Evolution of Simplicial UniverseEvolution of Simplicial UniverseShinichi HORATA and Tetsuyuki YUKAWA

Hayama center for Advanced Studies, Sokendai Hayama, Miura, Kanagawa 240-0193, Japan

Some of the topics have been already appeared inS.Horata, S.Horata, and T.Yukawa T.Yukawa : Making a Universe.hep-th/0611076.

e-mail address: yukawa@soken.ac.jp

KEK-WS

03/14/2007

Motivated by the Observation of CMB Motivated by the Observation of CMB anisotropies anisotropies WMAP (Wilkinson Microwave Anisotropy Probe) 2003

COBE (Cosmic Background Explorer) 1996 (2006 Nobel Prize)

Temperature fluctuation

)10(/),( 5TT T~2.7KT~2.7K

Fundamental Problems :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 :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.Axioms for creating the universe. 1. Existence of the elementelement ‘ ‘d-simplex’d-simplex’.. 2. Existence of the neighborsneighbors 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 S2 manifold

Simplicial Quantum GravitySimplicial Quantum Gravity

Space QuantizationSpace Quantization = =

Collection of all the possible triangulated (simplicial) manifolds

Appendix Appendix 1.1.

dimple phase

S.Horata,T.Y.(2002)

K-J.Hamada(2000)

Phase transition

{1,2}

{1,0}

Extension to open Extension to open topologytopology

(p,q) moves of S2 topology {V,S } moves of D3 topology

(1,3)

(2,2)

Example S2 to D3

{1,-2}

(3,1)

(1,3)

(3,1)

(2,2)

Quantum UniverseQuantum Universe: : ColCollection of all possible d-simplicial manifoldlection of all possible d-simplicial manifold

Evolution of the 2d quantum universe in Evolution of the 2d quantum universe in computercomputer

Start with an elementary triangle, and create a Markov chain by selecting moves randomly under the condition of detailed balance.

abb

bba

a

a wn

pw

n

p

pa: a priori probability weight for a configuration a,na: number of possible moves starting from a configuration a

)exp(~ Ap 12

~NNA B

with the volume V= 　　　 and the area S= a1~N

: the (lattice) cosmological constant B : the (lattice) boundary cosmological constant

22

4

3Na

(global and additive)

Simplest universes at the early stage NN22

1

~N

A universe with N2=19,N~

1=18

a lot of trees and bushes ->

Tutte algorithm

Appendix2. (Appendix2. (Old Old ) ) Matrix Matrix ModelModel Generating

function kl

klkl gjfgjF

, ,),(

32

2

1exp

1

1][),( gtrMtrM

jMtrMDgjF

BIPZ(1978)

)3

1exp()exp()exp(

25.05.2

,

)(

k

llklkf c

Bc

kl

s

k: # of triangles, l: # of boundary links

Conjecture from the singularity analysis

kl

lksklB

BeefZ,

)(,),(

]ˆ,ˆ[log2 BZN

]ˆ,ˆ[log~

1 BB

ZN

~

2

1

N

N12

~, NN diverges at 0ˆ

3

2ˆ 2/1 B

continuous limit

cBB

cB

c

ˆ

ˆ

-1 -0.5 0 0.5

-0.8

-0.6

-0.4

-0.2

0

ImIm[[ZZ((g,jg,j)])]

<<NN22>>

< >< >

3 Phases of the 2-dimensional universe

-1-0.5

0

0.5

-0.8

-0.6

-0.4

-0.2

0

0

1

2

3

-1-0.5

0

0.5

-0.8

-0.6

-0.4

-0.2

0

0

1

2

3

-1-0.5

00.5

-0.8

-0.6

-0.4

-0.2

0

0

1

2

3

-1-0.5

00.5

-0.8

-0.6

-0.4

-0.2

0

0

1

2

3

0ˆ3

2ˆ 2/1 B

1

~N

Defining the Defining the Physical Physical timetime tt with a dimensional factor c by '.)'()( dttSctV

t

Monte Carlo time and physical time t are related as

.'

)'(

)'(

1'

d

dV

Sdct

In the expanding phase computer simulation shows, V~V0 , S~S0 , thus we have

)exp( t

which means the inflation in t :

V

cS

)exp(~ 0 tSS

N.B. t becomes negative when the volume decreases.

t

S(t)

V(t)

0ˆ3

2ˆ 2/1 B( on )

~

2

1

N

Nmatrix model

The Liouville theoryThe Liouville theory

deQk

xdeS bB

baBL )

2())(

4

1( 222

Q: background charge (=b+b-1)

: the cosmological constant

B: the boundary cosmological constant

Liouville action with a boundaryLiouville action with a boundary

Appendix 3.Appendix 3.

A

l

blAlAZ

2

22/12/5

sin4

1exp~],[

~

Partition Partition functionfunction

Fateev,A.&Al.Zamolodchikovhep-th0001012

b2=2/3 for pure gravity

0 0],[

~],[ lA

BBelAZdldAZ

BBB bbZ 2

2/3

24/5 sin3

21sin1],[

3

2

4/3sin4

12

2

2

a

a

b)

3

1exp()exp()exp(

25.05.2

,

)(

k

llklkf c

Bc

kl

s

In the classical limitIn the classical limitClassical Liouville equation for the expanding regionClassical Liouville equation for the expanding region

bbe222 4

Line element expands Line element expands asas

tltl cosh)0()(

ˆ)4( 2/12 b

Physical time t and the conformal time

)cos(/1)cosh( t

dedt b )(

InflationInflation

Homogeneous solution Homogeneous solution ( 00=const.=const. )

)cos( 0

0)(

b

bb

e

ee

4/3

ˆ2a

N.B. Our definition of the physical time coincide with this physical time.

The boundaryThe boundary two point correlation wo point correlation functionfunction

2

1)}0(exp{)}(exp{

xx

Conformal theoryConformal theory predicts predicts

The boundary metric densityThe boundary metric densityexpexpbbxx

thethenumber of triangles shearing a boundary vertex number of triangles shearing a boundary vertex x+n-th neighborsx+n-th neighbors

IdentifyingIdentifying the the distancedistance

Geodesic distance DD = = Smallest number of links connecting two vertices

x ~ geodesic distance D

quantum+ ensemble averages

x +

1st neighbors

1b

Evolution of the correlation functionEvolution of the correlation function

dPfa ll )(cos)(

)(2

tL

D

)0()()( bxb eexf

Boundary 2-point functionBoundary 2-point function

L(t) = boundary length atboundary length at t

Angular power spectraAngular power spectra

Large angle correlationLarge angle correlation

Measured on one universe.

2

la

The power spectrum of the 2-point correlation function on a last scattering surface lss (S2) in S3 of D4

N.B. Normalized at l=10

(preliminary)

Future Problems :Future Problems :• Extension to the 4-dimensionExtension to the 4-dimension

• Inclusion of MatterInclusion of Matter

• Creation of Dynamical LawsCreation of Dynamical Laws