BRANEWORLD COSMOLOGICAL PERTURBATIONS Roy Maartens University of Portsmouth Tokyo IT October 2003

testing the braneworld scenario cosmology as a probe of theory braneworld observational signature?

RS braneworld t 5<0 y x3 >0 why does gravity not leak into 5D? cosmological constant in bulk 5 < 0 how is brane protected against 4<0 ? brane tension >0  + 5  4 =0 Minkowski brane in anti de Sitter bulk 5D gravitons – effective mass m on brane (massive KK modes) nonlocal KK effects y x3

field equations gravitational action RS solution – 4-Minkowski in 5-AdS

massive KK modes metric perturbation TT-gauge (4D) perturbed 5D field equation separate into modes

RS1: m > 0 discrete spectrum solution zero mode massive modes RS1: m > 0 discrete spectrum RS2: m > 0 continuous spectrum gravitational potential m=0 m>0

general braneworld Gauss equation Codazzi equation junction equations

high or low energy induced 4D Einstein tensor high-energy high or low energy 5D graviton - massive KK effects KK/ Weyl anisotropic stress – must be determined by 5D equations

KK stresses from brane matter matter obeys brane and bulk do not exchange energy not true if scalar field/radiation in bulk Bianchi identity KK stresses sourced by perturbations - inhomogeneity and anisotropic stress

inflation on the brane 4D inflaton, high-energy inflation high-energy assists slow-roll brane slow-roll parameters new possibility - steep inflation

brane matter perturbations decouple from bulk metric perturbation (large scales) curvature perturbation can be found (large scales) then

tensor perturbations from inflation perturbed de Sitter brane wave equation separable (H constant)

solutions mass gap above 0-mode massive modes decay during inflation 0-mode has increased amplitude at high energy but tensor/ scalar is reduced!

spectrum of normalizable states discrete zero mode (4D) m=0 massive KK continuum m>3H/2 only zero-mode excited during inflation evolution after inflation 0-mode re-enters Hubble – KK modes generated (5D gravitons bulk) loss of energy – damping (Koyama’s talk)

spectral indices scalar perturbations tensor perturbations same form as GR tensor perturbations compare GR but consistency condition has the same form

RS2 + induced gravity quantum correction to gravitational action brane matter / bulk graviton coupling quantum correction to gravitational action curvature term induced on brane modifies gravity at large scales/ low energies but – also removes RS high-energy correction early universe at high energy = GR + …

scalar perturbations less power since need to check curvature perturbations

tensor perturbations from inflation same bulk equations – same modes but boundary conditions different: giving

RS2+Gauss-Bonnet gravity most general 5D action with 2nd order equations quantum/ stringy correction to gravity modifies gravity at high energies suggests lower scalar perturbations but curvature perturbation must be checked

tensor perturbations from inflation bulk equation different but bulk wave equation has same form but boundary conditions different junction conditions cubic in extrinsic curvature! but same form of boundary condition as in IG: giving

Low energy approximation gradient expansion curvature radius on the brane curvature radius in the bulk To find - need boundary conditions - shadow/ regulator brane

background Friedmann equations dark radiation radion

radion low-energy solution

effective equations on +ve tension brane scalar-tensor theory

cosmological perturbations on large scales Weyl anisotropic stress given by radion

define new variables then

physical meaning of variables brane displacements bulk anisotropic perturbation

simple toy model Weyl anisotropic stress completely compensates entropy perturbation

further work one-brane case: choose physical shadow matter dark radiation in background one-brane case: needs suitable boundary/ initial conditions