people are studying string theory.
However, string theory is rather large framework and not yet mature to discuss cosmology.
Hence, it is worth seeking an alternative to string theory.
Horava’s idea
Horava 2009
In order to avoid ghosts, we can use spatial derivatives to kill UV divergence
1
1
1
1
2z


Gk

2
2 2
 2  c 2 k 2  Gk 2 z  2  c 2 k 2  2  c 2 k 2
 c k
The price we have to pay is that,
in the UV limit, we lose Lorentz symmetry.

1
1
1
2 2

c
k

2
2z
2
2z
2
2z
  Gk
  Gk
  Gk
Is the symmetry breakdown at UV strange?
No! We know lattice theory as such.
In fact, Horava found a similarity between his theory
and causal dynamical triangulation theory.
Lifshitz-like anisotropic scaling
Horava 2009
In order to get a renormalizable theory, we need the anisotropic scaling
t  bzt
x bx
t   z
 x  1
Because of this, we do not have 4-d diffeomorphism invariance.
foliation preserving diffeomorphism
t  t (t )
xi  xi ( x j , t )
ADM form
lapse
3d metric
shift
ds 2   N 2 (t )dt 2  gij ( x k , t )  dx i  N i ( x k , t )dt  dx j  N j ( x k , t ) dt 
N  0
extrinsic curvature
Kij 
 g ij   0
 N i   z  1

1   gij


N


N

i
j
j i
2 N  t

 K ij   z
Horava gravity – kinetic term
The kinetic term should be
SK 
2
2
3
ij
2
dtd
x
g
N
K
K


K


ij

Since the volume has dimension  dtd 3 x    z  3
   0
  
Coupling constants
In the case
 1
z 3
2

and
z3

run under the renormalization.
, we have an extra scalar degree of freedom.
In the IR limit, we should have
 1
Horava gravity – potential term
detailed balance condition
Horava 2009
SV   dtd x g N E Gijkm E
3
ij
Orlando & Reffert 2009
km
W  gkm 
 gij
1

  gik g jm  gim g jk  
gij g km
2
1  3
gE ij 
Gijkm
This guarantees the renormalizability of the theory beyond power counting.
The power counting renormalizable action with relevant deformation reads
W
1
2 n p m 
3
ijk  m
p
3
d
x
g










d
x g  R  2 w 
ip
jm
kn 
 i p j km
2 

w
3


z = 3 UV gravity
 w  0
   1
Cotton tensor
 w   2
1


C ij   ikmk  R j m   mj R 
4


relevant deformation
Horava gravity
S HG
2
 2 ij
ij
2
  dtd x g N  2  K Kij   K   4 C Cij
2w

3
z=3 UV gravity

 2 2
2 w2
 Rim j R
ijk

 2  2 1  4 2
2 

R Rij 
R


R

3


w
w 
8
8 1  3   4

 2 2
m
k
ij
z=1 IR gravity
comments
A parity violating term is required for the theory to be renormalizable!
We have a negative cosmological constant which must be compensated
by the energy density of the matter.
Cosmological constant problem!
To recover the general relativity, we need rescale
x 0  ct
The speed of light and Newton constant are emergent quantities
c
 2
4
w
1  3
2
GN 
32 c
c  2
Inflation in Horava Gravity
We consider a scalar field
 1 2 1 i

SM   dtd x g N  2      i  V ( ) 
2
 2N

3
In the slow roll phase, we can approximate it as
S M    dtd 3 x g NV
V  const.
In this case, we have de Sitter solution
ds  dt  e
2
2
2H t
 dx
2
 dy  dz
2
2

3 2  2  2w 
H  V 

12 
16

2
2 
Polarized Gravitational waves
Tensor perturbation
ds 2  a 2 ( )  d 2   ij  hij  dxi dx j 
hij , j  hi i  0
Because of the parity violation, we need a different basis to diagonalize the action
Polarization state
hij ( , xi )
Circular polarization
2

d 3k
  2
 
A R , L
 kA ( ) eik x pijA
3
ks s j A
 r pij  i  A priA
k
Left-handed circular polarization
i
i
polarization tensor
 R  1,
 L  1
Right-handed circular polarization
hR  h  ih
hL  h  ih
Action for gravitational waves
2
5
 1
  2k 6
 2  2  w k 2  A 2 
 2  2k 4

A 2
A  k
 S    dt
a  2 k   4 6  


k 
3
2 5
4
2
2

8
w
a
8
w
a
32
a
32
1

3

a

 
A R , L

 2  


d 3k
2
v  a
A
k
A
k
3
dt
d 
a
 4  2w
 
16(1  3 )
2
   2
y   k
 kA 
k A
v
 k
d 2 kA
2
A


(
y
)

k 0
2
dy
  H2
1  3
 w 2
 H
      0
 y    kA   0
 2 ( y)  1   y 2 (1   A y) 2 
2
y2
2
w2 
Chiral PGWs
Adiabatic vacuum

A
k
 y

 
exp  i   ( y ')dy '
2 ( y )
 yi

CA


 D A y2
y
y 0
1
k 
3
degree of circular polarization
| C R |2  | C L |2
 R 2
| C |  | C L |2
A 2
k
 2H 2 A 2

C
3

<TB> correlation in CMB
Iij (nˆ), i, j  1, 2
intensity tensor
Stokes parameter
nˆ
Q
1
1
 I11  I 22  , U  I12
4
2
dirction on the sky
Q  iU  nˆ    a(m2) 2Y m (nˆ)
,m
a Em 
tensor harmonics
1 (2)
a m  a (m2) 

2
a Bm  
1 (2)
a m  a (m2) 

2i
If parity symmetry is not violated
TB   TB  0
r=0.1
 r 
  0.35 

 0.05 
Saito et al. 2007
0.61
Direct detection of Chiral PGW
“Stokes” parameter
 h  f , n  h*  f ', n '

 h  f , n  h*  f ', n '

GW 
4 2 f 3
c
h  f , n  h*  f ', n '  1
 I f , n   Q  f , n  U  f , n   iV  f , n  
   2  n  n '   f  f '  

U
f
,
n

iV
f
,
n
I
f
,
n

Q
f
,
n
2
h  f , n  h*  f ', n ' 










Vf 
I f 
c
GW  f    f 
4 2 f 3
With three detectors or two well designed detectors, we can measure V.
1
    SNR 
  0.08  GW

15  
 10   5 
Seto 2007
Cooray 2005
What can be expected for BH?
Chiral Hawking radiation
c2 
Quantum
fluctuations
BH
r=0
r=2M
d
 k2
dk
Conclusion
• We have looked beyond the Planck scale via Horava
gravity and found that
the spacetime is chiral,
which can be tested by observing a circular polarization
of primordial gravitational waves.
This is a robust prediction of Horava gravity!
The renormalizability yields parity violation,
which is reminiscent of CKM parity violation.

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