1

Circular Polarization of

Gravitational Waves

in String Cosmology

MIAMI, 200７ .12.14

Jiro SodaKyoto University

work with Masaki Satoh & Sugumi Kanno arXiv:0706.3585

2

Polarization of Gravitational Waves

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

h h

41

64ij ij

ij ijS d x h h h hG

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

Action for GW

Any linear combination of these polarization can be a basis of GW.

3

Circular polarization of GWLeft-handed circular polarization Right-handed circular polarization

Rh h ih Lh h ih

Without a parity violating process, the circular polarization of primordial GW does not exist.

4

Motivation of our work

In the effective action of superstring theory, gravitational Chern-Simons term,

which violates the parity invariance, often appears.Hence, it may produce Circular polarization of primordial GW

Slow roll inflation does not produce circular polarization

Gauss-Bonnet term also appears in superstring theory

Known result S.Alexander & J.Martin, Phys.Rev.D71, 063526 (2005)

Our observation

We should study the primordial GW in the context of Gauss-Bonnet-Chern-Simons gravity.

5

Summary of our result4 41 1

( )2 2

S d x g R d x g V

4 2 41 1( ) ( )

16 16GBd x g R d x g R R

2 24GBR R R R R R 1

2R R R R

8 1G

This term is not relevant tobackground dynamics,but could produce the circular polarization of gravitational waves

Inflaton drives the slow-roll inflation

This term induces the super-inflation,and the instabilityof gravitational waves

These effects produce 100 % circular polarization of GW.

Moreover, the amplitude is also enhanced by the factor .310

Hence, the effect is detectable by DECIGO/BBO or even by LISA.

6

Outline of my talk

Inflation in Gauss-Bonnet-Chern-Simons Gravity

A mechanism to produce circular polarization

Two field inflation & detectability

Conclusion

7

Inflation in Gauss-Bonnet -Chern-Simons Gravity

8

2 2 2( ) i jijds a d dx dx

2 2 2 2 22

1 1 13 1 ' '

2 2 2H H m a

a

2 2 2,2

3'' 2 ' ' 0

2H H H m a

a

Cosmological background space-time

Homogeneous and isotropic universe

Friedman equation

Scalar field equation

H

a '

a

4 For concreteness, we take a simple model

'd

d

The equations can be cast into the autonomous system

There exists a region 　　　　　　　　　 where super-inflation occurs.

9

Numerical Result

Slow roll regime

Super-inflation regime

GB term drives the super-inflation.It indicates the violation of weak energy condition.

10

5/ 615 1/ 6

a 1

6H

0

3 2 23 'H a

Analytic solution in Super-inflation regime

2,2

3'' 2 ' ' 0

2H H H

a

In the super-inflationary regime, the system can be well described by Gauss-Bonnet dominant equations

0H expandingdecreasing

It is not difficult to obtain an analytic solution

What can we expect for the gravitational waves in this background?

11

A mechanism to produce circular polarization

12

Gravitational waves in GB-CS gravity

d 2kA

d2 1

H '

zA2

''

2zA2

k 2

zA

zA

kA 0

2 2

' '( ) 1

2 2A

A

Hz a k

a a

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

sj 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

A A Ak k kz

, 0ij ij ih h Tensor perturbation

Polarization state

Circular polarization 1, 1R L

With the transformation , we get

GB CS

polarization tensor

Right-handed and left-handed waves obey different equations!

13

GW in Super inflationary regime

zA2 1/3

5 152

18 4 /3

1 6A k 125 4 /3 A k

H ' 5 152

9 4 /3 '' 70 25 4 /3

d 2kA

d2 1

H '

zA2

''

2zA2

k 2

zA

zA

kA 0

d 2kA

d2 k 2 1 A 8

3

1

k

kA 0

In super-inflationary regime 1

Both GB and CS contribute here

Thus, we have

k 1

6and on the scales

14

d 2ukA

d2 k 2 1 A 8

3

1

k

u

kA 0

† *A A A A Ak k k k ka u a u

| 0 0Aka

2 20 | | 0A A

k ku

1

2A ikku e

k

2

3 8exp 2

8 3

A AAk

ku A k

k

Instability induces Polarization

quantization

vacuum fluctuations

E.O.M. on sub-horizon scales

1/ 6k

8 / 3k

Left-handed circular polarization mode is simply oscillating,Right-handed circular polarization mode is exponentially growing.

1, 1R L

15

Schematic picture of evolution

H 1

k

Bunch-Davisvacuum

instability freeze

right-handed

k 1

6 k

8

3

16

Degree of Polarization

2 2

2 2( ) 1R Lk k

R Lk k

u uk

u u

2

2

exp 32 / 32980

exp 8 / 3R

L

u

u

1 8

6 3k The instability continues during

8exp 2

3k

The growth factor gives

Hence, we have the degree of circular polarization

The string theory could produce 100 percent circularly polarized GW!

Note that the amplitude is also enhanced by the instability.

17

However, we have to consider the scalar curvature perturbationsfor which we also expect the very blue power spectrum

Everything seems to go well.

Fortunately, it is possible to circumvent this difficulty.

18

Two field inflation & detectability

19

　　　 Primordial GW

Inflation origin

BBN bound

CMB bound

Pulsar timing

(Maggiore 2000)

LISA

DECIGO/BBO

LIGO II

There is almost no constraint in this frequency range!

f 2

f 0

20

Two-field inflation

4 41 1 1( , )

2 2 2S d x g R d x g V

4 2 41 1( ) ( )

16 16GBd x g R d x g R R

At the onset of the second inflation, GB term induces the super-inflation

In principle, it is possible to observe the circular polarization of GWby LISA, if the onset of the second inflation lies in the appropriate period.

The amplitude of GW is enhanced there and the circular polarization is created.

field drives the first inflation where CMB spectrum is relevantfield drives the second inflation where GB and CS are important

21

A concrete realization 22 2 2 2 2 21 1

2 2V m m a b

610m 72

103

m

1110 3000a 0.04b

22

DetectabilityWe thus have the following schematic picture.

It should be stressed that our model is completely consistent withcurrent observations.

0.08 GW / 10 15 SNR / 5

SNR

GW

10 13

Seto 2006

1Hzat

Assuming 10 years observational time

GW / 10 8 SNR / 5

For LIGO and LCGT, we have

Taruya&Seto 2007

23

Conclusion

24

Observe the circular polarization of primordial gravitational waves!

It must be easier than that we have thought before.Because the amplitude is enhanced by several orders!

It strongly supports the superstring theory.At least, it indicates the existence of gravitational Chen-Simons term.

That might be a signature of the superstring theory!