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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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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 inflation:

-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 inflation:

  • 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 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 inflation:

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


Design a maximum with small curvature with polynomial eqs. inflation:

Needs to be tuned for inflation


Design a wide (symmetric) plateau with polynomial eqs. inflation:

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 inflation:

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


Designing flat features for inflation for string moduli: inflation: 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


Why creating a flat feature is not so easy (cont.) inflation:

  • extrema

    • min: WT = 0

    • max: WTT = 0

    • distance DT

Example: 2 exponentials

WT = 0 

WTT = 0 

For (a2-a1)<<a1,a2


Amount of tuning inflation:

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 inflation:

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 inflation:

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


Lessons for models of inflation
Lessons for models of inflation 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 inflation:

  • 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 inflation:

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 inflation:

  • 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 inflation: W. H. Kinney, E. W. Kolb, A. Melchiorri, A. Riotto,hep-ph/0305130

See also Boubekeur & Lyth


Simple example: inflation:


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

For example: inflation:

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: inflation:

Quadratic maximum

End of inflation determined by higher order terms

Unobservable!


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 inflation:

  • 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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