Cosmological Perturbations in the brane worlds

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Cosmological Perturbations in the brane worlds

• Kazuya Koyama
• Tokyo University
• JSPS PD fellow

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• Randall Sundrum model (Randall and Sudrum. 99)
• Simplest model for brane world
• Can we find the brane world signatures in
  cosmological observations such as CMB, GW ?

AdS
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• Brane world

CMB
bulk
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• Bulk inflaton model
• Exact solutions for perturbations
• K.K. and K. Takahashi Phys. Rev. D 67
  104011(2003)
• K.K. and K. Takahashi Phys. Rev. D in press
  (hep-th/0307073)
• Tensor perturbations
• Numerical calculations
• H. Hiramatsu, K.K. and A. Taruya,
• Phys. Lett. B in press ( hep-th/0308072)
• CMB anisotropies
• Low energy approximation
• T.Shiromizu and K.K. Phys. Rev. D 67 084022
  (2003)
• K.K Phys. Rev. Lett. in press
  (astro-ph/0303108)

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1.Cosmological Gravitational Waves
0-mode
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• Two ways to see the bulk
• Gaussian normal coordinate
• Brane is located at fixed value of the coordinate
• Bulk metric is non-separable with respect to t
  and y

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• Static coordinate
• Bulk metric is separable with respect to t and y
• Brane is moving

Poincare coordinate
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1-1 Gaussian Normal coordinate
Hiramatsu, Koyama, Taruya, Phys.Lett. B
(hep-th/0308072)

• Metric
• Friedmann equation
• Parameter

(Binetruy et.al)
( horizon crossing)
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• Wave equation
• Initial condition
• near brane/low energy metric is separable
• 0-mode KK-modes
• No known brane inflation model predicts
  significant
• KK modes excitation during inflation
• KK-modes are decreasing at super-horizon scales
• We adopt 0-mode initial
  condition at
• super-horizon scales (
  hconst. )

(Easther et. al., Battye et. al.)
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• Boundary condition
• There is a coordinate singularity in the bulk

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Results ( )
Time evolution on the brane
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• Amplitude of GW decreases due to KK modes
  excitation
• suppression at ?

damping
Damping factor
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1-2 Poincare coordinate (work in progress)

• Coordinate singularity in GN coordinate
• high energy region is difficult to treat
• Poincare coordinate is well behaved
• The general solutions for GW are easily derived

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• Brane is moving

Motion of the brane
low energy
high energy
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• Junction condition
• Particular solutions
• We should determine unknown coefficients from
  initial
• conditions and boundary conditions

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• Recovery of 0-mode solution
• Due to the moving of the brane, 0-mode on
  FRW brane does NOT correspond to m0
• Junction condition at late times

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• Numerical solution for
• Naïve boundary and initial conditions
• no incoming radiation at Cauchy horizon
• Initial condition

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• Numerical results for low energy
• The resultant solution

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• Initial/boundary conditions

De Sitter brane (GN coordinate)
(Battye et. al.)
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• We define a vacuum state during inflation
• Mode mixing
• Initial condition

(Gorbunov et. al, T. Kobayashi et. al.)
Quantum theory
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• Prediction of GW at high frequencies
• (in the near future)

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2. CMB anisotropies

• KK modes
• At decoupling time,
• KK modes are unlikely to be excited
• Dark radiation
• In homogeneous and isotropic universe, the bulk
  BH can
• affect the dynamics of the brane at late times
• Effects of dark radiation
  perturbation on CMB ?

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Cosmology with dark radiation

• Creation of the dark radiation
• Emission to the bulk creates dark radiation
• bulk field (reheating in bulk inflaton model)
• graviton emission in high energy era
• Cosmological observations
• BBN constraints as
• Dark radiation induces isocurvature
  perturbation
• Results of WMAP on CMB anisotropies strongly
  restrict
• the existence of isocurvature modes

(Himemoto, Tanaka)
(Langlois, Sorbo)
(Ichiki et. al.)
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• CMB anisotropies (SW effect)
• for adiabatic perturbation
• Longitudinal metric perturbations
• Curvature perturbation can be calculated without
  solving
• bulk perturbations but anisotropic stress cannot
  be
• predicted unless bulk perturbations are known

Red shift
photon
(Langlois, Maartens, Sasaki, Wands)
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Large scale perturbations - view from
the bulk -
Isotopic and homogeneous bulk
AdS-Schwarzshild
Anisotropy on the brane anisotropy in the
bulk anisotropy on the brane Bulk and brane
is coupled
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• Gaussian-Normal coordinate for Ads-Schwarzshild
• Consider the perturbation of dark radiation
  

AdS spacetime perturbations
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• Solutions for trace part
• Equations for

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• In this gauge, the brane location is perturbed
• Perform infinitesimal coordinate transformation
  and impose junction conditions

Matter perturbations
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• Junction condition relates matter perturbation on
  the brane to and bulk perturbations
• Adiabatic condition on matter perturbations
• equation for

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• Solution for integration
  constant
• 
  
• Metric perturbations on the brane
• Curvature perturbation

(Koyama. 02)
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• Curvature perturbation is determined only by
• Curvature perturbation brane dynamics
• ( FRW equation brane daynamics)
• Solution for curvature perturbation can be
  derived exactly
• at large scales (including , at high
  energies)
• Anisotropic stress
• anisotropic stress on the brane
• coupled to anisotropic shear in the bulk
  

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• Equations for anisotropic shear
• 
  
• Junction condition
• The problem is to solve the wave equations
  for
• with source and junction conditions

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• Low energy/gradient expansion
• Assuming
• Solution
• Junction condition
• Anisotropic stress

Integration constant
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Two branes model at low energies

• Junction condition completely determine
• This is equivalent to use the effective theory

(Shiromizu, Koyama)
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• Prediction of CMB anisotopies
• in two branes model

K.K Phys. Rev. Lett. in press (astro-ph/0303108)
Detailed analysis of CMB anisotropies in
two branes model (work in progress)
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One brane model

• In Gaussian Normal coordinate it is again
  difficult to address the boundary condition due
  to the coordinate singularity
• Formulation in Poincare
  coordinate
• Anisotropic stress depends on boundary/initial
  condition on bulk gravitational field
• What is the natural boundary/initial
  condition
• with dark radiation ? ( de
  Sitter vacuum for GW)

(Koyama, Soda 00)
(Cf. )
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• We should understand the relation between the
• choice of the boundary condition in the bulk and
• the behavior of anisotropic stress on the brane
• Anisotropic stress boundary condition
• Numerically
• Toy model where this relation can be
• analytically examined in a whole bulk
  spacetime

(Koyama, Takahashi, hep-th/0307073)
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