Horava-Lifshitz??????????

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Horava-Lifshitz??????????
2009.6.5 ??????????

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Ref. Chiral Primordial Gravitational Waves from
a Lifshitz Point
T.Takahashi J.Soda, arXiv0904.0554 hep-th,
to appear in Phys.Rev.Lett.
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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, we may be able to
observe primordial gravitational waves! Hence, in
this talk, I will mostly discuss an inflation.
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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?
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A brief review of Inflation
general relativity
FRW universe
conformal time
dynamics
We will consider a chaotic inflation.
deSitter universe
slow roll
All of the observations including CMB data are
consistent with an inflationary scenario!
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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
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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!!
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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.
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Horavas 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.
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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
shift
3d metric
extrinsic curvature
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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
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Horava gravity potential term
detailed balance condition
Orlando Reffert 2009
Horava 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
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Horava gravity
z3 UV gravity
z1 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
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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
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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
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Action for gravitational waves
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Chiral PGWs
Adiabatic vacuum
degree of circular polarization
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ltTBgt correlation in CMB
intensity tensor
Stokes parameter
dirction on the sky
tensor harmonics
If parity symmetry is not violated
r0.1
Saito et al. 2007
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Direct detection of Chiral PGW
Stokes parameter
With three detectors or two well designed
detectors, we can measure V.
Seto 2007
Cooray 2005
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What can be expected for BH?
Chiral Hawking radiation
Quantum fluctuations
BH
r0
r2M
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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.