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

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

The universe is so transparent for GW! 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? The reaction rate is much smaller than the expansion rate in the cosmic history. Hence, PGW can carry the information of the Planck scale. reaction rate expansion rate reaction rate

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

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

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

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

Horava’s idea 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. Horava 2009 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.

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

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

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

Horava gravity 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 comments A parity violating term is required for the theory to be renormalizable! z=3 UV gravity z=1 IR gravity

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

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

Action for gravitational waves

Chiral PGWs Adiabatic vacuum degree of circular polarization

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

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

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

Conclusion We have looked beyond the Planck scale via Horava gravity and found that the spacetime is chiral, 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.