EXPANDING UNIVERSE FROM
THE LORENTZIAN MATRIX MODEL
SANG-WOO KIM (OSAKA U., A01)
QUCS2012 @ NARABASED ON 1108. 1540 (PRL 108 011601,2012)
1110.4803 (PRD86 027901,2012)
1208.0711 (JHEP 1210 147,2012)
BY SWK, JUN NISHIMURA, ASATO TSUCHIYA

Main spirit of lattice is
1st principle calculation
with numerical method

Nonperturbative aspectsof QCD

 It is now very well estabilshed thanks to the efforts ofearly pioneers like Wilson.

Topic of this talk aims
1stprinciple calculation
with numerical method

Nonperturbative aspects,
low energy implication ofstring theory

 Not established yet. Rely on matrix model conjectures.
 This topic is an exploration for future directions
which is waiting for more courageous pioneers.

Our first target is cosmology,
where there are many interesting questions
Initial singularity problem
Spacetime dimensionality
Inflaton, reheating, …
CMB spectrum
Dark energy/matter , etc

along with related experiments/observations.
...

Many analytic works/insights so far
 Quantum cosmology based on Wheeler-DeWitt eq,
Vilenkin (’82), Hartle, Hawking (’83), …

 String gas cosmology,Brandenberger, Vafa (’89), …

 D-brane + Perturbative analysis, etc.
Herdeiro, Hirano, Kallosh (’01), …

What we obtained in our study isMonte Carlo study in Lorentzian signature.
Spontaneous breaking of rotational symmetry : SO(9)Expanding 3d spaces emerge after SSB.
Expansion is consistent with exponential behavior.

Matrix Model (review)Matrix Model Our starting point is type IIB matrix model in 0d.
Ishibashi, Kawai, Kitazawa, Tsuchiya (’96)

A nonperturbative formulation conjectured forsuperstring theory.
 cf. 1d Matrix Quantum Mechanics, 2d Matrix String Theory
Banks, Fischler, Shenker, Susskind (’96)

Dijkraaf, Verlinde, Verlinde (’97)

Symmetries

If eigenvalues of bosonic matrix = spacetime coordinate,Relation to string theory
 IIB matrix model can be considered as a worldsheet
regularization of Green-Schwarz action in Schild gauge.

 Wilson loop correlator is consistent with light-cone
string field theory. Fukuma, Kawai, Kitazawa, Tsuchiya (98)

Difficulties and problems
 Interpretation of matrix : coordinate , momentum, or else ?
 The action look like around 10d flat spacetime.
Can we describe curved geometry such as AdS ?
 Double scaling limit is not clear.
When we take large-N limit, how should we take coupling
constant as a function of N ?
 Emergence of 3d spacetimein Euclidean action.

d=3 has minimum
free energy
Nishimura, Okubo, Sugino (11)Lorentzian IIB Matrix ModelLorentzian IIB Matrix Model Let’s avoid Wick rotation to study real time evolution.
 Pfaffian is real.
 Noncompact temporal direction requires an IR cutoff.

 We want to study SSB of remaining SO(9) symmetryby Monte Carlo method.

Regularization of the model We can regularize oscillating phase bya) introduce damping term,b) insert identity,c) and integrate out scale withd) Need to introduce L :Lorentzian Matrix Model We study following model by Monte Carlo method : Note that

Lorentziannoncommutative : Lie algebraic, …
Euclidean
commutativeComparison with lattice regularization
Lattice

Lorentzian matrix modelAfter continuum limit (N) and infinite volume limit (L),only one parameter (kappa) remains.

Monte Carlo results

Time eigenvalues
Letbe eigenvalues of.

00Thanks to SUSY, their interaction cancels.

Band diagonal structureBand diagonal structure appears dynamically forin’s diagonal basis.
small

1/2

1|QIJ/QN/2,N/2|



small

(I+J)/2=2(I+J)/2=4(I+J)/2=6(I+J)/2=80.5

0.25

0.125

-16-12-8-40I-J

481216subblock matrix represents space structure at given time.

