BraneWorld Inflation

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Brane-World Inflation

• Alex Buchel and A. G
• PI, Canada and IPM, Iran
• Hep-th/0404151
• Phys. Rev. D70126008, 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 N2 (Pilch-Warner)
  model.

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Inflation from String Theory

• Strings live in 19 dimensions We live in
  13 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
compactificationsGiddings-Kachru-Polchinski
hep-th/0105097

• Warp Solutions

t
D3-brane
O3-brane
Wrapped D7-brane
Throat
y
x
Compactification Manifold
Electric Flux
Magnetic Flux
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• 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.

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• Towards inflation in string theory
• KKLMMT hep-th/0308055

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KKLMMT Model KS throat with slow rolling
De-Sitter deformed KS throat Buchel-Roiban
hep-th/0311154
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 N1 susy in four
  dimensions.
• Here F1 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-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
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Probe Dynamics of D5-branesin de-Sitter deformed
MN background
S-duality
4-dim de-Sitter
2-sphere
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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 N2 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 D7 (hep-th/0106117)
• 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/0003286 (Cvetic, Lu
and Pope) we can uplift D7 to D10 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_infinity1 the next leading term
for slow rolling parameter will be important
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Inflation in de-Sitter deformed N2 throats

• The relevant throat geometry is that of the
  supergravity dual to N2 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

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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
IR
Anti D3-brane
N2 Throat
KS Throat
D3-brane
UV
6 dim Compactification Manifold
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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 N2
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.

UV
IR
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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 ngt1
• 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 N2
  throat accept an inflationary model with small
  slow rolling parameters.