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Ram Brustein. אוניברסיטת בן-גוריון. 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

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slide1

Ram Brustein

אוניברסיטת בן-גוריון

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
stabilizing closed string moduli
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)
generic properties of moduli potentials
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
cosmological stability
Cosmological stability

hep-th/0408160

  • The overshoot problem
proposed resolution role of other sources

r

r

r

Proposed resolution:role 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!
example different phases
Example: different phases

== potential

== kinetic

== radiation

example trapped field
Example: trapped field

== potential

== kinetic

== radiation

using cosmological stabilization for designing models of inflation
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

my preferred models of inflation small field models

2

-1

f/mp

-4 -2 0 2 4

  • –wall thickness in space

(D/d)2 ~ L4 Inflation d H > 1 D > mp

H2~1/3 L4/mp2

D

(My) preferred models of inflation: small field models
  • “Topological inflation”: inflation off a flat feature

Guendelman, Vilenkin, Linde

Enough inflation V’’/V<1/50

results and conclusions preview
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

designing flat features for inflation
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
designing flat features for inflation in sugra
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 < 1)

For the purpose of finding local properties V can be treated as a polynomial

slide14

Design a wide (symmetric) plateau with polynomial eqs.

In practice creates two minima @ +y,-y

(*)

A simple solution: b2=0, b4=0, b1=1,b3=h/6,

b5 determined approximately by (*)

designing flat features for inflation in sugra1
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

b2=0, b4=0, b1=1,b3=h/6,

Need 5 parameters:

V’(0)=0,V(0)=1,V’’/V=h

DTW(-y), DTW(+y) = 0

b5 y4(y2+5) + y2+1=0

h =6 b1 b3 – 2(b0)2

slide16

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

slide17

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)<<a1,a2

slide18

Amount of tuning

For (a2-a1)<<a1,a2

To get h ~ 1/100 need tuning of coefficients

@ 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

designing flat features with exponential superpotentials
Designing flat features with exponential superpotentials

Trick: compare exponentials to polynomials by expanding about T = T2

Linear equations for the coefficients of

Need N > K+1 (K=5N=7!) unless linearly dependent

numerical examples
Numerical examples

7 (!) exponentials + tuning ~ h (aT)2 = UGLY

lessons for models of inflation
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
phenomenological consequences
Phenomenological consequences
  • Push inflationary region towards the central region
  • Consequences:
    • High scale for inflation
    • Higher order terms are important, not simply quadratic maximum
models of inflation background

Models of inflation: Background

de Sitter phase r + p << 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

models of inflation perturbations

Models of inflation:Perturbations

  • 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

wmapping inflationary physics w h kinney e w kolb a melchiorri a riotto hep ph 0305130
Wmapping Inflationary Physics W. H. Kinney, E. W. Kolb, A. Melchiorri, A. Riotto,hep-ph/0305130

See also Boubekeur & Lyth

the minimal model quadratic maximum end of inflation determined by higher order terms

For example:

The “minimal” model:

Quadratic maximum

End of inflation determined by higher order terms

Sufficient inflation

Qu. fluct. not too large

Minimal tuning  minimal inflation, N-efolds ~ 60

 “largish scale of inflation” H/mp~1/100

the minimal model quadratic maximum end of inflation determined by higher order terms1
The “minimal” model:

Quadratic maximum

End of inflation determined by higher order terms

Unobservable!

slide29

WMAP

1/2 < 25|V’’/V| < 1 

Expect for the whole class of models



Detecting any component of GW in the foreseeable future will rule out this whole class of models !

summary and conclusions
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
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