Horava-Lifshitz 重力理論とはなにか？

1. 2009.6.5 大阪市立大学セミナー Horava-Lifshitz重力理論とはなにか？ 早田次郎 　京都大学理学研究科 Ref. Chiral Primordial Gravitational Waves from a Lifshitz Point T.Takahashi & J.Soda, arXiv:0904.0554 [hep-th], to appear in Phys.Rev.Lett.

2. How to get to Planck scale? There are two well known the paths to reach the Planck scale. Hawking radiation GW Exponential red shift Exponential red shift inflation Quantum fluctuations BH Quantum fluctuations In reality, it would be difficult to observe Hawking radiation. However, wemay be able to observe primordial gravitational waves! Hence, in this talk, I will mostly discuss an inflation.

3. The universe is so transparent for GW! The reaction rate is much smaller than the expansion rate in the cosmic history. reaction rate reaction rate expansion rate Hence, PGW can carry the information of the Planck scale. Namely, one can see the very early universe! Indeed, we can indirectly observe PGW through CMB or directly observe PGW by LISA or DECIGO. What kind of smoking gun of the Planck scale can be expected?

4. A brief review of Inflation general relativity FRW universe conformal time dynamics We will consider a chaotic inflation. slow roll deSitter universe All of the observations including CMB data are consistent with an inflationary scenario!

5. PGW must exist if you assume inflation! Tensor perturbation action for GW GW propagating in the z direction can be written in the TT gauge as polarization Gravitational waves in FRW background are equivalent to two scalar fields with Wavelength of fluctuations length scale Super-horizon Bunch-Davis vacuum Sub-horizon Power spectrum t Quantum fluctuations

6. Is general relativity reliable? Length Scale horizon size k Initial conditions are set deep inside the horizon galaxy scale For GUT scale inflation Planckian region t We are looking beyond the Planck scale! We need quantum gravity!!

7. Quantum Gravity and Renormalizability A difficulty UV divergence in general relativity can not be renormalized Higher curvature improves the situation but suffers from ghosts That is why many people are studying string theory. However, string theory is rather large framework and not yet mature to discuss cosmology. Hence, it is worth seeking an alternative to string theory.

8. Horava’s idea Horava 2009 In order to avoid ghosts, we can use spatial derivatives to kill UV divergence The price we have to pay is that, in the UV limit, we lose Lorentz symmetry. Is the symmetry breakdown at UV strange? No! We know lattice theory as such. In fact, Horava found a similarity between his theory and causal dynamical triangulation theory.

9. Lifshitz-like anisotropic scaling Horava 2009 In order to get a renormalizable theory, we need the anisotropic scaling Because of this, we do not have 4-d diffeomorphism invariance. foliation preserving diffeomorphism ADM form lapse 3d metric shift extrinsic curvature

10. Horava gravity – kinetic term The kinetic term should be Since the volume has dimension Coupling constants and run under the renormalization. In the case , we have an extra scalar degree of freedom. In the IR limit, we should have

11. Horava gravity – potential term detailed balance condition Horava 2009 Orlando & Reffert 2009 This guarantees the renormalizability of the theory beyond power counting. The power counting renormalizable action with relevant deformation reads z = 3 UV gravity relevant deformation Cotton tensor

12. Horava gravity z=3 UV gravity z=1 IR gravity comments A parity violating term is required for the theory to be renormalizable! We have a negative cosmological constant which must be compensated by the energy density of the matter. Cosmological constant problem! To recover the general relativity, we need rescale The speed of light and Newton constant are emergent quantities

13. Inflation in Horava Gravity We consider a scalar field In the slow roll phase, we can approximate it as In this case, we have de Sitter solution

14. Polarized Gravitational waves Tensor perturbation Because of the parity violation, we need a different basis to diagonalize the action Polarization state polarization tensor Circular polarization Left-handed circular polarization Right-handed circular polarization

15. Action for gravitational waves

16. Chiral PGWs Adiabatic vacuum degree of circular polarization

17. <TB> correlation in CMB intensity tensor Stokes parameter dirction on the sky tensor harmonics If parity symmetry is not violated r=0.1 Saito et al. 2007

18. Direct detection of Chiral PGW “Stokes” parameter With three detectors or two well designed detectors, we can measure V. Seto 2007 Cooray 2005

19. What can be expected for BH? Chiral Hawking radiation Quantum fluctuations BH r=0 r=2M

20. Conclusion • We have looked beyond the Planck scale via Horava gravity and found that the spacetime ischiral, which can be tested by observing a circular polarization of primordial gravitational waves. This is a robust prediction of Horava gravity! The renormalizability yields parity violation, which is reminiscent of CKM parity violation.