Horava-Lifshitz 重力理論とはなにか？

早田次郎　京都大学理学研究科

2009.6.5 大阪市立大学セミナー

T.Takahashi & J.Soda, arXiv:0904.0554 [hep-th], to appear in Phys.Rev.Lett.

Ref. Chiral Primordial Gravitational Waves from a Lifshitz Point

Quantumfluctuations

Hawking radiation

Exponentialred shift

Quantum fluctuationsBH

How to get to Planck scale?

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.

There are two well known the paths to reach the Planck scale.

inflation Exponentialred shift

GW

The universe is so transparent for GW!

3

1p

T

H M

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?

The reaction rate is much smaller than the expansion rate in the cosmic history.

Hence, PGW can carry the information of the Planck scale.

1/ 2 188 2.4 10 GeVp NM G

reaction rate

H expansion rate

T reaction rate

A brief review of Inflation

( ) H ta t e

222

1(

3 2)

1

p

H VM

3 '( ) 0H V

2 2 2 2 2 2 2 2( ) ( ) i jijds dt a t dx dy dz a d dx dx

aH

a

FRW universe

dynamics

deSitter universeslow roll

24 4 1

( )2 2pM

S d x gR d x g V general relativity

conformal time

We will consider a chaotic inflation.

All of the observations including CMB data are consistent with an inflationary scenario!

Ap

Hh

M

/ 2A p AM h

24

8p ij ij

ij ij

MS d x h h h h action for GW

2

2 2

2h

p

HP

M

polarization

length scale

t

Wavelength of fluctuations

1H

Quantum fluctuations

1

2ike

a k

decaying modec

Sub-horizon

Super-horizon

a

k

2'' 2 ' 0A A A

ak

a

PGW must exist if you assume inflation!

h h

2 2 2 2 2(1 ) (1 ) 2ds dt dz h dx h dy h dxdy

GW propagating in the z direction can be written in the TT gauge as

Bunch-Davis vacuum

2A

H

Gravitational waves in FRW background are equivalentto two scalar fields

,A with

Power spectrum

2 2 2( ) i jij ijds a d h dx dx , 0ij i

j ih h Tensor perturbation

Is general relativity reliable?

Planckian region

Length Scale

t

k

3310 cm

2810 cm

2410 cm 1Mpc

2110 cm 1kpc

16N

Initial conditions are set deep inside the horizon

271/ 10H cm

3410 cm

We are looking beyond the Planck scale!

horizon size

1H For GUT scale inflation

galaxy scale

We need quantum gravity!!

Quantum Gravity and Renormalizability

2n nN

n

G k

4 4 42 2 2 2 2 2 2 4

1 1 1 1 1 1 1N N N

N

G k G k G kk k k k k k k G k

2 4 22

1 1 11

N

N

k G k k kG

2 2 2 2k c k

UV divergence in general relativity can not be renormalized

Higher curvature improves the situation

That is why many people are studying string theory.

However, string theory is rather large framework and not yet mature to discuss cosmology.

2NG

but suffers from ghosts

A difficulty

Hence, it is worth seeking an alternative to string theory.

Horava’s idea

2 2 2 2

1zc k Gk

2 2

2 2 2 2 2 2

1 1 1z z z

c kGk Gk Gk

2

2 2 2 2 2 2 2 2 2

1 1 1zGkc k c k c k

In order to avoid ghosts, we can use spatial derivatives to kill UV divergence

The price we have to pay is that,in the UV limit, we lose Lorentz symmetry.

Horava 2009

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

foliation preserving diffeomorphism

( )t t t ( , )i i jx x x t

In order to get a renormalizable theory, we need the anisotropic scaling

x xb zt b t

22 2( ) ( , ) ( , ) ( , )k i i k j j kijds dt g x t dx N x t dt dx N t tN t x d

x 1 t z

0N

Horava 2009

0ijg 1iN z

1

2ij

ij i j j i

gK N N

N t

extrinsic curvature

ijK z

ADM form

Because of this, we do not have 4-d diffeomorphism invariance.

lapse shift3d metric

Horava gravity – kinetic term

3 22

2 ijK ijS dtd x gN K K K

Since the volume has dimension 3 3dtd x z

The kinetic term should be

3

2

z 3z

1 In the IR limit, we should have

0

In the case 1 , we have an extra scalar degree of freedom.

