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KEK-WS 03/14/2007. Evolution of Simplicial Universe Shinichi HORATA and Tetsuyuki YUKAWA Hayama center for Advanced Studies, Sokendai Hayama, Miura, Kanagawa 240-0193, Japan. Some of the topics have been already appeared in S.Horata, and T.Yukawa : Making a Universe.hep-th/0611076. - PowerPoint PPT Presentation
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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: [email protected]
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