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A mathematical model of the distribution function for the discrete 3-disk is proposed in order to utilize in the statistical evolution equation of the 3-dimensional Universe. The model distribution is constructed based on analyses in known exact solutions of recursion equations for the generating functions of the discrete 2-disk.The proposed distribution function is compared with numerical simulations of the dynamical triangulation with $ S^3 $, and $ D^3 $ topologies.The model distribution exhibits three types of phases characterized by geometrical nature of the disk with either 1, 2, or 3- dimensional structure.Transitions between those phases are either cross-over, 1st order, or 2nd order depending on the parameters, which reflect the type of discretization and matter fields coupled to space., Comment: 29 pages, 15 figures

We study the inflation process of universe based on the renormalizable quantum gravity formulated as a conformal field theory (CFT). We show that the power-law CFT spectrum approaches to that of the Harrison-Zel'dovich-Peebles type as the amplitude of gravitational potential gradually reduces during the inflation. The non-Gaussanity parameter is preserved within order of unity due to the diffeomorphism invariance. Sharp fall-off of the angular power spectrum of cosmic microwave background (CMB) at large scale is understood as a consequence of the existence of dynamical scale of the quantum gravity \Lambda_QG (\simeq 10^{17})GeV. The angular power spectra are computed and compared with the WMAP5 and ACBAR data with a quality of \chi^2/dof \simeq 1.1., Comment: 20 pages, 5 figures, added more sentences, references, and an appendix to explain our inflation model

Inflationary scenario based on a renormalizable model of conformal gravity is proposed and primordial spectrum is derived. The sharp fall off of the angular power spectra at low multipoles in the COBE and WMAP observations are explained by a dynamical scale of quantum gravity. At this scale, the universe would make a sharp transition from the quantum spacetime with conformal invariance to the classical spacetime., 4 pages, 1 figure, Talks presented by T.Y. at the Sapporo Winter School in Niseko '04, Jan 8-12, 2004, Niseko Annupuri, Hokkaido, Japan; K.H. at the 12th International Conference on Supersymmetry and Unification of Fundamental Interactions, June 17-23, 2004, Epochal Tsukuba, Tsukuba, Japan; at the 17th International Conference on General Relativity and Gravitation, July 18-23, 2004, RDS, Dublin, Ireland