Cosmological Perturbations in the brane worlds Kazuya Koyama Tokyo University JSPS PD fellow

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

Brane world Brane world CMB bulk

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

0-mode 1.Cosmological Gravitational Waves Kaluza-Klein(KK) mode ？

Two ways to see the bulk Two ways to see the bulk Gaussian normal coordinate Brane is located at fixed value of the coordinate Brane is located at fixed value of the coordinate Bulk metric is non-separable with respect to t and y Bulk metric is non-separable with respect to t and y Gaussian-Normal coordinate y= const.

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

1-1 Gaussian Normal coordinate Metric Metric Friedmann equation Friedmann equation Parameter Parameter Hiramatsu, Koyama, Taruya, Phys.Lett. B (hep-th/ ) ( horizon crossing) (Binetruy et.al)

Wave equation Wave equation Initial condition Initial condition near brane/low energy metric is separable near brane/low energy metric is separable 0-mode + KK-modes 0-mode + KK-modes No known brane inflation model predicts significant No known brane inflation model predicts significant KK modes excitation during inflation KK modes excitation during inflation KK-modes are decreasing at super-horizon scales KK-modes are decreasing at super-horizon scales We adopt 0-mode initial condition at We adopt 0-mode initial condition at super-horizon scales ( h=const. ) super-horizon scales ( h=const. ) (Easther et. al., Battye et. al.)

Boundary condition Boundary condition There is a coordinate singularity in the bulk There is a coordinate singularity in the bulk Physical brane Regulator brane

Results ( ) Time evolution on the brane KK-mode 0-mode Solution on t = const. surface

Amplitude of GW decreases due to KK modes excitation Amplitude of GW decreases due to KK modes excitation suppression at ? suppression at ? damping Damping factor

Coordinate singularity in GN coordinate Coordinate singularity in GN coordinate high energy region is difficult to treat high energy region is difficult to treat Poincare coordinate is well behaved Poincare coordinate is well behaved The general solutions for GW are easily derived 1-2 Poincare coordinate (work in progress)

Brane is moving Brane is moving Motion of the brane high energy low energy

Junction condition Junction condition Particular solutions Particular solutions We should determine unknown coefficients from initial conditions and boundary conditions

Recovery of “ 0-mode ” solution Recovery of “ 0-mode ” solution Due to the moving of the brane, “ 0-mode ” on FRW brane does NOT correspond to m=0 Due to the moving of the brane, “ 0-mode ” on FRW brane does NOT correspond to m=0 Junction condition at late times Junction condition at late times Non-local terms ( “ CFT ” part in the context of AdS/CFT correspondence)

Numerical solution for Numerical solution for Na ï ve boundary and initial conditions Na ï ve boundary and initial conditions “ no incoming radiation ” at Cauchy horizon “ no incoming radiation ” at Cauchy horizon Initial condition Initial condition

Numerical results for low energy Numerical results for low energy The resultant solution The resultant solution 1 4D

Initial/boundary conditions Initial/boundary conditions De Sitter brane (GN coordinate) Initial condition (Battye et. al.)

We define a vacuum state during inflation We define a vacuum state during inflation “ Mode mixing ” “ Mode mixing ” Initial condition Initial condition Quantum theory (Gorbunov et. al, T. Kobayashi et. al.)

Prediction of GW at high frequencies Prediction of GW at high frequencies (in the near future) (in the near future)

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

Cosmology with dark radiation Creation of the dark radiation Creation of the dark radiation Emission to the bulk creates dark radiation bulk field (reheating in bulk inflaton model) bulk field (reheating in bulk inflaton model) graviton emission in high energy era graviton emission in high energy era Cosmological observations Cosmological observations BBN constraints as BBN constraints as Dark radiation induces isocurvature perturbation Dark radiation induces isocurvature perturbation Results of WMAP on CMB anisotropies strongly restrict Results of WMAP on CMB anisotropies strongly restrict the existence of isocurvature modes the existence of isocurvature modes (Himemoto, Tanaka) (Langlois, Sorbo) (Ichiki et. al.)

CMB anisotropies (SW effect) CMB anisotropies (SW effect) for adiabatic perturbation for adiabatic perturbation Longitudinal metric perturbations 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)

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

Gaussian-Normal coordinate for Ads-Schwarzshild Gaussian-Normal coordinate for Ads-Schwarzshild Consider the perturbation of dark radiation Consider the perturbation of dark radiation AdS spacetime + perturbations

Solutions for trace part Solutions for trace part Equations for Equations for

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

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

Solution for : integration constant Solution for : integration constant Metric perturbations on the brane Metric perturbations on the brane Curvature perturbation Curvature perturbation (Koyama. ’ 02)

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

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

Low energy/gradient expansion Low energy/gradient expansion Assuming Assuming Solution Solution Junction condition Junction condition Anisotropic stress Anisotropic stress : Integration constant

Junction condition completely determine Junction condition completely determine This is equivalent to use the effective theory This is equivalent to use the effective theory (Shiromizu, Koyama) Two branes model at low energies

Prediction of CMB anisotopies Prediction of CMB anisotopies in two branes model in two branes model K.K Phys. Rev. Lett. in press (astro-ph/ ) Detailed analysis of CMB anisotropies Detailed analysis of CMB anisotropies in two branes model (work in progress) in two branes model (work in progress) Generalizations Generalizations Comparison with observations Stabilization mechanism Stabilization mechanism

In Gaussian Normal coordinate it is again difficult to address the boundary condition due to the coordinate singularity In Gaussian Normal coordinate it is again difficult to address the boundary condition due to the coordinate singularity Formulation in Poincare coordinate Formulation in Poincare coordinate Anisotropic stress depends on boundary/initial condition on bulk gravitational field Anisotropic stress depends on boundary/initial condition on bulk gravitational field What is the “ natural ” boundary/initial condition What is the “ natural ” boundary/initial condition with dark radiation ? ( de Sitter vacuum for GW) with dark radiation ? ( de Sitter vacuum for GW) One brane model (Koyama, Soda 00) (Cf. )

We should understand the relation between the We should understand the relation between the choice of the boundary condition in the bulk and choice of the boundary condition in the bulk and the behavior of anisotropic stress on the brane the behavior of anisotropic stress on the brane Anisotropic stress boundary condition Anisotropic stress boundary condition Numerically Numerically Toy model where this relation can be Toy model where this relation can be analytically examined in a whole bulk spacetime analytically examined in a whole bulk spacetime (Koyama, Takahashi, hep-th/ )