renormalizable quantum gravity and its cosmological...
TRANSCRIPT
KEK 6/2、2008
Renormalizable Quantum Gravity and
Its Cosmological Implications
• K. Hamada, S. Horata and T. Yukawa, “Focus on Quantum Gravity Research”(Nova Science Publisher, NY, 2006), Chap.1
• K. Hamada, S. Horata, N. Sugiyama and T. Yukawa, arXiv:0705.3490[astro-ph]• K. Hamada, S. Horata and T. Yukawa, Phys.Rev.D74(2006)123502 • K. Hamada and T. Yukawa, Mod. Phys. Lett. A20 (2005) 509• K. Hamada, A. Minamizaki and A. Sugamoto, Mod. Phys. Lett. A23 (2008) 237
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The end of quantum gravity
is to understand beyond Planck scale phenomena
The starting point of quantum gravity is to give up graviton picture!
Quantum gravity = quantization of space-time
= quantization of graviton
Key idea
Conformal invariance/Background metric independence no scale and no singularity
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Evolution of fluctuation (CFT to CMB)Planck phenomena (CFT) space-time transition (big bang) today
CMB spectrumconsistent with WMAP
0.000.050.10
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60 0.00 0.05
0.10 0.15
0.00
0.05
0.10
Bardeen Potential Φ(b1=10, m=0.05)
1
proper time τ
k [Mpc-1]
58.0
58.5
59.0
59.5
60.0 0.00 0.05
0.10 0.15
1 × 10-53 × 10-55 × 10-57 × 10-59 × 10-5
proper time τ
k [Mpc-1]
CFT spectrum at Planck time From Planck length to Hubble distance
293059 101010 +=inflation
inflation Friedmann
Big bang
Fluctuations decreasing during inflation
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TT p
ower
spec
trum
, l(l+
1)C l /
2π
multipole l
Ωb = 0.045Ωcdm = 0.22Ωvac = 0.735τe = 0.1
WMAP-TT version 2.0 (March 2006)b1=15,m=0.05,u=0.0,h=0.77,r=0.7b1=15,m=0.05,u=0.1,h=0.77,r=0.3b1=20,m=0.05,u=0.0,h=0.75,r=0.5b1=20,m=0.05,u=0.1,h=0.75,r=0.3
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K. H., S. Horata, N. Sugiyama and T. Yukawa, arXiv:0705.3490[astro-ph]
The Basis of Quantum Gravity I
IntegrabilityRenormalizable ActionConformal InvariancePhysical States
K.H. and S. Horata, hep-th/0307008;K.H., hep-th/0402136
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Integrability and actionbare action(conf. anomaly)Conformal variation of effective action
(=path integral by conf. mode)
Integrability condition
Weyl action and Euler combination (no R^2)Integrable Action
asymptotically free dimensionless coupling constant
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Path integral quantization
Lowest term of S for the coupling t
Jacobian=Wess-Zumino actionfor conformal anomaly
cf. Liouville action
Dynamics of conformal mode is induced from the measure!
higher order WZ terms dimensional regularization
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Diffeomorphism invariance: gauge parameter
Mode decomposition
coupling const.no coupling const.
Conformal and traceless modes are completely decoupled !
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Conformal invariance as diffeomorphisminvariance at t 0
• gauge symmetry 1
• gauge symmetry 2
( )
: conformal Killing vector
conformal symmetry on (# is fixed)
otherwise=0
c.f.
(note analogy )
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Conformal algebra on
Conformal algebra
radiation gauge using degrees of freedom
residual gauge symmetry = conformal symmetry
Isometry of S^3=
: Hamiltonian
: S^3 rotation
: special conf. + dilatation transf.[=4 vectors of SO(4)]
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Conformal charges for gravitational fieldsScalar harmonics = (J,J) representation of
Hamiltonian for conformal mode
Special conformal + dilatation transformation
SU(2)^2 Clebsch-Gordan coeff. of SSS type
4 vector
Wigner D function
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Vector harmonics = rep. withTensor harmonics = rep. with
(polarizations)
SU(2)^2 CG coeff.
