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

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The Universe a sphere, a donut, or a fractal?

• Andrei Linde

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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

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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?
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Closed, open or flat universe
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Big Bang Theory
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Inflationary Universe
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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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Inflation as a theory of a harmonic oscillator
Eternal Inflation
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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 f
large H
large friction
field f 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.
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Generation of Quantum Fluctuations
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WMAP and the temperature of the sky
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Name Recognition
Stephen Hawking
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A photographic image of quantum fluctuations
blown up to the size of the universe
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WMAP and spectrum of the cosmic microwave
background anisotropy
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Add a constant to the inflationary potential -
obtain inflation and acceleration
acceleration
inflation
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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)

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Chaotic inflation in supergravity
Main problem
Canonical Kahler potential is
Therefore the potential blows up at large f,
and slow-roll inflation is impossible
Too steep, no inflation
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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 f, 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
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
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Volume stabilization
Kachru, Kallosh, A.L., Trivedi 2003
Basic steps of the KKLT scenario

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

AdS minimum
Metastable dS minimum
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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

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String Theory Landscape
Perhaps 10100 - 101000 different minima
Lerche, Lust, Schellekens 1987
Bousso, Polchinski Susskind Douglas, Denef,
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Inflation in string theory
KKLMMT brane-anti-brane inflation
D3/D7 brane inflation
Racetrack modular inflation
DBI inflation
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Example Racetrack Inflation
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
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 lt m23/2.
V
Modification of V at large H
VAdS
Constraint on the Hubble constant in this class
of models
H lt m3/2
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In the AdS minimum in the KKLT construction
Therefore
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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 gtgt 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
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
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If the universe initially had a Planckian size
(the smallest possible size), then within the
cosmological time t gtgt 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
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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
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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.

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Self-reproducing Inflationary Universe
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Populating the Landscape
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Landscape of eternal inflation