Introduction: The case for

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Models of Modular Inflation
with I. Ben-Dayan, S. de Alwis To
appear Based on hep-th/0408160 with
S. de Alwis, P. Martens hep-th/0205042 with S.
de Alwis, E. Novak

• Introduction The case for
• closed string moduli as inflatons
• Cosmological stabilization of moduli
• Designing inflationary potentials (SUGRA, moduli)
• The CMB as a probe of model parameters

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Stabilizing closed string moduli

• Any attempt to create a deSitter like phase will
  induce a potential for moduli
• A competition on converting potential energy to
  kinetic energy, moduli win, and block any form
  of inflation

Inflation is only 1/100 worth of tuning
away! (but see later)
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Generic properties of moduli potentials

• The landscape allows fine tuning
• Outer region stabilization possible
• Small ve or vanishing CC possible
• Steep potentials
• Runaway potentials towards decompactification/weak
  coupling
• A mini landscape near every stable mininum
  additional spurious minima and saddles

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Cosmological stability

• The overshoot problem

hep-th/0408160
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Proposed resolutionrole of other sources

• The 3 phases of evolution
• Potential push jump
• Kinetic glide
• Radiation/other sources
• parachute opens

Previously Barreiro et al tracking Huey et al,
specific temp. couplings
Inflation is only 1/100 worth of tuning away!
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Example different phases
potential kinetic radiation
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Example trapped field
potential kinetic radiation
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Using cosmological stabilization for designing
models of inflation

• Allows Inflation far from final resting place
• Allows outer region stabilization
• Helps inflation from features near
• the final resting place

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(My) preferred models of inflation small field
models

• Topological inflation inflation off a flat
  feature

Guendelman, Vilenkin, Linde
Enough inflation ? V/Vlt1/50
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Results and Conclusions preview

• Possible to design fine-tuned models in SUGRA and
  for string moduli
• Small field models strongly favored
• Outer region models strongly disfavored
• Specific small field models
• Minimal number of e-folds
• Negligible amount of gravity waves all models
  ruled out if any detected in the foreseeable
  future
• ? Predictions for future CMB experiments

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Designing flat features for inflation

• Can be done in SUGRA
• Can be done with steep exponentials alone
• Can (??) be done with additional (???)
  ingredients (adding Dbar, const. to potential see
  however .. )
• Lots of fine tuning, not very satisfactory
• Amount of tuning reduces significantly towards
  the central region

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Designing flat features for inflation in SUGRA
Take the simplest Kahler potential and
superpotential
Always a good approximation when expanding in a
small region (f lt 1)
For the purpose of finding local properties V can
be treated as a polynomial
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Design a maximum with small curvature with
polynomial eqs.
Needs to be tuned for inflation
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Design a wide (symmetric) plateau with
polynomial eqs.
In practice creates two minima _at_ y,-y
()
A simple solution b20, b40, b11,b3h/6,
b5 determined approximately by ()
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Designing flat features for inflation in SUGRA
A numerical example
The potential is not sensitive to small changes
in coefficients Including adding small higher
order terms, inflation is indeed 1/100 of tuning
away
b20, b40, b11,b3h/6,
Need 5 parameters V(0)0,V(0)1,V/Vh DTW(-y),
DTW(y) 0
b5 y4(y25) y210
h 6 b1 b3 2(b0)2
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Designing flat features for inflation for string
moduli Why creating a flat feature is not so easy

• An example of a steep superpotential

• An example of Kahler potential

Similar in spirit to the discussion of
stabilization
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Why creating a flat feature is not so easy (cont.)

• extrema
• min WT 0
• max WTT 0
• distance DT

Example 2 exponentials WT 0 ? WTT 0 ?
?
For (a2-a1)ltlta1,a2
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Amount of tuning
For (a2-a1)ltlta1,a2
To get h 1/100 need tuning of coefficients _at_
1/100 x 1/(aT)2 The closer the maximum is to the
central area the less tuning. Recall we need to
tune at least 5 parameters
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Designing flat features with exponential
superpotentials
Trick compare exponentials to polynomials by
expanding about T T2
Linear equations for the coefficients of
Need N gt K1 (K5?N7!) unless linearly dependent
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Numerical examples
7 (!) exponentials tuning h (aT)2 UGLY
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Lessons for models of inflation

• Push inflationary region towards the central
  region
• Consequences
• High scale for inflation
• Higher order terms are important, not simply
  quadratic maximum

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Phenomenological consequences

• Push inflationary region towards the central
  region
• Consequences
• High scale for inflation
• Higher order terms are important, not simply
  quadratic maximum

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Models of inflation Background
de Sitter phase r p ltlt r ? H const.
Parametrize the deviation from constant H
by the value of the field
Or by the number of e-folds
Inflation ends when e 1
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Models of inflationPerturbations

• Spectrum of scalar perturbations
• Spectrum of tensor perturbations

Spectral indices
r C2Tensor/ C2Scalar (quadropole !?)
Tensor to scalar ratio (many definitions) r is
determined by PT/PR , background cosmology,
other effects r 10 e (current canonical r
16 e)
CMB observables determined by quantities 50
efolds before the end of inflation
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Wmapping Inflationary Physics W. H. Kinney, E.
W. Kolb, A. Melchiorri, A. Riotto,hep-ph/0305130
See also Boubekeur Lyth
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Simple example
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• The minimal model
• Quadratic maximum
• End of inflation determined by higher order
  terms

For example
Sufficient inflation
Qu. fluct. not too large
Minimal tuning ? minimal inflation, N-efolds
60 ? largish scale of inflation H/mp1/100
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• The minimal model
• Quadratic maximum
• End of inflation determined by higher order
  terms

Unobservable!
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Expect for the whole class of models
??
Detecting any component of GW in the foreseeable
future will rule out this whole class of models !
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Summary and Conclusions

• Stabilization of closed string moduli is key
• Inflation likely to occur near the central region
• Will be hard to find a specific string
  realization
• Specific class of small field models
• Specific predictions for future CMB experiments