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MIAMI, 200 ７ .12.14. Circular Polarization of Gravitational Waves in String Cosmology. Jiro Soda. Kyoto University. work with Masaki Satoh & Sugumi Kanno arXiv:0706.3585. Polarization of Gravitational Waves. Action for GW.
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Action for GW
GW propagating in the z direction can be written in the TT gauge as
Any linear combination of these polarization can be a basis of GW.
Right-handed circular polarization
Left-handed circular polarization
Without a parity violating process, the circular polarization of primordial GW does not exist.
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
S.Alexander & J.Martin, Phys.Rev.D71, 063526 (2005)
Slow roll inflation does not produce circular polarization
Gauss-Bonnet term also appears in superstring theory
We should study the primordial GW
in the context of Gauss-Bonnet-Chern-Simons gravity.
Inflaton drives the slow-roll inflation
This term is not relevant to
but could produce the circular
polarization of gravitational waves
This term induces
and the instability
of gravitational waves
These effects produce 100 % circular polarization of GW.
Moreover, the amplitude is also enhanced by the factor .
Hence, the effect is detectable by DECIGO/BBO or even by LISA.
Homogeneous and isotropic universe
Scalar field equation
For concreteness, we take a simple model
The equations can be cast into
the autonomous system
There exists a region
where super-inflation occurs.
Slow roll regime
GB term drives the super-inflation.
It indicates the violation of weak energy condition.
In the super-inflationary regime, the system can be well described
by Gauss-Bonnet dominant equations
It is not difficult to obtain an analytic solution
What can we expect for the gravitational waves in this background?
With the transformation
, we get
Right-handed and left-handed waves obey different equations!
In super-inflationary regime
and on the scales
Thus, we have
Both GB and CS contribute here
E.O.M. on sub-horizon scales
Left-handed circular polarization mode is simply oscillating,
Right-handed circular polarization mode is exponentially growing.
The instability continues during
The growth factor
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.
However, we have to consider the scalar curvature perturbations
for which we also expect the very blue power spectrum
Fortunately, it is possible to circumvent this difficulty.
There is almost no constraint in this frequency range!
field drives the first inflation where CMB spectrum is relevant
field drives the second inflation where GB and CS are important
At the onset of the second inflation, GB term induces the super-inflation
The amplitude of GW is enhanced there and the circular polarization is created.
In principle, it is possible to observe the circular polarization of GW
by LISA, if the onset of the second inflation lies in the appropriate period.
We thus have the following schematic picture.
Assuming 10 years observational time
For LIGO and LCGT, we have
It should be stressed that our model is completely consistent with
of primordial gravitational waves!
That might be a signature of the superstring theory!
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.