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

Closed, open or flat universe

Big Bang Theory

Inflationary Universe

Why do we need inflation? Why do we need inflation? What was before the Big Bang? homogeneous Why is our universe so homogeneous (better than 1 part in 10000) ? isotropic Why is it isotropic (the same in all directions)? Why all of its parts started expanding simultaneously? flat 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? What was before the Big Bang? homogeneous Why is our universe so homogeneous (better than 1 part in 10000) ? isotropic Why is it isotropic (the same in all directions)? Why all of its parts started expanding simultaneously? flat 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? Problems of the standard Big Bang theory:

Inflation as a theory of a harmonic oscillator Eternal Inflation

Einstein: Klein-Gordon: Einstein: Klein-Gordon: Equations of motion: Compare with equation for the harmonic oscillator with friction:

Logic of Inflation: Large φ large H large friction 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.

Inflation makes the universe flat, homogeneous and isotropic In this simple model the universe typically grows times during inflation. Now we can see just a tiny part of the universe of size ct = light yrs. That is why the universe looks homogeneous, isotropic, and flat.

Generation of Quantum Fluctuations Generation of Quantum Fluctuations

WMAP and the temperature of the sky

Name Recognition Stephen Hawking

A photographic image of quantum fluctuations blown up to the size of the universe

WMAP and spectrum of the cosmic microwave background anisotropyWMAP

Add a constant to the inflationary potential - obtain inflation and acceleration inflation acceleration

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 ) 2)Inflationary perturbations should be gaussian and adiabatic, with flat spectrum, n s = 1+ O(10 -1 ) Observations: perturbations are gaussian and adiabatic, with flat spectrum, n s = 1 + O(10 -2 )

Chaotic 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: shift 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:

Inflation in String Theory The 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 2003 Burgess, 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 2001

Volume stabilization Basic steps of the KKLT scenario: AdS minimum Metastable dS minimum Kachru, Kallosh, A.L., Trivedi ) 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: n 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 years n Apparently, vacuum stabilization can be achieved in different ways. This means that the potential energy V of string theory may have minima where we (or somebody else) can enjoy life

String Theory Landscape Perhaps different minima Bousso, Polchinski; Susskind; Douglas, Denef,… Lerche, Lust, Schellekens 1987

Inflation in string theory KKLMMT brane-anti-brane inflation Racetrack modular inflation D3/D7 brane inflation DBI inflation

Example: Racetrack Inflation 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 |V AdS | =m 2 3/2. The inflationary potential V infl cannot be much higher than the height of the barrier. Inflationary Hubble constant is given by H 2 = V infl /3 < m 2 3/2. Constraint on the Hubble constant in this class of models: H < m 3/2 V V AdS Modification of V at large H STRING COSMOLOGY AND GRAVITINO MASS

In the AdS minimum in the KKLT construction Therefore

A new class of KKLT models Kallosh, A.L. hep-th/ Small mass of gravitino, no correlation with the height of the barrier and 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!

The size of the torus (our universe) grows as t 1/2, 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/

Closed versus compact flat universe in 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 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 universe looks like a torus. 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.

Self-reproducing Inflationary Universe

Populating the Landscape Populating the Landscape

Landscape of eternal inflation