Quantum Gravity and CMB Anisotropies in Hawaii

1. Hawaii 10/30, 2006 Space-time Evolution and CMB Anisotropies from Quantum Gravity Ken-ji Hamada KEK with S. Horata and T. Yukawa Based on astro-ph/0607586 (to appear in PRD) and Contribution to chapter 1 of the book “Focus on Quantum Gravity Research” (Nova Science Publisher, NY, 2006)

2. 1. Introduction 2 Why we need quantum gravity! Quantization of gravity Scale invariant space-time lose the concept of distance resolve space-time singularities and divergences, because no fixed scale and no special point in space. Can break the wall of the Planck scale ! • The end of this talk • construct such a scale invariant space-time • create the universe itself from quantum space-time • explain results of WMAP experiment.

3. A consequence of scale invariant space-time 3 In very early universe, space-time fluctuations are so great that geometry lose its classical meaning, and scale invariant space-time picture will emerge. On the other hand, in the present universe, the metric acquires a physical significance for measuring time and distance. In very early universe, there is a space-time transition Quantum Space-time Classical Space-time (with a scale) (scale invariant) Novel Dynamical Scale A: quantum B: classical A correlation length B

4. There is a possibility to observe the instance of the transition, because we can trace the past guided by the known physical laws as far as the classical spacetime exists. 4 If we believe the idea of inflation, CMB anisotropies provide us information about dynamics beyond the Planck scale. Initial conditions would be given by quantum gravity Trans-Plankian problem: This value is necessary to solve flatness problem universe inflation : scale factor

5. 5 Plan of talk • Introduction • Renormalizable Quantum Gravity • Evolutional Scenario of The Universe • Linear Perturbation Theory • CMB Angular Power Spectra • Conclusion

6. 6 2. Renormalizable Quantum Gravity conformal invariant No restriction on conf. mode Perturbation about conformal flat ( ) Conformalmode and traceless-tensormode are treated separately! Renormalization (QED+gravity) : conformal mode is not renormalized K.H., hep-th/0203250

7. 7 Running coupling constant [asymptotic freedom] :comoving momentum defined on where with Physical momentum : Dynamical scale : Confromal mode increasing => running coupling getting large!

8. 8 Conformal mode is treated non-perturbatively The partition function Distler-Kawai, David, Antoniadis-Mazur-Mottola. Sugino-K.H., K.H. Jacobian = Wess-Zumino action Dynamics of conformal mode is induced from the measure: Conformal Field Theory (CFT) Higher order of the coupling conformal inv. : , and thus

9. 9 Asymptotic freedom for the traceless mode The configuration dominates at very high energy, and thus singularities with are removed quantum mechanically. This is a prominent feature of renormalizable gravity On the other hand, the singular configuration cannot be excluded in the Einstein theory, because such a configuration has the vanishing scalar curvature so that its quantum weight in the path integral is unity: Fluctuations of the conformal mode dominate, and thus the space-time dynamics is described by CFT at very high energies. Primordial spectrum is simply given by CFT : a deviation from CFT

10. 3. Evolutional Scenario of the Universe 10 3.1 Inflationary phase Starobinsky, Yukawa-K.H. Consider the orderings Effective action (WZ + Einstein) yields WZ action Conformal symmetry starts to be broken at the Planck scale Inflationary universe with (No logarithmic violation at , because dynamics is still described by CFT) growing up and energies are reducing ( increasing) Conformal symmetry is completely broken at the dynamical scale Friedmann universe

11. 11 Dynamical Model of Space-time Phase Transition The running coupling constant => time-dependent average proper time Dynamical coefficient in front of WZ action: where Dynamical coefficient vanishes at dynamical time scale

12. 12 Evolution equation Energy conservation where Inflation era [stable] Einstein era [unstable] (discussed next) : matter density (red line) Extra gravitational degrees of freedom decay to matter fields at the transition point.  Big Bang Dotted line denotes Friedmann solution

13. 13 3.2 Einstein phase ( ) Low energy effective action (derivative expansion) tree + 1-loop tree cf. chiral perturbation theory Here, we restrict effective action up to the fourth order, and thus using lowest Einstein’s equation , a variety of four-derivative actions is reduced, which is merely given by with running effect :phenomenologically determined Higher-derivative terms are irrelevant!

