1

Circular Polarization of

Gravitational Waves

in String Cosmology

KITPC, 200７ .11.23

Jiro SodaKyoto University

work with Masaki Satoh & Sugumi Kanno arXiv:0706.3585

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　弦理論的宇宙論　円偏極重力波生成

KITPC, 200７年 11月23日

早田　次郎京都大学理学研究科

3

Why primordial GW?

In other words, one can see the very early universe through GW!

Of course, due to the weakness of gravity, it would be difficult to see GW.

Typically, we need to see a very small number 2210h

However, it is not impossible thanks to the current technology!!

Hence, taking look at the beginning of the universe is exciting and challenging.

Because the gravitational interaction is so weak, gravitational waves can propagate freely even from the very early universe.

That’s why we are so fascinated by the primordial GW.

4

Plan of the talk

Basics of GW Primordial GW generated during slow roll Inflation

Inflation in Chern-Simons-Gauss-Bonnet Gravity A mechanism to produce circular polarization of GW Two field model & detectability Conclusion

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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.

6

Circular polarization

Left-handed circular polarization

Right-handed circular polarization

Rh h ih

Lh h ih

7

Astrophysical sources

Free fall time scale 1/fft G f Gfrequency

Ex. NS binary M M 10kmR

3011 4

3 4 3

106.6 10 10 Hz

10

Mf G

R

Ex. White dwarf binary 0.6M M 510 kmR

310 Hzf

Ex. Giant BH binary 610M M 710 kmR

310 Hzf

LISA range

LIGO range

Assuming the distance to be 100Mpc, the amplitude is about 2210h

8

Cosmological sourcesf G H

The observed frequency is redshifted to obsf Ha a

EW scale 210 GeVT 310 Hzobsf

2

442

10p p

T Tf H

M M

For cosmological source, the typical frequency would be

In the thermal case, we have

2

440 10obsp

T Tf Ha

T M

Ex.

In the thermal case, we have

log obsf inflation

CMB

LISA

LIGO

log a

Annoying degeneracy

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How to quantify GW?Energy density of GW

( )1

32 logij GW

GW ij

d fdfh h

G f d f

20

( )8( )

3 logGW

GW

d fGf

H d f

22 2( )

32c

GW

h ff

G

1201.5 10

100Hzc GW

fh

310 Hzf 1410GW 2210ch

LISA1110GW

BBO1510GW

at 0.1 Hz

Ultimate DECIGO 2010GW at 0.1 Hz

at 1 mHz

Let us define ch by

Density parameter

It allows us to compare the amplitude of point sources and cosmological ones.

Ex.

Detector sensitivity

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Slow roll inflation

( ) H ta t e

22 1

2

8( )

3

GH V

3 '( ) 0H V

2 2

2

1 '4 1

16

VG

G V H

0.01 2 21

2V m

Slow roll parameters

2 2 2 2 2 2( )ds dt a t dx dy dz

aH

a

metric

dynamics

quasi-deSitter universe

slow roll 2 8( )

3

GH V

3 '( ) 0H V

Ex.

11

Origin of fluctuations

length

t

Wavelength of fluctuations

1H

Quantum fluctuations

22 0a

ka

1

2ike

a k

decaying modec

A free scalar field

Sub-horizon

Super-horizon

a

k

12

Amplitude of fluctuations

2 1c

pl

H HR H t H

M

pl

Hh

M

curvature perturbations

gravitational waves

aH k

32

Hc

k 22 3 20 0 kk H

Matching at

plM h

gives

The relation implies

The tensor to the scalar ratio2

2 16 ( 0.16)T

S c

hPr

P R

GW 10 4 10 10 10 14

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　　　 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

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Motivation of our workSuperstring theory may induce Gravitational Chern-Simons term

which may produce Circular polarization of GW

Slow roll inflation does not produce circular polarization

Gauss-Bonnet term is also predicted by superstring theory

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

Our observation

Then, the purpose of our work is to study the primordial GW in the context of Chern-Simons-Gauss-Bonne gravity.

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String Inspired Model4 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

Combined effect produces the 100 % circular polarization.

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

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

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Details

17

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

This could accelerate the scalar field

Friedman equation

Scalar field equation

H

a '

a

This could be dominant

4 For concreteness, we take a simple model

'd

d

18

5/ 615 1/ 6

a 1

6H

0

3 2 23 'H a

Super-inflation regime

2,2

3'' 2 ' ' 0

2H H H

a

GB term produces the kinetic energy dominant stage where the systemcan be well described by

0H expandingdecreasing

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

It is not difficult to obtain an analytic solution

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Exit to slow-roll inflationary phase

As we will see, subsequently, the slow-roll inflation will commence.

2 2 2 2 22

1 1 13 1 ' '

2 2 2H H m a

a

5/3

At some point, the asymptotic solution ceases to be valid.

Fortunately, super-inflation does not continue forever in generic cases.

If super-inflation does not end, we encounter the singularity.

Exit from the super-inflation can be seen more preciselyin the phase diagram.

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Dynamical Flow in the phase space

3

2H 3,

3

2H 3,

2

6H 2 2V

Using the cosmic time, we have

2 H,

3

4H 4,

2

H

5H 2 1 1

2H,

H 2 1 1

2,

2

2V

1

2H 2, 3H V '()

3

2H 4,

f (, H )

H g(, H )autonomoussystem

Here, H is the physical Hubble.

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Numerical Result

Slow roll regime

Super-inflation regime

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

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Gravitational waves

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!

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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

For super-inflationary regime 1

Both GB and CS contribute here

Thus, we have

k 1

6and on the scales

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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 mode is simply oscillating,right handed-mode is exponentially growing

1, 1R L

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Schematic picture of evolution

H 1

k

Bunch-Davisvacuum

instability freeze

right-handed

k 1

6 k

8

3

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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.

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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

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

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.

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A concrete realization 22 2 2 2 2 21 1

2 2V m m a b

610m 72

103

m

1110 3000a 0.04b

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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

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SummaryObserve 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!