1. Brane-World Inflation
Alex Buchel and A. G
PI, Canada and IPM, Iran
Hep-th/
Phys. Rev. D70:126008, 2004

2. 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.

3. Inflation from String Theory
Strings 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 fields
Stable or Fixed
Flat Potential for Slow Rolling

4. Hierarchies from fluxes in string compactifications Giddings-Kachru-Polchinski hep-th/0105097
Warp Solutions: t
D3-brane
O3-brane
Wrapped
D7-brane
Throat y x
Compactification
Manifold
Electric Flux
Magnetic Flux

5. De-Sitter vacua in string theory
KKLT hep-th/
Lifting Ads vacua to ds vacua + Moduli stabilization by putting an anti D3-brane at the tip of the KS throat.

6. Towards inflation in string theory
KKLMMT hep-th/

7. De-Sitter deformed KS throat Buchel-Roiban hep-th/0311154
KKLMMT Model
KS throat with
slow rolling
De-Sitter deformed KS throat
Buchel-Roiban hep-th/
Small Slow Rolling ?

8. 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.

9. SU(2) left invariant one form
Metric
SU(2) left invariant one form
On 3-sphere
SU(2) gauge fields on 2-sphere
NS-NS
3 form
field
Dilaton

10. We de-Sitter deform MN background by changing the four dimensional Minkowski space-time to a de-Sitter in 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

11. Probe Dynamics of D5-branes in de-Sitter deformed MN background
S-duality
4-dim de-Sitter
2-sphere

12. 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

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

14. 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

15. 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/ )
The Metric and gauge field in this background are
And a, f, F, x and y are functions of radial coordinate

16. Equations of motions are

17. Using the method in hep-th/ (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

18. 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

19. 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

20. We start with the case (a), the similar arguments works for case (b)
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

21. 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

22. In the UV region we also have a power series solution
Where

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

24. 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).

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

26. Inflation in de-Sitter deformed N=2* throats
The relevant throat geometry is that of the supergravity dual to N=2* susy gauge theory constructed in (Pilch and Warner hep-th/ ). 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

27. 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.

28. 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

29. Phenomenology IR Anti D3-brane N=2* Throat KS Throat D3-brane UV
6 dim Compactification Manifold

30. The effective potential for this scenario is the sum of two terms
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

31. calculate some properties of our model.
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

32. 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,

33. 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

34. 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.