: STT type: STV type: SVV type
Negative-metric modes are necessary to close conformal algebra (=Wheeler-DeWitt algebra)
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Physical sates
Consider composite creation op. R_n satisfying vacuum state
then
pure imaginaryn=even integer
n=0 : cosmological constant (=physical metric field)n=2 : scalar curvature
“Real” confromal field positive two-point function !Initial spectrum of the universe
The Basis of Quantum Gravity II
Dimensional RegularizationRenormalizationConformal Anomaly
K.H., hep-th/0203250
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Euclidean sign.Renormalizable actionD dimensional integrability bare action
Renormalization factors
( )
: conformal mode is not renormalized
Ward-Takahashi identity
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On regularization methods
• DeWitt-Schwinger method
one-loop order
conformal anomaly(= effective action)heat kernel
• Dimensional regularization
all orders, diffeomorphism invariant
conformal anomaly comes frombetween D and 4 dimensions
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Conformal anomaly [WZ action]residues x_1, x_2
beta function
Bare action vertices and counterterms
ordinary counterterms
new vertices and new counterterms
Bare Weyl action Wess-Zumino actionfor conformal anomaly
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Laurent expansion of b
Euler term
counterterms
Wess-Zumino actionsand their counterterms
Conformal modedynamics
Kinetic term is induced
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Beta functions
b_n coefficients
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Non-renormalization of conf. mode [ ]
= finite+
no term
z: small mass [IR cutoff]not gauge invariant cancel out !
Since the Einstein term is composite field such as , it is not mass term
power-law dependence of M_P
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Vertex function ( ) of e^6Two-point function of e^4
These are renormalized bythe condition
And also, two-point function of e^6
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Effective action
Beta function conf. anomaly
Running coupling const.
physical momentum
Asymptotic freedom comoving momentum
At high energy beyond Planck scale
Singularities with divergent Riemann curvatureare excluded quantum mechanically
CFT to CMB
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Evolution scenario
inflation
CFT
baryogenesiscorrelation length:
Number of e-foldings scale factor
K.H., Minamizaki, SugamotoarXiv:0708.2127[hep-ph]
Planck length at Planck time
grows up tothe Hubble distance
today today
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Calculation of CMB multipoles
The evolution of scalar curvature fluctuation
(CFT)Big Bang Friedmanninflation
Simple estimation of the amplitude
de Sitter curvature
At the big bang
Linear perturbation is applicablefor
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Constraint equation initially
finally
Evolution equation for gravitational potentials
Inflationary background Dynamical factor
: matter density
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Spectrum of quantum gravity (2-pt. function)
Initial QG fluctuations = CFT (scale invariant)
Scalar spectral index
Size of fluctuation we considerSize of Planck length at Planck time
at the transition point, the size is much more extended than the correlation lengthnot disturbed by the dynamics of transition
comoving Planck const.
coeff. of Wess-Zumino action
HZ spectrum
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CMB multipoles
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0 200 400 600 800 1000
TT p
ower
spe
ctru
m, l
(l+1)
Cl /
2π
multipole l
Ωb = 0.045Ωcdm = 0.22Ωvac = 0.735τe = 0.1
WMAP-TT version 2.0 (March 2006)b1=15,m=0.05,u=0.0,h=0.77,r=0.7b1=15,m=0.05,u=0.1,h=0.77,r=0.3b1=20,m=0.05,u=0.0,h=0.75,r=0.5b1=20,m=0.05,u=0.1,h=0.75,r=0.3
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For the region ,non-linear effects (CFT)become effective.(in progress)
Inflation era Einstein era
proper time
Hub
ble
varia
ble
H
Space-time transition
Matter density
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Conclusion
Repulsive effect in quantum gravityinduce inflation. origin of expanding universe.fluctuations decrease during inflationprevent black hole from collapsing to a point
Asymptotic freedom of traceless tensor modenovel dynamical scale:space-time phase transition
Quantum gravity spectrumgiven by conformal field theory (non-Gaussian)can explain sharp fall-off of low multi-pole componentsby appearance of dynamical scale
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Appearance of QG effects?Limitation of classical relativity
energy dependence of speed of lightVery high energy particle is no longer point-like, it is dressed by quantum gravity and then space-time around it might be locally deformed gamma-ray burst, provided such a effect is given by the order of .Black hole evaporation
extremely high energy gamma rayAt the final stage of evaporation, Hawking temperature becomes extremely high. Then, horizon disappears and thus BH vanishes explosively.
Black holehorizon
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Tensor perturbation
Initial CFT spectrum
Tensor fluctuation is initially small because of asymptotic freedom, which is preserved to be small during inflation.