Horava-Lifshitz 重力理論とはなにか?
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Transcript of Horava-Lifshitz 重力理論とはなにか?
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
TH 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
aHa
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
HhM
/ 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
HPM
polarization
length scale
t
Wavelength of fluctuations
1H
Quantum fluctuations
12
ikea k
decaying modec
Sub-horizon
Super-horizon
ak
2'' 2 ' 0A A Aa ka
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
2AH
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 cm2410 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 22 2 2 2 2 2
1 1 1z z z
c kGk Gk Gk
22 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
12
ijij 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
32z
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 2 23
ijk m p n p mi p j k m i p jm k n wW d x g d x g R
w
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
14
ij ikm j jk m mC R R
Cotton tensor
Orlando & Reffert 2009Horava 2009
12 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
22
ij ijHG ij ijS dtd x gN K K K C C
w
A parity violating term is required for the theory to be renormalizable!
2 2 2 22 2
22
21 4 342 8 8 1 3
ijij w
ijk mim j k wR RR R R R
w
z=3 UV gravity
z=1 IR gravity
Inflation in Horava Gravity
3 22
1 1 ( )2 2
iM iS dtd x gN V
N
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
k p i pk
3
3,
( , )( )
2 2
ii
iij ik xA A
k ijA R L
h x d k e 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 ( ) 0A
Akk
d ydy
2 2 23 2 6 2 5 2 2 42 22 3
3 2 4 6 2 5 4 2,
12 8 8 32 32 1 32
A A Awk k
A R L
kd k k k kS dt aw a w a a a
A Ak kv a dtd
a
4 22
16(1 3 )w
22
1 3
w
H
2
2Hw
2 2 22
2( ) 1 (1 )Ay y yy
y k A Ak k
k v
2 0
0Aky
Chiral PGWs
2 2
2 2
| | | || | | |
R L
R L
C CC C
Adiabatic vacuum1 exp ( ') '
2 ( )i
y
y
i y dyy
0 2A
yA Ak
C D yy
2 22 23
3A Ak
Hk C
degree of circular polarization
<TB> correlation in CMB
r=0.1
Saito et al. 2007
0.61
0.350.05r
If parity symmetry is not violated
0TB TB
( 2)2
,
ˆ ˆ( )m mm
Q iU n a Y n
Stokes parameter 11 22 121 1,4 2
Q I I U I
(2) ( 2)12
Em m ma a a
(2) ( 2)12
Bm 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
SNR0.0810 5
GW
* *
* *
, ', ' , ', '
, ', ' , ', '
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
f I 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 kdk
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.