Inflation, String Theory,

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Inflation, String Theory,
and Origins of Symmetry

• Andrei Linde

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Contents

• Inflation as a theory of a harmonic oscillator
• Inflation and observations
• Inflation in supergravity
• String theory and cosmology
• Eternal inflation and string theory landscape
• Origins of symmetry moduli trapping

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Various models of inflation

• Starobinsky model Starobinsky 1979
• Old Inflation Guth 1981
• New Inflation A.L., Albrecht,
  Steinhardt 1982
• Chaotic Inflation A.L. 1983
• Hybrid Inflation A.L. 1991

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Inflation as a theory of a harmonic oscillator
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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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Phase portrait of chaotic inflation
Blue lines universe with scalar field only Red
lines universe with scalar field and
radiation Almost all trajectories approach
inflationary attractor, corresponding to the
overdamped oscillator Inflation ends when the
field becomes small, and the phase portrait
becomes similar to the one of the underdamped
harmonic oscillator
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Add a constant to the energy of a harmonic
oscillator - obtain 2 stages of inflation
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WMAP and the temperature of the sky
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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 microwave
background anisotropy
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Comparing different inflationary models

• Chaotic inflation can start in the smallest
  domain of size 10-33 cm with total mass Mp
  (less than a milligram) and entropy O(1)
• New inflation can start only in a domain with
  mass 6 orders of magnitude greater than Mp and
  entropy greater than 109
• Cyclic inflation can occur only in the domain of
  size greater than the size of the observable part
  of the universe, with mass gt 1055 g and entropy
  gt 1087

Solves flatnes, mass and entropy problem
Not very good with solving flatnes, mass and
entropy problem
Does not solve flatnes, mass and entropy problem
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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 legitimate 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 Consider a
potential of the 4d theory obtained by
compactification in string theory of type IIB
Here
is the dilaton field, and
describes volume of the compactified space
The potential with respect to these two fields is
very steep, they run down, and V vanishes
Giddings, Kachru and Polchinski 2001
The problem of the dilaton stabilization was
solved in 2001,
but the volume stabilization problem was most
difficult and was solved only recently (KKLT
construction)
Kachru, Kallosh, Linde, Trivedi 2003
Burgess, Kallosh, Quevedo, 2003
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Volume stabilization
Basic steps

• Warped geometry of the compactified space and
  nonperturbative effects
  AdS space (negative vacuum energy) with
  unbroken SUSY and stabilized volume
• Uplifting AdS space to a metastable dS space
  (positive vacuum energy) by adding anti-D3 brane
  (or D7 brane with fluxes)

AdS minimum
Metastable dS minimum
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Implications for dark energy
Two models The simplest and
the most advanced
The simplest model linear potential with a small
slope
Acceleration followed by collapse
A.L. 1986
The most advanced model metastable de Sitter
state in string theory
Acceleration followed by decay to 10d space
KKLT, 2003
None of these theories lead to the standard
picture of eternal acceleration
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Inflation with stabilized volume

• Use KKLT volume stabilization
• Kachru, Kallosh, Linde, Maldacena,
  McAllister, Trivedi 2003
• Introduce the inflaton field with the potential
  which is flat due to shift symmetry
• Break shift symmetry either due to superpotential
  or due to radiative corrections

Hsu, Kallosh , Prokushkin 2003 Koyama, Tachikawa,
Watari 2003 Firouzjahi, Tye 2003 Hsu, Kallosh
2004
Alternative approach Modifications of kinetic
terms in the strong coupling regime
Silverstein and Tong, 2003
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String inflation and shift symmetry
Hsu, Kallosh , Prokushkin 2003
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Why shift symmetry?
It is not just a requirement which is desirable
for inflation model builders, but, in a certain
class of string theories, it is an unavoidable
consequence of the mathematical structure of the
theory
Hsu, Kallosh, 2004
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String Theory Landscape
100
Perhaps 10 different vacua
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Landscape of eternal inflation
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Self-reproducing Inflationary Universe
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Finding the way in the landscape
Anthropic Principle Love it or hate it
but use it Vacua counting Know where
you can go Moduli trapping Live in
the most beautiful valleys
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Two possible regimes

• Resurrection From any dS minimum one can always
  jump back with probability e?S, and experience a
  stage of inflation
• Eternal youth A much greater fraction of the
  total volume is produced due to eternal jumps in
  dS space at large energy density, and subsequent
  tunneling followed by chaotic inflation

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Solutions for Stochastic Equations
Stationary probability distribution with a
maximum at some particular values of the fields
A.L., D. Linde, A Mezhlumian, 1994
A probability distribution which grows
exponentially faster for particular values of
parameters
Garcia-Bellido, A.L., 1994 Vilenkin, 1994
Problem ambiguity of probability measure
(comparing infinities)
Knowledge of the total number of different vacua
and their volume is insufficient
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From discretuum to continuum
Restricted versus unrestricted freedom
Do we have only a finite number of discrete
choices? This may be insufficient to fine-tune
the cosmological constant 10-120
Two possibilities
Nonperturbative quantum effects such as a
continuous dependence of the - vacuum energy
on the parameter
Slow roll regime, as in inflation (quintessence)
Dark Energy implies unrestricted freedom
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Beauty is Attractive
Kofman, A.L., Liu, McAllister, Maloney,
Silverstein hep-th/0403001
also Silverstein and Tong, hep-th/0310221

• Quantum effects lead to particle production which
  result in moduli trapping near enhanced symmetry
  points
• These effects are stronger near the points with
  greater symmetry, where many particles become
  massless
• This may explain why we live in a state with a
  large number of light particles and
  (spontaneously broken) symmetries

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Basic Idea
is related to the theory of
preheating after inflation
Kofman, A.L., Starobinsky 1997
Consider two interacting moduli with potential
It can be represented by two intersecting valleys
Suppose the field f moves to the right with
velocity . Can it create particles f ?
Nonadiabaticity condition
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How does this process occur?
Uncertainty relations imply that during this time
one can have particle production with momenta
Number density of produced particles
Each of these particles has energy gf (for
large f), so the energy density is
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When the field f passes the (red) nonadiabaticity
region near the point of enhanced symmetry, it
created particles ? with energy density
proportional to f. Therefore the rolling field
slows down and stops at the point when
Then the field falls down and reaches the
nonadiabaticity region again
V
f
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When the field passes the nonadiabaticity region
again, the number of particles ? (approximately)
doubles, and the potential becomes two times more
steep. As a result, the field becomes trapped at
distance that is two times smaller than before.
V
f
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Each time the field passes the point of extended
symmetry, the trapping distance decreases twice,
so the field exponentially rapidly falls to f
0. At this point both fields f and ? become
massless.
V
3
2
4
1
f
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Trapping of a real scalar field
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Interesting features of moduli trapping

• ESP with greater symmetry (with larger number of
  fields becoming massless at these points) are
  more attractive
• Symmetry may grow step by step
• Example

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Thus anthropic and statistical considerations are
supplemented by a dynamical selection mechanism,
which may help us to understand the origin of
symmetries in our world.
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Anthropic principle says that we can live only in
those parts of the universe where we can
survive
Moduli trapping is a dynamical mechanism which
may help us to find places where we can
live well