אוניברסיטת בן - גוריון Ram Brustein  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 with I. Ben-Dayan, S. de Alwis To appear ============ Based on: hep-th/ with S. de Alwis, P. Martens hep-th/ with S. de Alwis, E. Novak Models of Modular Inflation

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: 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 The overshoot problem hep-th/

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 == potential == kinetic == radiation

Example: trapped field == potential == kinetic == radiation

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  – wall thickness in space     Inflation   H > 1   > m p H 2 ~1/3    m p 2 2   m p “ Topological inflation”: inflation off a flat feature Guendelman, Vilenkin, Linde Enough inflation  V’’/V<1/50

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 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 Take the simplest Kahler potential and superpotential Always a good approximation when expanding in a small region (  < 1) For the purpose of finding local properties V can be treated as a polynomial

Design a maximum with small curvature with polynomial eqs. Needs to be tuned for inflation

Design a wide (symmetric) plateau with polynomial eqs. A simple solution: b 2 =0, b 4 =0, b 1 =1,b 3 =  /6, b 5 determined approximately by (*) (*) In practice creates two +y,-y

Designing flat features for inflation in SUGRA The potential is not sensitive to small changes in coefficients Including adding small higher order terms, inflation is indeed 1/100 of tuning away A numerical example: b 2 =0, b 4 =0, b 1 =1,b 3 =  /6, b 5 y 4 (y 2 +5) + y 2 +1=0  b 1 b 3 – 2(b 0 ) 2 Need 5 parameters: V’(0)=0,V(0)=1,V’’/V=  D T W(-y), D T W(+y) = 0

An example of a steep superpotential An example of Kahler potential Similar in spirit to the discussion of stabilization Designing flat features for inflation for string moduli: Why creating a flat feature is not so easy

extrema min: W T = 0 max: W TT = 0 distance  T Why creating a flat feature is not so easy (cont.) Example: 2 exponentials W T = 0  W TT = 0   For (a 2 -a 1 )<<a 1,a 2

Amount of tuning To get  ~ 1/100 need tuning of 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 Need N > K+1 (K=5  N=7!) unless linearly dependent Trick: compare exponentials to polynomials by expanding about T = T 2 Linear equations for the coefficients of

Numerical examples 7 (!) exponentials + tuning ~  (aT) 2 = UGLY

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 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 de Sitter phase  + p <<    const. Parametrize the deviation from constant H Or by the number of e-folds by the value of the field Inflation ends when  = 1

Models of inflation:Perturbations Spectrum of scalar perturbations Spectrum of tensor perturbations Tensor to scalar ratio (many definitions) r is determined by P T /P R, background cosmology, & other effects r ~ 10  (“current canonical” r =16  ) Spectral indices r = C 2 Tensor / C 2 Scalar (quadropole !?) 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/ See also Boubekeur & Lyth

Simple example:

The “minimal” model: – Quadratic maximum – End of inflation determined by higher order terms Minimal tuning  minimal inflation, N-efolds ~ 60  “largish scale of inflation” H/m p ~1/100 Sufficient inflation Qu. fluct. not too large For example:

The “minimal” model: – Quadratic maximum – End of inflation determined by higher order terms Unobservable!

 Expect for the whole class of models 1/2 < 25|V’’/V| < 1  WMAP Detecting any component of GW in the foreseeable future will rule out this whole class of models !

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