Cosmology from matrix model
formulation of superstring theory
Jun Nishimura (KEK & SOKENDAI)
Invited talk at Tsukuba Global Science Week 2018
Tsukuba International Congress Center, Tsukuba, Japan
September 21, 2018

Big Bang cosmology

Our universe is expanding since it was bornas an invisibly tiny point 13.7 billion years ago.

3 evidences for standard Big Bang cosmology• Discovery of cosmic expansion• Theory of nucleosynthesis• Discovery of Cosmic Microwave Background

HubbleGamov(Hubble 1929)(Alpher-Bethe-Gamov 1948)(Penzias-Wilson 1965)

Penzias, Wilson

History of our UniverseHot
Birth of ourUniverse

110
sec. sec.

1000
sec.

0.3 million
years

nucleosynthesis
neutrinos decouplingfrom thermal equilibrium

generation ofdensity fluctuation

structureformation

recombinationinflation
pair annihilation ofelectrons and positronstransparent
to radiation

present
time

growth ofdensity fluctuation
CosmicMicrowave
BackgroundUnknown Region !
established by standard cosmological modelSingularities (space-time curvature diverges)
Black hole

Big bang

singularity (curvature diverges)
General Relativity becomes invalid!
（Quantum effects become non-negligible.）

We need to go beyond Einstein !

Quantum Cosmology• “Creation of Universes from nothing” Vilenkin (’82, ’84)
tunneling effects discussed with imaginary time

• “Wave function of the Universe” Hartle-Hawking (’83)“no boundary” proposal in path-integral formulation
The problem of UV divergences in quantum gravity
was ignored by restriction to a uniform isotropic universe.

a consistent theory of quantum gravity : superstring theory
We investigate the beginning of our Universe
using its matrix formulation.

Plan of the talk
0.
1.
2.
3.
4.
5.

IntroductionSuperstring theory as a consistent theoryof quantum gravity
Matrix Model for superstring theory
The beginning of our Universe
Expanding behaviors in simplified models
Summary1. Superstring theory as a consistenttheory of quantum gravity

Superstring theory

1974 Sherk-Schwarz, Yoneya
1984 Green-Schwarz

Quantum gravity is non-renormalizable due to severe UV divergences.
This problem can be solved by describing particles byextended objects like strings !
photongluonetc.
graviton,in particular
Various vibration modes correspond to various particles.

Unified description of 4 forces including gravity

The goals of superstring theory
particle physics• space-time dimensionality puzzle
critical dimension is (9+1), but we live in (3+1)d• particle contentsgauge group :matter contents : 3 generations+ Higgs• coupling constants in the Standard Modelcosmology• the birth of our Universe and “inflation”• the fate of our Universe(dark energy, cosmological constant problem)• the interior structure of a black holeA big obstacle:
non-perturbative definition is not yet established !
c.f.) quark confinement in QCD

non-perturbativecalculations

lattice gauge theory (Wilson, 1974)

perturbativecalculations

Compactification• Superstring theory is naturally defined in (9+1)dim.
unitarity + Lorentz invariance• (3+1)-dimensional space-time is expected to appear due tosome non-perturbative dynamics.
not well understood yet.
• Search for perturbative vacua with compactified 6d.
good : One can obtain Standard Model-like models.
bad : Too many vacua. (“Landscape”)
Understanding the non-perturbative dynamics of compactificationis crucial to understand our real world !

2. Matrix model for superstring theory

Matrix model for superstring theory in 10d• IKKT matrix model

Ishibashi-Kawai-Kitazawa-Tsuchiya (’97)

proposed as a non-perturbative formulation of superstring theory

raised and lowered by the metric

The action has manifest SO(9,1) symmetry and maximal SUSY.

Connection to the perturbative formulation• worldsheet theory (1st quantization)Natural realization of 2nd quantization

Each of these blocks

disconnected worldsheet

many-body states of strings naturally included (2nd quantization !)c.f.) Schwinger-Dyson equations in the IKKT model Hamiltonian of String Field Theory Fukuma-Kawai-Kitazawa-Tsuchiya (’98)

3. The beginning of our Universe

Kim-J.N.-Tsuchiya, PRL 108 (2012) 011601

The definition of the partition function
connection to the worldsheet theory

pure phase factor

polynomial in A

We have to regularize the model by introducing IR cutoffs.

opposite signs !

How to extract the real-time evolution
from the matrix configurations

averagesmall

We observeband-diagonal structure.
small

represents the state
at fixed time t.

Determination of the block size

We take.

Spontaneous breaking of SO(9)represent the extent of space-time in each direction.
SSB“critical time”

Mechanism of SSB (a simple example)

Lagrange multipliersolution :representation matrices of a compact semi-simple Lie algebrawith d generators

d=3 !

Examples of spontaneous symmetry breaking
in theoretical physics
 superconductor (Bardeen-Cooper-Schrieffer theory)
SSB of U(1) sym. due to formation of Cooper pairs
 the origin of constituent quark mass in QCD (Nambu)
SSB of chiral sym. due to formation of chiral condensate
pions as Nambu-Goldstone bosons
Likewise, we may say :The origin of our Universe :SSB of SO(9,1) in the IKKT matrix model

4. Expanding behaviors in simplified models

Ito, Kim, Koizuka, J.N., Tsuchiya, PTEP 2014 (2014) no.8, 083B01
Ito, J.N., Tsuchiya JHEP 1511 (2015) 070

Exponential expansion at early times
results of a simplified model for early time behaviors
Ito, Kim, Koizuka, J.N., Tsuchiya
PTEP 2014 (2014) no.8, 083B01

12

2R (t)/R (tc)-C

10

exponential expansion

B=35, E= 7, z=3.56
B=29, E= 8, z=3.24
B=24, E= 9, z=3.18
B=20, E=10, z=3.16
B=16, E=11, z=3.20
B=12, E=12, z=3.41
original(N=64, z=1.00)

0.1

reminiscent of thecosmic inflation

0.01
-0.5

00.5
1
1.5(t-tc)/(z R(tc))

22.5

Note: No inflaton, no potential
introduced by hand.

Power law expansion at late times
results of a simplified model for late time behaviors
Ito, J.N., Tsuchiya JHEP 1511 (2015) 070

35

R2(t)/R2(tc)

30
25

N=128
N=256
N=384
N=512

power law expansion

20reminiscent of the FriedmannRobertson-Walker Universe in
the radiation dominated era

1510𝑡𝑐50-0.500.5

11.5(t-tc)/R(tc)

22.5

3

The expected scenario
for the full IKKT matrix model

FRW universein the radiation dominated era

inflation

The value of E-folding
may be determined dynamically !

(Big Bang)

5. Summary and future prospects

Summary Superstring theory

 strings naturally solve the problem ofsevere UV divergences in quantum gravity
 unified theory of all particles (both forces and matters)
 however, too many vacua (“landscape”)
due to variety of compactifications from 10d to 4d
 fully non-perturbative formulation is crucial
 Matrix model formulation

 analogous to lattice gauge theory for QCD IKKT model : non-perturbative formulation of superstrings
 Monte Carlo simulation revealed
SSB of SO(9) down to SO(3) at some critical time.
 expanding behavior in simplified models
exponential/power-law

Future prospects
 Do Standard Model particles (gauge bosons, quarks, leptons, Higgs)
appear at later times ?

Monte Carlo simulation &studies of classical solutions (+ quantum corrections)Classical eq. of motion becomes valid at late times
due to the expansion of space.

We hope the investigations of the IKKT matrix model
will provide a new perspective onparticle physics beyond the standard model (incl. dark matter)cosmological models for inflation, dark energy, etc..