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

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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.

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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

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

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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.

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Towards inflation in string theory KKLMMT hep-th/

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

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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.

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

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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

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

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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

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

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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

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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

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Equations of motions are

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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

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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

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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

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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

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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

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

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

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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).

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

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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

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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.

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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

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

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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

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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. UVIR

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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,

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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

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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.

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