The Universe:The Universe: a sphere, a donut, or a a sphere, a donut, or a

fractal?fractal?

The Universe:The Universe: a sphere, a donut, or a a sphere, a donut, or a

fractal?fractal?

Andrei Linde

Andrei Linde

Contents:Contents: Contents:Contents:

From the Big Bang theory to Inflationary Cosmology and the theory of Dark Energy

Inflation as a theory of a harmonic oscillator Inflation in string theory Initial conditions for inflation Does our universe looks like a sphere or like a bagel? Eternal inflation and string theory landscape: From a

bagel to a fractal

From the Big Bang theory to Inflationary Cosmology and the theory of Dark Energy

Inflation as a theory of a harmonic oscillator Inflation in string theory Initial conditions for inflation Does our universe looks like a sphere or like a bagel? Eternal inflation and string theory landscape: From a

bagel to a fractal

Two major cosmological discoveries:Two major cosmological discoveries: Two major cosmological discoveries:Two major cosmological discoveries:

The new-born universe experienced rapid acceleration (inflation)

A new (slow) stage of acceleration started 5 billion years ago (dark energy)

How did it start, and how it is going to end?How did it start, and how it is going to end? How did it start, and how it is going to end?How did it start, and how it is going to end?

Closed, open or flat universeClosed, open or flat universe Closed, open or flat universeClosed, open or flat universe

Big Bang TheoryBig Bang Theory Big Bang TheoryBig Bang Theory

Inflationary UniverseInflationary Universe Inflationary UniverseInflationary Universe

Why do we need inflation?Why do we need inflation? Why do we need inflation?Why do we need inflation?

What was before the Big Bang?

Why is our universe so homogeneoushomogeneous (better than 1 part in 10000) ?

Why is it isotropicisotropic (the same in all directions)?

Why all of its parts started expanding simultaneously?

Why it is flatflat? Why parallel lines do not intersect? Why it contains so many particles? Why there are so many people in this auditorium?

What was before the Big Bang?

Why is our universe so homogeneoushomogeneous (better than 1 part in 10000) ?

Why is it isotropicisotropic (the same in all directions)?

Why all of its parts started expanding simultaneously?

Why it is flatflat? Why parallel lines do not intersect? Why it contains so many particles? Why there are so many people in this auditorium?

Problems of the standard Big Bang theory:Problems of the standard Big Bang theory:Problems of the standard Big Bang theory:Problems of the standard Big Bang theory:

Inflation as a theory of a harmonic oscillatorInflation as a theory of a harmonic oscillator Inflation as a theory of a harmonic oscillatorInflation as a theory of a harmonic oscillator

Eternal InflationEternal Inflation

Einstein:

Klein-Gordon:

Einstein:

Klein-Gordon:

Equations of motion:Equations of motion:Equations of motion:Equations of motion:

Compare with equation for the harmonic oscillator with friction:

Logic of Inflation:Logic of Inflation: Logic of Inflation:Logic of Inflation:

Large φLarge φ large H large H large friction large friction

field φ moves very slowly, so that its potential energy for a long time remains nearly constantfield φ moves very slowly, so that its potential energy for a long time remains nearly constant

No need for false vacuum, supercooling, phase transitions, etc.No need for false vacuum, supercooling, phase transitions, etc.

Inflation makes the universe flat, Inflation makes the universe flat, homogeneous and isotropichomogeneous and isotropic

Inflation makes the universe flat, Inflation makes the universe flat, homogeneous and isotropichomogeneous and isotropic

In this simple model the universe typically grows 101000000000000 times during inflation.

Now we can see just a tiny part of the universe of size ct = 1010 light yrs. That is why the universe looks homogeneous, isotropic, and flat.

Generation of Quantum FluctuationsGeneration of Quantum Fluctuations Generation of Quantum FluctuationsGeneration of Quantum Fluctuations

QuickTime™ and a decompressor

are needed to see this picture.

