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The Universe: a sphere, a donut, or a fractal?. Andrei Linde. 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

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The universe a sphere a donut or a fractal l.jpg
The Universe:a sphere, a donut, or a fractal?

  • AndreiLinde


Contents l.jpg
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


Two major cosmological discoveries l.jpg
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?





Why do we need inflation l.jpg
Why do we need inflation?

Problems of the standard Big Bang theory:

  • What was before the Big Bang?

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

  • Why is it isotropic (the same in all directions)?

  • Why all of its parts started expanding simultaneously?

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



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Equations of motion:

  • Einstein:

  • Klein-Gordon:

Compare with equation for the harmonic oscillator with friction:


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Logic of Inflation:

Large φ

largeH

largefriction

field 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.


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Inflation makes the universe flat, homogeneous 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.




Name recognition l.jpg

Name Recognition

Stephen Hawking



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WMAP to the size of the universe

and spectrum of the cosmic microwave background anisotropy


Add a constant to the inflationary potential obtain inflation and acceleration l.jpg
Add a constant to the inflationary potential - obtain to the size of the universeinflationandacceleration

acceleration

inflation


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Predictions of Inflation: to the size of the universe

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)


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Chaotic inflation in supergravity to the size of the universe

Main problem:

Canonical Kahler potentialis

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

Too steep, no inflation…


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A solution: to the size of the universeshift symmetry

Kawasaki, 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:


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Inflation in String Theory to the size of the universe

The volume stabilization problem:

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

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

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.

Giddings, Kachru, Polchinski 2001

Dilaton stabilization:

Kachru, Kallosh, A.L., Trivedi 2003

Volume stabilization: KKLT construction

Burgess, Kallosh, Quevedo, 2003

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


Volume stabilization l.jpg
Volume stabilization to the size of the universe

Kachru, Kallosh, A.L., Trivedi 2003

Basic steps of the KKLT scenario:

Start with a theory with runaway potential discussed above

Bend this potential down due to (nonperturbative) quantum effects

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

AdS minimum

Metastable dS minimum


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The results: to the size of the universe

  • 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


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String Theory Landscape to the size of the universe

Perhaps 10100 - 101000 different minima

Lerche, Lust, Schellekens 1987

Bousso, Polchinski; Susskind; Douglas, Denef,…


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Inflation in string theory to the size of the universe

KKLMMT brane-anti-brane inflation

D3/D7 brane inflation

Racetrack modular inflation

DBI inflation


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Example: Racetrack Inflation to the size of the universe

waterfall from the saddle point


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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


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STRING COSMOLOGY AND GRAVITINO MASS 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

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 < m23/2.

V

Modification of V at large H

VAdS

Constraint on the Hubble constant in this class of models:

H < m3/2


Slide29 l.jpg

In the AdS minimum in the KKLT construction 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

Therefore


A new class of kklt models l.jpg
A new class of KKLT models 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

Kallosh, A.L. hep-th/0411011

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

Inflation in the new class of KKLT models can occur at H >> m3/2

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


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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?

Yes we can!

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.


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The size of the torus (our universe) grows as in such models starts relatively late. 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 horizont


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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 homogeneousand 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 in quantum cosmology l.jpg
Closed versus compact flat universe in quantum cosmology possible size), then within the cosmological time

tunneling

Closed universe

Wave function is exponentially suppressed at large scale factora

Compact flat universe

Wave function is not exponentially suppressed


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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 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


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This does not necessarily mean that our 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.