Inflationary multiverse and string theory Andrei Linde

Contents: Inflation as a theory of a harmonic oscillator Inflation in string theory Vacuum stabilization and the gravitino mass 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

Inflation as a theory of a harmonic oscillator Eternal Inflation

Equations of motion: Einstein: Klein-Gordon: 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 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 Fluctuations

WMAP and the temperature of the sky

Name Recognition Stephen Hawking

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:

The volume stabilization problem: Inflation in String Theory 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 Basic steps of the KKLT scenario: 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

3H2 = VINFLATION < VBARRIER ~ VAdS/3 = m3/2 Inflation and the gravitino mass V = + D/3 VAdS = - 3 eK |W|2 m3/2 == eK |W|2 =VAdS/3 2 3H2 = VINFLATION < VBARRIER ~ VAdS/3 = m3/2 2

Inflationary Unplifting Vinfl < 3Vbarrier ~ 3VAdS ~ m23/2 minflaton << H ~ m3/2

Perhaps 10100 - 101000 different minima String Theory Landscape Perhaps 10100 - 101000 different minima Lerche, Lust, Schellekens 1987 Bousso, Polchinski; Susskind; Douglas, Denef,…

Example: Racetrack Inflation Vbarrier = Vinflation waterfall from the saddle point

A new class of KKLT models 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

One of the problems 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.

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 in quantum cosmology tunneling Closed universe Wave function is exponentially suppressed at large scale factor a Compact flat universe Wave function is not exponentially suppressed

Spheres are expensive, bagels are free 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.

What is good in 4D may be bad in 10D Consider creation of a 3D box (torus) with a large volume modulus field  From the 4D point of view, we create a universe which may have Planck size in all directions when created at the Planck density - should be easy to do it! 3D 3D 6D Meanwhile from the 10D perspective, this box at the Planck time has a large size in 6 directions, corresponding to the large volume modulus  This means that the universe consists of MANY independent Planck size domains which should have similar properties; difficult to create. This is a new version of the horizon problem: The probability must be exponentially suppressed.

We want to start with the universe which has a Planckian (or stringy) size in all 9 space directions. But this implies starting at small , which may lead to overshooting, and decompactification: V 

Second uplifting in D3/D7 model

Inflationary potential at as a function of S and Shift symmetry is broken only by quantum effects

Potential of hybrid inflation with a stabilized volume modulus Unlike in the brane-antibrane scenario, inflation in D3/D7 model does not require fine-tuning

Is D3/D7 inflation eternal? Condition for a slow-roll eternal inflation: For D3/D7 model it implies that the inflaton field S should be greater than 104 g. But S describes the distance between the branes; it must be much smaller than the CY size. Thus eternal inflation is possible only for very small g. One may improve the situation by increasing V without changing V’. This requires an additional uplifting. The idea is that by changing fluxes one can increase V without altering the flatness of the potential, which remains protected by shift symmetry.

The resulting scenario: 1) The universe eternally jumps from one dS vacuum to another. Each bubble containing a new dS vacuum from the point of view of an internal observer is an infinite open dS universe. This process definitely occurs, but the bubbles contain no particles unless this process ends by a stage of a slow-roll inflation. Here is how: 2) At some stage the universe is in dS state with large V but with the flat D3/D7 direction. Quantum fluctuations during eternal inflation in this state push the inflaton field S in all directions along the D3/D7 valley. 3) Eventually, this state decays, and a bubble is produced which is an open inflationary universe containing all possible values of the inflaton field S in its different parts. D3/D7 slow-roll inflation begins and makes this universe flat. It produces particles, galaxies, and the participants of this conference.

Self-reproducing Inflationary Universe

Landscape of eternal inflation

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