BRANEWORLD COSMOLOGICALPERTURBATIONS

Roy Maartens

University of PortsmouthTokyo ITOctober 2003

testing the braneworld scenario cosmology as a probe of theory

braneworld observational signature?

RS braneworldwhy does gravity not leak into 5D? cosmological constant in bulk 5 < 0how is brane protected against 40 + 5 4 =0 Minkowski brane in anti de Sitter bulk

5D gravitons effective mass m on brane (massive KK modes)

nonlocal KK effects

t>05

field equationsgravitational action

RS solution 4-Minkowski in 5-AdS

massive KK modes metric perturbation

TT-gauge (4D)

perturbed 5D field equation

separate into modes

solution

zero mode

massive modes

RS1: m > 0 discrete spectrumRS2: m > 0 continuous spectrumgravitational potential

m=0 m>0

general braneworldGauss equation

Codazzi equation

junction equations

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 bulkBianchi identity

KK stresses sourced by perturbations - inhomogeneity and anisotropic stress

standard 4D perturbation picture

t

inflation

brane-worldperturbations

bulk

0-mode + KK modes

KK anisotropy

inflation on the brane4D inflaton, high-energy inflation

high-energy assists slow-rollbrane 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 inflationperturbed de Sitter brane

wave equation separable (H constant)

solutions

mass gap above 0-mode

massive modes decay during inflation0-mode has increased amplitude at high energy

but tensor/ scalar is reduced!

spectrum of normalizable statesdiscrete zero mode (4D) m=0massive KK continuum m>3H/2 only zero-mode excited during inflationevolution after inflation0-mode re-enters Hubble KK modes generated (5D gravitons bulk)loss of energy damping(Koyamas talk)

spectral indices scalar perturbations same form as GR

tensor perturbations

compare GR

but consistency condition has the same form

RS2 + induced gravitybrane matter / bulk graviton couplingquantum correction to gravitational actioncurvature term induced on brane

modifies gravity at large scales/ low energiesbut also removes RS high-energy correctionearly universe at high energy = GR +

scalar perturbations

less power since need to check curvature perturbations

tensor perturbations from inflationsame bulk equations same modesbut boundary conditions different:

giving

RS2+Gauss-Bonnet gravitymost general 5D action with 2nd order equationsquantum/ stringy correction to gravity

modifies gravity at high energies

suggests lower scalar perturbationsbut curvature perturbation must be checked

tensor perturbations from inflationbulk equation different

but bulk wave equation has same form

but boundary conditions differentjunction conditions cubic in extrinsic curvature!but same form of boundary condition as in IG:

giving

curvature perturbation

matter conservation

curvature perturbation determined (on large scales)

CMB scalar anisotropies

Sachs-Wolfe

adiabatic (matter) +isocurvature (dark radiation) + Weyl anisotropic stress terms

gradient expansion curvature radius on the brane curvature radius in the bulk

To find - need boundary conditions - shadow/ regulator brane Low energy approximation

background

Friedmann equationsdark radiationradion

radionlow-energy solution

effective equations on +ve tension branescalar-tensor theory

cosmological perturbations

on large scales

Weyl anisotropic stress given by radion

define new variablesthen

brane displacements

bulk anisotropic perturbation physical meaning of variables

simple toy model

Weyl anisotropic stress

completely compensates entropy perturbation

further work

choose physical shadow matter

dark radiation in background

one-brane case: needs suitable boundary/ initial conditions