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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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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Add a constant to the inflationary potential -
obtain two stages of inflation
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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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A photographic image of quantum fluctuations
blown up to the size of the universe
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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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How important is the gravitational wave
contribution?
For these two theories the ordinary scalar
perturbations coincide
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Is the simplest chaotic inflation natural?

• Often repeated (but incorrect) argument

Thus one could expect that the theory is
ill-defined at
However, quantum corrections are in fact
proportional to
and to
These terms are harmless for sub-Planckian masses
and densities, even if the scalar field itself
is very large.
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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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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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The Potential of the Hybrid D3/D7 Inflation Model
is a hypermultiplet
is an FI triplet
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In many F and D-term models the contribution of
cosmic strings to CMB anisotropy is too large

• This problem disappears for very small coupling
  g
• Another solution is to add a new hypermultiplet,
  and a new global symmetry, which makes the
  strings semilocal and topologically unstable

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Semilocal Strings are Topologically Unstable

Achucarro, Borill, Liddle, 98
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D3/D7 with two hypers
Dasgupta, Hsu, R.K., A. L., Zagermann,
hep-th/0405247

• Detailed brane construction - D-term inflation
  dictionary

? Brane construction of generalized D-term
inflation models with additional global or local
symmetries due to extra branes and
hypermultiplets.
? Resolving the problem of cosmic string
production additional global symmetry, no
topologically stable strings, only semilocal
strings, no danger Confirmation of Urrestilla,
Achucarro, Davis Binetruy, Dvali, R. K.,Van
Proeyen
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Bringing it all together Double Uplifting
KKL, in progress
First uplifting KKLT
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Second uplifting in D3/D7 model
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Inflationary potential at as a
function of S and
Shift symmetry is broken only by quantum effects
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Potential of hybrid inflation with a stabilized
volume modulus
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For two hypers
Inflaton potential
Symmetry breaking potential
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Can we have eternal inflation in such models?
Yes, by combining these models with the ideas of
string theory landscape
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String Theory Landscape
100
Perhaps 10 different vacua
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de Sitter expansion in these vacua is eternal. It
creates quantum fluctuations along all possible
flat directions and provides necessary initial
conditions for the low-scale inflation
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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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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 c ?
Nonadiabaticity condition
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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 c (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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Trapping of a real scalar field
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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.