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/0106174PRD 64 (2001), hep-th/0002087

hep-th/0205042hep-th/0212344

with S. de Alwis, E. Novak

Moduli stabilization, SUSY breaking and Cosmology

HO

IIIB

IIA

HWMS1

HE

String Theories and 11D SUGRA

HW=11D SUGRA/I1

MS1=11D SUGRA/S1

T

T

S

S

“S”

N=1 (10D)

N=2 (10D)

“S”

String Moduli Space

HO

IIIB

IIA

HWMS1 HE

Requirements•D=4•N=1 SUSY N=0•CC<(m 3/2)4

•SM (will not discuss)•Volume/Coupling moduli T S

Central region“minimal computability”

Outer regionperturbative

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

– Practical Cosmological Constant Problem

– Runaway potentials (not solved by duality)

BPS Brane-instanton SNP’s

Euclidean wrapped branesPotential V~e-action

Complete under duality

From hep-th/0002087

Outer RegionModuli – chiral superfields of N=1 SUGRA,

K=K(S,S*), W=W(S)

N=1 SUGRASteep 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 pointIn 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 potentialAnsatz

Solution

No slow-roll for real steep potential

realistic steep potential

Central Region

• Parametrization with D=4, N=1 SUGRA

• Stabilization by SNP effects @ string scale

• Continuously adjustable parameter

• SUSY breaking @ lower scale by FT effects

• PCCP o.k. after SUSY breaking

Our proposal:

VADIM: CAN YOU HAVE A CONTINOUSLYADJUSTABLE PARAMETER THAT IS NOT A MODULUS? ARE 2 AND 3 CONSISTENTOFER: KACHRU ET AL CENTRAL REGION.DISCRETE PARAMETER

Stable SUSY breaking minimum

Two Moduli, S (susy breaking direction), T (orthogonal) , m3/2/MP=~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 O(1) @ 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 MS

8 V6 (M119 V7 in M).

Curvature term multiplied by same factors“Calibrate” using 4D Newton’s const. 8GN=mp

-2

Typical distances are O(mp)

22421

pp mRgmxd

• The scale of the potential

)/(

)(

4214

42214

p

p

mVxd

Vmxd

4263 / pSS mMVMW

Numerical examples:

4/34/3

25/116106.8 gGeV YM

I

2/36/16

4/1

25/116 )4(106.7

VMGeV GUTHWYM

NO VOLUME FACTORS!!!Banks

• The shape of the potential

mp

outer region

-4 -2 0 2 4

2

-1

V(MS6mp

-2

outer regioncentral region

zero CC min. & potential vanishes @ infinity intermediate max.

Inflation: constraints & predictions

• Topological inflation

– wall thickness in spaceInflation H > 1 > mp H2~1/3 mp

2

mp-4 -2 0 2 4

2

-1

V(MS6mp

-2

|)(''|252/|)(''|105.6 max4/1

max162 VExpVGeV

CMB anisotropies and the string scale

For consistency need |V’’|~1/25

dNend

pmend )(

121),(

)(''~,2~ max2

max4

2

Vmp

|(''|

22

max)

2

V

mp

Slow-roll parameters

Number of efolds

The “small” parameter

Sufficient inflation 2max )/(120 pinit mExp

6/1)('' max2max VHinit Qu. fluct. not too large

0

)1|)(''|25(08.92. max

r

VnS CMB

CMBCMBS

r

n

7.13

241

1/3 < 25|V’’| < 3 0

97.76.

r

nS

0)(''

0~)/'(

max

2

V

VV

CMB

CMBFor our model

If consistent:

)1(6/11

3/1

10

17 325

4

2/1

)(107.1 S

S

nn

P

S eGeVM

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