WMAP and the temperature of the skyWMAP and the temperature of the sky WMAP and the temperature of the skyWMAP and the temperature of the sky

Name RecognitionName RecognitionName RecognitionName Recognition

Stephen Hawking

A photographic A photographic image of quantum image of quantum

fluctuations blown up fluctuations blown up to the size of the to the size of the

universeuniverse

A photographic A photographic image of quantum image of quantum

fluctuations blown up fluctuations blown up to the size of the to the size of the

universeuniverse

WMAPWMAP

and spectrum of the cosmic microwave background anisotropy

WMAPWMAP

and spectrum of the cosmic microwave background anisotropy

Add a constant to the inflationary potential

- obtain inflation and acceleration

Add a constant to the inflationary potential

- obtain inflation and acceleration

inflation

acceleration

Predictions of Inflation:Predictions of Inflation:Predictions of Inflation:Predictions of Inflation:

1) The universe should be homogeneous, isotropic and flat, = 1 + O(10-4) [

Observations: the universe is homogeneous, isotropic

and flat, = 1 + O(10-2)

• Inflationary perturbations should be gaussian and adiabatic, with flat spectrum, ns = 1+ O(10-1)

Observations: perturbations are gaussian and adiabatic,

with flat spectrum, ns = 1 + O(10-2)

Chaotic inflation in supergravityChaotic inflation in supergravity Chaotic inflation in supergravityChaotic inflation in supergravity

Main problem:

Canonical Kahler potential is

Therefore the potential blows up at large |φ|, and slow-roll inflation is impossible:

Too steep, no inflation…

A solution:A solution: shift symmetryshift symmetry A solution:A solution: shift symmetryshift symmetryKawasaki, Yamaguchi, Yanagida 2000

Equally good Kahler potential

and superpotential

The potential is very curved with respect to X and Re φ, so these fields vanish.

But Kahler potential does not depend on

The potential of this field has the simplest form, without any exponential terms:

Inflation in String TheoryInflation in String Theory Inflation in String TheoryInflation in String TheoryThe volume stabilization problem:

A potential of the theory obtained by compactification in string theory of type IIB:

The volume stabilization problem:

A potential of the theory obtained by compactification in string theory of type IIB:

The potential with respect to X and Y is very steep, these fields rapidly run down, and the potential energy V vanishes. We must stabilize these fields.

Volume stabilization: KKLT construction Kachru, Kallosh, A.L., Trivedi 2003Kachru, Kallosh, A.L., Trivedi 2003Burgess, Kallosh, Quevedo, 2003Burgess, Kallosh, Quevedo, 2003

Maloney, Silverstein, Strominger, in non-critical string theory

X and Y are canonically normalized field corresponding to the dilaton field and to the volume of the compactified space; is the field driving inflation

Dilaton stabilization: Giddings, Kachru, Polchinski 2001Giddings, Kachru, Polchinski 2001

Volume stabilizationVolume stabilization Volume stabilizationVolume stabilization

Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:

AdS minimumAdS minimum Metastable dS minimumMetastable dS minimum

Kachru, Kallosh, A.L., Trivedi 2003Kachru, Kallosh, A.L., Trivedi 2003

1) Start with a theory with runaway potential discussed above

2) Bend this potential down due to (nonperturbative) quantum effects

3) Uplift the minimum to the state with positive vacuum energy by adding a positive energy of an anti-D3 brane in warped Calabi-Yau space

The results:The results:The results:The results:

It seems possible to stabilize internal dimensions, and to obtain an accelerating universe. Eventually, our part of the universe will decay and become ten-dimensional, but it will only happen in 1010120 years

Apparently, vacuum stabilization can be achieved in 10100 - 101000 different ways. This means that the potential energy V of string theory may have 10100 - 101000 minima where we (or somebody else) can enjoy life

String Theory LandscapeString Theory Landscape String Theory LandscapeString Theory Landscape

Perhaps 10Perhaps 10100100 - 10 - 1010001000 different minimadifferent minima

Perhaps 10Perhaps 10100100 - 10 - 1010001000 different minimadifferent minima

Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…

Lerche, Lust, Schellekens 1987Lerche, Lust, Schellekens 1987 Lerche, Lust, Schellekens 1987Lerche, Lust, Schellekens 1987

Inflation in string theoryInflation in string theory Inflation in string theoryInflation in string theory

