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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How did it start, and how it is going to end?
Problems of the standard Big Bang theory:
Compare with equation for the harmonic oscillator with 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.
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.
A photographic image of quantum fluctuations blown up to the size of the universe
WMAP to the size of the universe
and spectrum of the cosmic microwave background anisotropy
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)
Canonical Kahler potentialis
Therefore the potential blows up at large |φ|, and slow-roll inflation is impossible:
Too steep, no inflation…
Kawasaki, Yamaguchi, Yanagida 2000
Equally good Kahler potential
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 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
Kachru, Kallosh, A.L., Trivedi 2003
Volume stabilization: KKLT construction
Burgess, Kallosh, Quevedo, 2003
Maloney, Silverstein, Strominger, in non-critical string theory
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
Metastable dS minimum
String Theory Landscape to the size of the universe
Perhaps 10100 - 101000 different minima
Lerche, Lust, Schellekens 1987
Bousso, Polchinski; Susskind; Douglas, Denef,…
KKLMMT brane-anti-brane inflation
D3/D7 brane inflation
Racetrack modular 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
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.
Modification of V at large H
Constraint on the Hubble constant in this class of models:
H < m3/2
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
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 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.
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
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
Wave function is exponentially suppressed at large scale factora
Compact flat universe
Wave function is not exponentially suppressed
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. 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.
Landscape of eternal inflation a torus.