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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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Circular polarization of gravitational waves in string cosmology

MIAMI, 2007.12.14

Circular Polarization of Gravitational Waves in String Cosmology

Jiro Soda

Kyoto University

work with Masaki Satoh & Sugumi Kanno


Polarization of gravitational waves
Polarization of Gravitational Waves

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.

Circular polarization of gw
Circular polarization of GW

Right-handed circular polarization

Left-handed circular polarization

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

Motivation of our work
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

Known result

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

Slow roll inflation does not produce circular polarization

Our observation

Gauss-Bonnet term also appears in superstring theory

We should study the primordial GW

in the context of Gauss-Bonnet-Chern-Simons gravity.

Summary of our result
Summary of our result

Inflaton drives the slow-roll inflation

This term is not relevant to

background dynamics,

but could produce the circular

polarization of gravitational waves

This term induces

the super-inflation,

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.

Outline of my talk
Outline of my talk

  • Inflation in Gauss-Bonnet-Chern-Simons Gravity

  • A mechanism to produce circular polarization

  • Two field inflation & detectability

  • Conclusion

Inflation in gauss bonnet chern simons gravity

Inflation in Gauss-Bonnet -Chern-Simons Gravity

Cosmological background space time
Cosmological background space-time

Homogeneous and isotropic universe

Friedman equation

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.

Numerical result
Numerical Result

Super-inflation regime

Slow roll regime

GB term drives the super-inflation.

It indicates the violation of weak energy condition.

Analytic solution in super inflation regime
Analytic solution in Super-inflation regime

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?

A mechanism to produce circular polarization

A mechanism to produce circular polarization

Gravitational waves in gb cs gravity
Gravitational waves in GB-CS gravity

Tensor perturbation

Polarization state

polarization tensor

Circular polarization

With the transformation

, we get



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

Gw in super inflationary regime
GW in Super inflationary regime

In super-inflationary regime

and on the scales

Thus, we have

Both GB and CS contribute here

Instability induces polarization
Instability induces Polarization


vacuum fluctuations

E.O.M. on sub-horizon scales

Left-handed circular polarization mode is simply oscillating,

Right-handed circular polarization mode is exponentially growing.

Schematic picture of evolution
Schematic picture of evolution






Degree of polarization
Degree of Polarization

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.

Everything seems to go well.

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.

Primordial gw
Primordial GW

(Maggiore 2000)

Pulsar timing

BBN bound

CMB bound




Inflation origin

There is almost no constraint in this frequency range!

Two field inflation
Two-field inflation

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

Seto 2006


For LIGO and LCGT, we have

Taruya&Seto 2007

It should be stressed that our model is completely consistent with

current observations.

Observe the circular polarization

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.