14. 14 Number of e-foldings: The universe grows up Inflation era : Einstein era : Planck length grows up to the present Hubble distance

15. 15 4. Linear Perturbation Theory Amplitudes of fluctuations are damping during stable inflation era, but not vanish. It is expected to survive small fluctuations due to the dynamics. Naively amplitude is estimated as follows: Near transition point, the running coupling gets large, and so . Since scalar curvature has two derivatives, the size of fluctuation is estimated to be the order of the square of dynamical scale: deSitter spacetime: Liner perturbation about stable inflationary solution becomes applicable!

16. Stability of fluctuation 16 Fluctuations getting smaller during inflation Inflationary solution (stable) Friedmann solution (unstable) fluctuations (=perturbations) Big bang Structure formation We compute transfer functions from Planck time to the big bang!

17. 17 Scalar perturbations Gauge invariant variables: determined by gravitational potentials gravitational potentials Scalar equation Constraint equation initially finally

18. 18 Tensor and Vector perturbations Gauge invariant variables: Tensor equation Vector equation

19. 19 5. CMB Angular Power Spectra Initial Power Spectra at Planck time (given by CFT) dimensionless From CFT: From asymptotic freedom: : coeff. of WZ action : comoving Planck constant

20. 20 Transfer functions from Planck time to transition time Scalar fluctuation gradually decrease. Primordial scalar spectrum transition point Primordial tensor spectrum Tensor fluctuation preserved to be small

21. 21 Primordial Spectra at the transition point scalar tensor The patterns of spectra are independent of the details of transition, because the fluctuation expands enough to the size much larger than the dynamical length scale at the transition point. This fact also justifies the linear approximation

22. 22 Transfer function from transition time to today is computed using CMBFAST code The obtained CMB angular power spectrum fit with WMAP3 data well Preliminary with N. Sugiyama

23. 6. Conclusion 23 • Quantum gravity scenario of inflation without any additional fields We proposed that a space-time phase transition occurs at scale, and then extra degrees of freedom in higher-derivative gravitational fields may decay to matter degrees of freedom, causing Big Bang. • CMB Angular power spectrum Primordial fluctuations are given by conformal fields (CFT). • The evolution of scalar, vector and tensor fluctuations about • inflationary background has been evaluated, and the primordial • spectra at the transition point were computed. • Angular power spectrum was computed using CMBFAST code, • which can explain the WMAP3 result well.

24. Future problems 24 • Repulsive effect in quantum gravity casts light on the question • why the universe has been expanding since it was born. • The repulsive effect may prevent black hole from collapsing • to a point, and play an important role to release matters in black • hole outside. •  Black hole explosion!? Asymptotic freedom for the traceless tensor mode • Spacetime singularities removed quantum mechanically. • Physical states are changed, which are governed by CFT. Conformal algebra: Antoniadis-Mazur-Mottola Horata-K.H., K.H. Physical states: No singularity and changing of physical states in quantum space-time are preferable features to resolve information paradox.

25. 25 Cylinder amplitude (transfer function) Friedmann Big bang inflation Disk amplitude computed by CFT, or dynamical triangulation  Horata’s talk

26. 26 4D Simplicial Quantum Gravity String susceptibility conformal gravity predicts (Sc only) (0.0028) (4.27) Dimple Phase Crumple Phase Horata,Egawa,Yukawa(2002)

27. Physical States are governed by Conformal Invariance 27 Conformal algebra and Physical states (on cylinder ): Antoniadis-Mazur-Mottola Horata-K.H., K.H. special conf. transfs. Hamiltonian rotation on S^3 M, N = vector index of SO(4) Conformal inv. vacuum = physical state satisfying Physical operators: cosmological const. scalar curvature Conformal charge: scaling behavior of physical operators

28. Evolution equation in Einstein phase Energy conservation Equations of motion are solved including running effect: Initial conditions given by the values at the transition :