Brane-World Inflation

Alex Buchel and A. G

PI, Canada and IPM, Iran

Hep-th/0404151

Phys. Rev. D70:126008, 2004

• An introduction to inflation in Klebanov-Strassler model.

• Inflation in wrapped brane-worlds.

1. Maldacena-Nunez

2. Gauntlett-Kim-Martelli-Waldram

• Inflation and slow rolling in N=2* (Pilch-Warner) model.

Inflation from String TheoryStrings live in 1+9 dimensions We live in 1+3 dimensions

Compactification

String theory Inflation

By compactification we could control shape and size of compactification manifold as well as string coupling

Moduli fieldsStable

or FixedFlat

Potential for Slow

Rolling

Hierarchies from fluxes in string compactificationsGiddings-Kachru-Polchinski hep-th/0105097

Warp Solutions:

dydyygexddte yAyA )()( )(22)(

D3-brane

WrappedD7-brane

Electric FluxMagnetic Flux

O3-brane

Throat

CompactificationManifold

x

t

y

De-Sitter vacua in string theory

KKLT hep-th/0301240

Lifting Ads vacua to ds vacua + Moduli stabilization by putting an anti D3-brane at the tip of the KS throat.

Towards inflation in string theory

KKLMMT hep-th/0308055

KKLMMT ModelKS throat with

slow rolling

3

2 scorrectionsmall

3

2

De-Sitter deformed KS throatBuchel-Roiban hep-th/0311154

?Small Slow Rolling 1

Maldacena-Nunez Background

This background is supergravity solution corresponding to a large number of NS-5 branes wrapped on a two sphere with N=1 susy in four

dimensions.

Here F=1 and n is the number of NS-5 branes.

Metric

SU(2) left invariant one formOn 3-sphere

SU(2) gauge fields on 2-sphere

NS-NS3 formfield

Dilaton

We de-Sitter deform MN background by changing the four dimensional Minkowski space-time to a de-Sitterin addition we let F be a nontrivial function of rho in order to have a warp solution. In order to find this background we need to solve the IIB supergravity equations of motion. By considering G, a and string coupling as a function of radial coordinate, rho, these equations are

Probe Dynamics of D5-branesin de-Sitter deformed MN background

2-sphere

4-dim de-Sitter

S-duality

If we consider D5-brane localized at a point in 3-sphere and radial coordinate rho as a function of four dimensional de-

Sitter space the effective action for D5-brane after integrating over 2-sphere will be

Where E is the Error function. Now if we write this effective action in a canonical form

By change of variable we write the action in a canonical form.

In order to calculate the inflation potential we need to know the behavior of different functions

appearing in equations of motion. Asymptotic large distance behavior of these functions are

Then the first leading term in

potential will be:

And the slow rolling parameter is 2

3)(''

3

12

H

V

GKMW Background

This background corresponds to solution of IIB supergravity equations of motion for wrapped NS-5 branes on two sphere with N=2 susy in four dimensions.

In order to find slow rolling parameter for de-Sitter deformed GKMW background we start from the effective Lagrangian for SO(4) gauged supergravity in D=7 (hep-th/0106117)

The Metric and gauge field in this background are

And a, f, F, x and y are functions of radial coordinate

Equations of motions are

Using the method in hep-th/0003286 (Cvetic, Lu and Pope) we can uplift D=7 to D=10 solutions so that equations of motions

now are compatible with IIB supergravity equations of motions

Again we probe the background with a D5-brane which is wrapped on 2-sphere and located on a point on 3-sphere and

we consider the radial coordinate as a function of four dimensional de-sitter coordinates.

By going to canonical form for the action we need to change the radial coordinate so that

Solutions to the equations of motion

By changing the variables as

And the following relations

There are two topologically distinct solutions for equations of motion

We start with the case (a), the similar arguments works for case (b). By changing the variables as

There are two power series solutions in IR and UV regions. In IR we have three initial arbitrary values.

Where are characterizing the size of two sphere, a circle inside three sphere and the size of de-Sitter space

Numerical solutions show that there is a critical value k_c for k_0 which above this value the radius of two sphere shrinks and makes singular solutions. If we sketch the radius of two sphere in terms of radial coordinate r

In the UV region we also have a power series solution

Where

For different initial IR values it is possible to find UV solutions regarding to the following numerical analysis

Slow rolling conditions

The inflationary potential and slow rolling parameter for this model is

For cases where k_infinity is less or bigger than 1 the inflationary potential has local minimum and is unbounded from blow so we have instability (tachyonic potential).

For the case k_infinity=1 the next leading term for slow rolling parameter will be important

Inflation in de-Sitter deformed N=2* throatsThe relevant throat geometry is that of the supergravity dual

to N=2* susy gauge theory constructed in (Pilch and Warner hep-th/0004063). Construction of de-Sitter deformed geometry is as before. We start with a five dimensional gauged supergravity and uplift it to ten dimensions. Here also there two region for power series solutions. The final results for slow rolling parameter for a D3-brane probe is

5

211

00

)(

)(

S

Mass

Mass

Bosonic

Fermionic

Turning on the Fermionic mass increases slow rolling parameter but from equations of motion it can be set to zero. But the Bosonic mass square can be either positive or negative.

There are two regimes with locally minimized potential energy leading to slow rolling

The important point here is that the Bosonic mass in UV region is related to IR mass rho_0

Phenomenology

KS Throat

Anti D3-brane

N=2* Throat

D3-brane

6 dim Compactification Manifold

UV

IR

The effective potential for this scenario is the sum of two terms. Cosmological constant term of the KS throat and inflationary potential of N=2* throat

• Supergravity approximations: Size of compactification manifold is much bigger than the string length and the string coupling is very small.

• D3-brane moving deep inside the throat far from UV and IR region where slow rolling parameter is very small.

• Also we need some

parameters in order to

calculate some properties

of our model.

UV IR

Using these data we can compute some properties of our inflationary model such as slow rolling parameters, the tilt in

the spectrum of the density perturbations, the scale of the adiabatic density perturbations and the power in the gravity

wave perturbations,

• Slow rolling

• Observation data for n>1

• Maximum Number of e-folding

• Hubble constant during the inflation (low scale inflation)

• Much below the level of detection

Conclusions

• Probe dynamics of wrapped D5-branes inside the MN or GKMW throats shows the same (large slow rolling parameter) as KS model.

• Probe dynamics of D3-branes inside the N=2* throat accept an inflationary model with small slow rolling parameters.