Coupling constants and run under the renormalization.

Horava gravity – potential term

3 32

1 22

3ijk m p n p m

i p j k m i p jm k n wW d x g d x g Rw

3 ij kmV ijkmS dtd x gNE G E

kmij

ij

W ggE

g

detailed balance condition

0w 1 2w

relevant deformationz = 3 UV gravity

This guarantees the renormalizability of the theory beyond power counting.

The power counting renormalizable action with relevant deformation reads

1

4ij ikm j j

k m mC R R

Cotton tensor

Orlando & Reffert 2009Horava 2009

1

2 1 3ijkm ik jm im jk ij kmG g g g g g g

Horava gravity

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 rescale0x ct

2

4 1 3wc

2c

2

32NG c

The speed of light and Newton constant are emergent quantities

comments

2

3 22 4

2

2ij ij

HG ij ijS dtd x gN K K K C Cw

A parity violating term is required for the theory to be renormalizable!

2 2 2 22 2

22

21 43

42 8 8 1 3ij

ij wijk m

im j k wR RR R R Rw

z=3 UV gravity

z=1 IR gravity

Inflation in Horava Gravity

3 22

1 1( )

2 2i

M iS dtd x gN VN

We consider a scalar field

In the slow roll phase, we can approximate it as

3MS dtd x gNV .V const

In this case, we have de Sitter solution

2 2 222 3

12 16wH V

2 2 2 2 2 2H tds dt e dx dy dz

Polarized Gravitational waves

2 2 2( ) i jij ijds a d h dx dx

s j A A Asr ij ri

kp i p

k

3

3,

( , )( )

2 2

ii

iij ik xA A

k ijA R L

h x d ke p

, 0ij ij ih h Tensor perturbation

Polarization state

Circular polarization 1, 1R L

polarization tensor

Left-handed circular polarization Right-handed circular polarization

Rh h ih

Lh h ih

Because of the parity violation, we need a different basis to diagonalize the action

Action for gravitational waves

22

2( ) 0

AAkk

dy

dy

2 2 23 2 6 2 5 2 2 4

2 22 33 2 4 6 2 5 4 2

,

1

2 8 8 32 32 1 32A A Awk k

A R L

kd k k k kS dt a

w a w a a a

A Ak kv a dt

da

4 2

2

16(1 3 )w

22

1 3

w

H

2

2Hw

2 2 22

2( ) 1 (1 )Ay y y

y

y k A Ak k

kv

2 0

0Aky

Chiral PGWs

2 2

2 2

| | | |

| | | |

R L

R L

C C

C C

Adiabatic vacuum1

exp ( ') '2 ( )

i

y

y

i y dyy

0 2A

yA Ak

CD y

y

2 22 23

3A Ak

Hk C

degree of circular polarization

<TB> correlation in CMB

r=0.1

Saito et al. 2007

0.61

0.350.05

r

If parity symmetry is not violated

0TB TB

( 2)2

,

ˆ ˆ( )m mm

Q iU n a Y n

Stokes parameter 11 22 12

1 1,

4 2Q I I U I

(2) ( 2)1

2Em m ma a a

(2) ( 2)1

2Bm m ma a a

i

intensity tensor ˆ( ) , , 1, 2ijI n i j

tensor harmonics

n̂ dirction on the sky

Direct detection of Chiral PGW

Seto 2007

1

15

SNR0.08

10 5GW

* *

* *

, ', ' , ', '

, ', ' , ', '

h f n h f n h f n h f n

h f n h f n h f n h f n

2 , , , ,1' '

, , , ,2

I f n Q f n U f n iV f nn n f f

U f n iV f n I f n Q f n

“Stokes” parameter

2 34

GWc

fI f

2 34c

GWV f f ff

With three detectors or two well designed detectors, we can measure V.

Cooray 2005

What can be expected for BH?

Quantumfluctuations

BH

Chiral Hawking radiation

r=0 r=2M

2 2dc k

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.