Post on 03-Feb-2016
description
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
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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.
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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.
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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.
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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.
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Outline of my talk
Inflation in Gauss-Bonnet-Chern-Simons Gravity
A mechanism to produce circular polarization
Two field inflation & detectability
Conclusion
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Inflation in Gauss-Bonnet -Chern-Simons Gravity
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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.
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Numerical Result
Slow roll regime
Super-inflation regime
GB term drives the super-inflation.It indicates the violation of weak energy condition.
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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?
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A mechanism to produce circular polarization
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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!
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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
In 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 circular polarization mode is simply oscillating,Right-handed circular polarization 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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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.
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Two field inflation & detectability
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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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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
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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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Conclusion
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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!