Evolution of Simplicial Universe Shinichi HORATA and Tetsuyuki YUKAWA Hayama center for Advanced Studies, Sokendai Hayama, Miura, Kanagawa , Japan Some of the topics have been already appeared in S.Horata, T.Yukawa S.Horata, and T.Yukawa : Making a Universe.hep-th/ address: KEK-WS 03/14/2007

Motivated by the Observation of CMB anisotropies WMAP (Wilkinson Microwave Anisotropy Probe) 2003 COBE (Cosmic Background Explorer) 1996 (2006 Nobel Prize) Temperature fluctuation T~2.7K

Fundamental Problems : How has the universe started ? Initial condition How has it evolved ? Cosmological dynamics What is the space-time of the universe? Direction and expanse How did the physical laws appear ? Physical reality Creation by rules, without laws Phenomenological Problem : Obtain the two point correlation function of temperature fluctuation ｓ in the CMB Correlation beyond the event horizon

A simplest example of the creation without laws : The Peano axioms (rules) for the natural number 1. Existence of the element ‘1’. 2. Existence of the successor ‘S (a )’ of a natural number ‘a’. Axioms for creating the universe. element ‘d-simplex’. 1. Existence of the element ‘d-simplex’. neighbors 2. Existence of the neighbors of simplicial complex. For example, creating a 2-dimensional universe 1. The element = an equilateral triangle 2. The neighbor = 2-d triangulated surfaces constructed under the manifold conditions : ii) Triangles sharing one vertex form a disk (or a semi-disk). i)Two triangles can attach through one link (face).

Simplicial S 2 manifold Simplicial Quantum Gravity Space Quantization = Collection of all the possible triangulated (simplicial) manifolds Appendix 1. dimple phase S.Horata,T.Y.(2002) K-J.Hamada(2000) Phase transition

{ 1,2 } { 1,0 } Extension to open topology ( p,q ) moves of S 2 topology {  V,  S } moves of D 3 topology ( 1,3 ) ( 2,2 ) Example S 2 to D 3 { 1,-2 } ( 3,1 ) ( 1,3 ) ( 3,1 ) ( 2,2 ) Quantum Universe : Collection of all possible d-simplicial manifold Quantum Universe : Collection of all possible d-simplicial manifold

Evolution of the 2d quantum universe in computer Start with an elementary triangle, and create a Markov chain by selecting moves randomly under the condition of detailed balance. p a : a priori probability weight for a configuration a, n a : number of possible moves starting from a configuration a with the volume V= and the area S= a  : the (lattice) cosmological constant  B : the (lattice) boundary cosmological constant (global and additive)

Simplest universes at the early stage N2N2N2N2 A universe with N 2 =19,N ~ 1 =18 a lot of trees and bushes -> Tutte algorithm

Appendix2. (Old ) Matrix Model Generating function BIPZ(1978) k: # of triangles, l: # of boundary links Conjecture from the singularity analysis diverges at continuous limit

Im[Z(g,j)] <N2><N2><N2><N2> 3 Phases of the 2-dimensional universe

Defining the Physical time t with a dimensional factor c by  Monte Carlo time  and physical time t are related as In the expanding phase computer simulation shows, V~V 0 , S~S 0 , thus we have which means the inflation in t : N.B. t becomes negative when the volume decreases. t S(t) V(t) ( on ) matrix model

The Liouville theory Q : background charge (= b+b -1 ) : the cosmological constant B : the boundary cosmological constant Liouville action with a boundary Appendix 3. Partition function Fateev,A.&Al.Zamolodchikovhep- th b 2 =2/3 for pure gravity

In the classical limit Classical Liouville equation for the expanding region Line element expands as Physical time t and the conformal time  Inflation Homogeneous solution  0 =const. Homogeneous solution (  0 =const. ) N.B. Our definition of the physical time coincide with this physical time.

The boundarywo point correlation function The boundary two point correlation function Conformal theory predicts The boundary metric density  exp  b  x   the  number of triangles shearing a boundary vertex x+n-th neighbors Identifying the distance D = Geodesic distance D = Smallest number of links connecting two vertices x ~ geodesic distance D quantum+ ensemble averages x + 1 st neighbors

Evolution of the correlation function Boundary 2-point function boundary length at L(t) = boundary length at t Angular power spectra Large angle correlation Measured on one universe.

The power spectrum of the 2- point correlation function on a last scattering surface lss (S 2 ) in S 3 of D 4 N.B. Normalized at l=10 (preliminary) Future Problems : Extension to the 4-dimension Extension to the 4-dimension Inclusion of Matter Inclusion of Matter Creation of Dynamical Laws Creation of Dynamical Laws