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

20

40

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.