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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

A master equation for the evolution of two-dimensional universe is derived based on the simplicial quantum gravity regarding the evolution as the Markov process of a space-time lattice. Three typical phases, expanding, elongating and collapsing phase, which have been found in the numerical simulation, are studied together with their boundaries, analytically. Asymptotic solutions of the evolution equation for statistical quantities, such as averaged area, boundary length, and correlation of fluctuations, are obtained for each phase and boundary.After introducing a physical time the cosmological significance of each phase is discussed.

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

The angular power spectra of cosmic microwave background are analyzed under the light of the evolutional scenario of the universe based on the renormalizable quantum theory of gravity in four dimensions. The equation of evolution is solved numerically fixing the power law spectrum predicted by the conformal gravity for the initial condition. The equation requires to introduce a dynamical energy scale about 10^{17}GeV, where the inflationary space-time evolution makes a transition to the big-bang of the conventional Friedmann universe. The quality of fit to the three-year data of WMAP implies the possibility to understand the observation by quantum gravity., Comment: 12 pages, 7 figures

For understanding the origin of anisotropies in the cosmic microwave background, rules to construct a quantized universe is proposed based on the dynamical triangulation method of the simplicial quantum gravity. A $d$-dimensional universe having the topology $ D^d $ is created numerically in terms of a simplicial manifold with $d$-simplices as the building blocks. The space coordinates of a universe are identified on the boundary surface $ S^{d-1} $, and the time coordinate is defined along the direction perpendicular to $ S^{d-1} $. Numerical simulations are made mainly for 2-dimensional universes, and analyzed to examine appropriateness of the construction rules by comparing to analytic results of the matrix model and the Liouville theory. Furthermore, a simulation in 4-dimension is made, and the result suggests an ability to analyze the observations on anisotropies by comparing to the scalar curvature correlation of a $ S^2 $-surface formed as the last scattering surface in the $ S^3 $ universe., Comment: 27pages,18figures,using jpsj.sty

We propose an evolutional scenario of the universe which starts from quantum states with conformal invariance, passing through the inflationary era, and then makes transition to the conventional Einstein space-time. The space-time dynamics is derived from the renormalizable higher-derivative quantum gravity developed on the basis of a renormalizable conformal gravity in four dimensions. Based on the linear perturbation theory in the inflationary background, we simulate evolutions of gravitational scalar, vector and tensor perturbations, and evaluate the spectra at the transition point located at the beginning of the big bang. The obtained spectra cover the range of the primordial spectra for explaining the tiny anisotropy in the homogeneous CMB., Comment: 30 pages, 6 figures. Version to appear in Phys. Rev. D

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., Comment: 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

A novel primordial spectrum with a dynamical scale of quantum gravity origin is proposed to explain the sharp fall off of the angular power spectra at low multipoles in the COBE and WMAP observations. The spectrum is derived from quantum fluctuations of the scalar curvature in a renormalizable model of induced gravity. This model describes the very early universe by the conformal field fluctuating about an inflationary background with the expansion time constant of order of the Planck mass., Comment: 12 pages, 2 figures

We study phases and fractal structures of three-dimensional simplicial quantum gravity by the Monte-Carlo method. After measuring the surface area distribution (SAD) which is the three-dimensional analog of the loop length distribution (LLD) in two-dimensional quantum gravity, we classify the fractal structures into three types: (i) in the hot (strong coupling) phase, strong gravity makes the space-time one crumpled mother universe with small fluctuating branches around it. This is a crumpled phase with a large Hausdorff dimension $d_{\mbox{\tiny H}} \simeq 5$. The topologies of cross-sections are extremely complicated. (ii) at the critical point, we observe that the space-time is a fractal-like manifold which has one mother universe with small and middle size branches around it. The Hausdorff dimension is $d_{\mbox{\tiny H}} \simeq 4$. We observe some scaling behaviors for the cross-sections of the manifold. This manifold resembles the fractal surface observed in two-dimensional quantum gravity. (iii) in the cold (weak coupling) phase, the mother universe disappears completely and the space-time seems to be the branched-polymer with a small Hausdorff dimension $d_{\mbox{\tiny H}} \simeq 2$. Almost all of the cross-sections have the spherical topology $S^2$ in the cold phase., Comment: 14 pages, latex file, 5 Postscript figures, use psfig.sty