Results : SSB of SO(9) symmetryorder parameter :eigenvalues of Tij(t)0.5

9x9 real sym.
Critical time

0.4
0.3
0.2
0.1
0-2.5-2-1.5
t-1-0.50Mechanism of SSB



Can we understand SSB in the large kappa ?

Maximizewith

Mechanism of SSBLarge kappa for fixed N is described by

EOM isWith an Ansatz

Mechanism of SSB


We can solve this problem without any Ansatz
by Monte Carlo method.
12

eigenvalues

0.5

LL3
i[L1,L2]+L30-0.5-112345678n-th

2x2 representation of SU(2) algebra gives the maximum,
which explains 3d expanding spaces.

Continuum / Infinite volume limit

1052R(t) /R(tc)3622R(t) /R(tc)

κ=2.0
κ=4.0
κ=8.0

82

4

N=8
N=12
N=162

4

1

200-2-10(t-tc)/R(tc)12-4-3-2-101(t-tc)/R(tc)

Different parameter values of N & kappaseem to converge to a single curve.

234Summary and Discussion

Summary0-d matrix modelwith SO(9,1)

Unique time history
 SSB from 9d to 3d spaces Noncommutative mechanism

Two IR cutoffs

10



κ=2.0
κ=4.0
κ=8.0

SO(3)

62

R(t) /R(tc)284SO(9)2Early Universe

0-4-3-2-101(t-tc)/R(tc)234

What happens in late time ?

3d spaces expand due
to Lie algebra type NC.

10
κ=2.0
κ=4.0
κ=8.0

R(t)2/R(tc)286Will approach to one
of classical solutions
due to the expansion.

No singularity.
420-4-3-2-101(t-tc)/R(tc)234

Candidates for future
Classical EOMAnsatz based on
Lie algebra

We classified all possible solutions within the Ansatz. There aremany expanding commutative classical solutions.
Even with quite simple Ansatz, we can easily find classicalsolutions with power law or exponential behavior for expansion.
With these candidates for future, we are now ready toproceed Monte Carlo study.

Effective modelEffective model forwill appear if we integrate
over matrix elements corresponding to.

smallsmall

By changing, we can obtain a RG flow to late time.

Quenched model

Ito, SWK, Koizuka, Nishimura, Tsuchiya(a work in progress)

 Simple quenching is not enough : We include fermionic repulsion to temporal eigenvalues.

 Interesting properties such as SSB to 3d, expansion are kept.
 RG flow to late time is carefully studied in this model.
 But in late time,will become more important.

 However, we study this model first as a basic reference.

A preliminary idea
 We may consider a parameter extension of the modelin a way to mimic the late time flow.

SSB without expansion
3 expanding spaces
SO(9) symmetric?
3 expanding spaces
SO(9) symmetricBackup (Classical solution)

General prescription
 variational function

 classical equations of motion

 commutation relations
Eq. of motion & Jacobi identity
Lie algebra
Unitary representation

classical solution

Ansatz
Ignore extra dimension
Assume commutative spaceSimplification

Lie algebrae.g.)d=1 caseTake a direct sum

can be distributed on a unit S3(3+1)D space-time

R×S3

A complete classification of d=1 solutions has been done.
Below we only discuss a physically interesting solution.

SL(2,R) solution SL(2,R) solution

 realization of the SL(2,R) algebra on

Space-time structure primary unitary series representation

tri-diagonal

 3K×3K diagonal block

Space-time noncommutativity disappears
in the continuum limit.

Cosmological implications
 the extent of spacecont. lim.



Hubble constant and the w parameterradiation dominantmatter dominantcosmological constantt0 is identified with the present time.

This part is considered to givethe late-time behavior of the matri
x modelpresent accelerated expansion
cosmological const.
a solution to the cosmological constant problem
Cosmological constant disappears in the future.