KKLMMT brane-anti-brane inflation

Racetrack modular inflation

D3/D7 brane inflation

DBI inflation

Example: Racetrack InflationExample: Racetrack Inflation Example: Racetrack InflationExample: Racetrack Inflation

waterfall from the saddle point

waterfall from the saddle point

Many versions of stringy inflation (KKLMMT, D3/D7) are similar to hybrid inflation. In such models inflation ends with a “waterfall,” which may result in production of cosmic strings. Gravitational waves produced by such strings may serve as a unique source of information about string theory

Tye et al 2002, KKLMMT 2003, Polchinski et al 2004

The height of the KKLT barrier is smaller than |VAdS| =m23/2. The

inflationary potential Vinfl cannot be much higher than the height of the barrier. Inflationary Hubble constant is given by H2 = Vinfl/3 < m2

3/2.

Constraint on the Hubble constant in this class of models:

H < m3/2 H < m3/2

V

VAdS

Modification of V at large H

STRING COSMOLOGY AND GRAVITINO MASSSTRING COSMOLOGY AND GRAVITINO MASSSTRING COSMOLOGY AND GRAVITINO MASSSTRING COSMOLOGY AND GRAVITINO MASS

In the AdS minimum in the KKLT construction

Therefore

A new class of KKLT modelsA new class of KKLT models A new class of KKLT modelsA new class of KKLT modelsKallosh, A.L. hep-th/0411011

Small mass of gravitino, no correlation with the height of the barrier Small mass of gravitino, no correlation with the height of the barrier and with the Hubble constant during inflationand with the Hubble constant during inflation

Inflation in the new class of KKLT models can occur at H >> m

3/2

One can obtain a supersymmetric Minkowski vacuum without any uplifting of the potential

One of the problem with string inflation is that inflation in such models starts relatively late. A typical closed universe will collapse before inflation begins. Open or flat universes would not collapse, but they are infinite, it is hard to make them...

Can we create a finite flat universe?

Take a box (a part of a flat universe) and glue its opposite sides to each other. What we obtain is a

torus, which is a topologically nontrivial flat universe.

Yes we can!Yes we can!Yes we can!Yes we can!

The size of the torus (our universe) grows as t1/2, whereas the mean free path of a relativistic particle grows much faster, as t

Therefore until the beginning of inflation the universe remains smaller that the size of the horizon t

If the universe initially had a Planckian size (the smallest possible size), then within the cosmological time t >> 1 (in Planck units) particles run around the torus many times and appear in all parts of the universe with equal probability, which makes the universe homogeneous and keeps it homogeneous until the beginning of inflation

Zeldovich, Starobinsky 1984; Cornish, Starkman, Spergel 1996; A.L. hep-th/0408164

Closed versus compact flat universe Closed versus compact flat universe

in quantum cosmologyin quantum cosmologyClosed versus compact flat universe Closed versus compact flat universe

in quantum cosmologyin quantum cosmology

Closed universe

Wave function is exponentially

suppressed at large scale factor a

Compact flat universe

Wave function is not exponentially suppressed

tunneling

Creation of a closed inflationary universe, and of

an infinite flat or open universe is exponentially less probable than creation of a compact topologically nontrivial flat or open universe

Spheres are expensive, bagels are freeSpheres are expensive, bagels are freeSpheres are expensive, bagels are freeSpheres are expensive, bagels are free

This generalizes the standard Kaluza-Klein idea that some spatial dimensions are compactified. Now it seems likely that all spatial dimensions are compactified. Some of them remain small (KKLT mechanism), whereas some other dimensions become large due to inflation

This does not necessarily mean that our This does not necessarily mean that our universe looks like a torus.universe looks like a torus. Inflation in string theory is always eternal, due to large number of metastable dS vacua (string theory landscape).

The new-born universe typically looks like a bagel, but the grown-up universe looks like an eternally growing fractal.

Self-reproducing Inflationary UniverseSelf-reproducing Inflationary Universe Self-reproducing Inflationary UniverseSelf-reproducing Inflationary Universe

Populating the LandscapePopulating the Landscape Populating the LandscapePopulating the Landscape

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Landscape of eternal inflation Landscape of eternal inflation