אוניברסיטת בן - גוריון Ram Brustein  Moduli space of effective theories of strings  Outer region of moduli space: problems!  “central” region:  stabilization  interesting cosmology PRL 87 (2001), hep-th/ PRD 64 (2001), hep-th/ hep-th/ hep-th/ with S. de Alwis, E. Novak Moduli stabilization, SUSY breaking and Cosmology

HO I IIB IIA HW MS1 HE String Theories and 11D SUGRA HW=11D SUGRA/I 1 MS1=11D SUGRA/S 1 T T S S “S” N=1 (10D) N=2 (10D) “S”

String Moduli Space HO I IIB IIA HW MS1 HE Requirements D=4 N=1 SUSY  N=0 CC<(m 3/2 ) 4 SM (will not discuss) Volume/Coupling moduli TS Central region “minimal computability” Outer region perturbative Perturbative theories = phenomenological disaster SUSY+msless moduli Gravity = Einstein’s Cosmology String universality ?

Cosmological moduli space

“Lifting Moduli” Perturbative –Compactifications –Brane Worlds Non-Perturbative –SNP = Brane instantons –Field-Theoretic, e.g., gaugino-condensation Generic Problems –P ractical C osmological C onstant P roblem –Runaway potentials (not solved by duality)

BPS Brane-instanton SNP’s Euclidean wrapped branes Potential V~e -action Complete under duality From hep-th/

Outer Region Moduli – chiral superfields of N=1 SUGRA, K=K(S,S*), W=W(S) N=1 SUGRA Steep potentials e.g: K=-ln(S+S*) Pert. Kahler 

Extremum: Min?, Max?, Saddle? (ii) Two types: (i) Outer Region Stabilization ?

Case (i) Case (i) is a minimum Case (ii) Case (ii) is a saddle point In general, max or saddle, but never min !

Outer Region Cosmology:Slow-Roll? S-duality 5D – same solutions! T-duality Without a potential: 4D, 5D, 10D, 11D : “fast-roll”

With a potential Use to find properties of solutions with real potential Ansatz Solution No slow-roll for real steep potential realistic steep potential

Central Region Parametrization with D=4, N=1 SUGRA Stabilization by SNP string scale Continuously adjustable parameter SUSY lower scale by FT effects PCCP o.k. after SUSY breaking Our proposal: VADIM: CAN YOU HAVE A CONTINOUSLY ADJUSTABLE PARAMETER THAT IS NOT A MODULUS? ARE 2 AND 3 CONSISTENT OFER: KACHRU ET AL CENTRAL REGION. DISCRETE PARAMETER

Stable SUSY breaking minimum Two Moduli, S (susy breaking direction), T (orthogonal), m 3/2 /M P =  ~10 -16

(a),(b),(e) & (2,3,4)  (b),(c ),(e) & (2)  (2) (3) (1) (4) (5)

Higher derivatives in S (> 3) and T (> 1), & mixed derivatives of order > 2 generically O(1). In SUSY limit, in T direction, V is steep, all derivatives > 2 generically min. In S direction, potential is very flat around min. Masses of SUSY breaking S moduli o(  in general masses of T moduli O(1). With more work

Simple example  Reasonable working models,  Additional SUSY preserving  <0 minima!

Scales & Shape of Moduli Potential The width of the central region In effective 4D theory: kinetic terms multiplied by M S 8 V 6 (M 11 9 V 7 in M). Curvature term multiplied by same factors “Calibrate” using 4D Newton’s const. 8  G N =m p -2  Typical distances are O(m p)

The scale of the potential Numerical examples: NO VOLUME FACTORS!!! Banks

The shape of the potential  m p outer region V(  M S 6 m p -2 outer region central region zero CC min. & potential infinity  intermediate max.

Inflation: constraints & predictions Topological inflation  – wall thickness in space     Inflation   H > 1   > m p H 2 ~1/3    m p 2   m p V(  M S 6 m p -2

CMB anisotropies and the string scale For consistency need |V’’|~1/25 Slow-roll parameters Number of efolds The “small” parameter Sufficient inflation Qu. fluct. not too large

 1/3 < 25|V’’| < 3  For our model If consistent: WMAP

Summary and Conclusions Stabilization and SUSY breaking –Outer regions = trouble –Central region: need new ideas and techniques –Prediction: “light” moduli Consistent cosmology: –Outer regions = trouble –Central region: –scaling arguments –Curvature of potential needs to be “smallish” –Predictions for CMB