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Transcript of 1 Circular Polarization of Gravitational Waves in String Cosmology KITPC, 200 7.11.23 Jiro Soda...
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
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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
5
Polarization of Gravitational Waves
2 2 2 2 2(1 ) (1 ) 2ds dt dz h dx h dy h dxdy
h h
4164
ij ijij ijS d x h h h h
G
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 sourcesFree fall time scale 1/fft G f Gfrequency
Ex. NS binary M M 10kmR
3011 4
3 4 3
106.6 10 10 Hz10
Mf GR
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 HM M
For cosmological source, the typical frequency would be
In the thermal case, we have
2
440 10obsp
T Tf HaT 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
( )132 log
ij GWGW ij
d fdfh hG f d f
20
( )8( )3 log
GWGW
d fGfH d f
22 2( )
32c
GW
h ff
G
1
201.5 10100Hzc GW
fh
310 Hzf 1410GW 2210ch
LISA 1110GW
BBO 1510GW 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 12
8 ( )3GH V
3 '( ) 0H V
2 2
2
1 ' 4 116
V GG V H
0.01 2 212
V m
Slow roll parameters
2 2 2 2 2 2( )ds dt a t dx dy dz
aHa
metric
dynamics
quasi-deSitter universe
slow roll 2 8 ( )3GH V 3 '( ) 0H V
Ex.
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Origin of fluctuations
length
t
Wavelength of fluctuations
1H
Quantum fluctuations
22 0a ka
12
ikea k
decaying modec
A free scalar field
Sub-horizon
Super-horizon
ak
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Amplitude of fluctuations
2 1c
pl
H HR H t HM
pl
HhM
curvature perturbations
gravitational waves
aH k
32
Hck
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
hPrP 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
14
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 2S 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
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2 2 2( ) i jijds a d dx dx
2 2 2 2 22
1 1 13 1 ' '2 2 2
H H m aa
2 2 2,2
3'' 2 ' ' 02
H H H m aa
Cosmological background space-timeHomogeneous 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
' dd
18
5/ 615 1/ 6a 16
H
0
3 2 23 'H a
Super-inflation regime
2,2
3'' 2 ' ' 02
H H Ha
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 2
H H m aa
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
32
H 3, 32
H 3,
2
6H 2 2V
Using the cosmic time, we have
2 H,
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H 4,2
H
5H 2 1 12
H,
H 2 1 12
, 2
2V
12
H 2, 3H V '() 32
H 4,
f (, H )
H g(, H )autonomoussystem
Here, H is the physical Hubble.
21
Numerical Result
Slow roll regime
Super-inflation regime
What can we expect for the gravitational waves in this background?
22
Gravitational waves
d 2kA
d2 1H 'zA
2 ''2zA
2
k 2
zA
zA
k
A 02 2
' '( ) 12 2
AA
Hz a ka 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 k e 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 1H 'zA
2 ''2zA
2
k 2
zA
zA
k
A 0
d 2kA
d2 k 2 1 A 83
1 k
k
A 0
For super-inflationary regime 1
Both GB and CS contribute here
Thus, we have
k 16and on the scales
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d 2ukA
d2 k 2 1 A 83
1 k
uk
A 0
† *A A A A Ak k k k ka u a u
| 0 0Aka
2 20 | | 0A A
k ku
12
A ikku e
k
2
3 8exp 28 3
A AAk
ku A k
k
Instability induces Polarizationquantization
vacuum fluctuations
E.O.M. on sub-horizon scales1/ 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 16
k 83
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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 86 3
k The instability continues during
8exp 23
k
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 2
S 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 10
3m
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!