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

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

3. 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 >  + 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

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

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

6. 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

7. general braneworld
Gauss equation
Codazzi equation
junction equations

8. 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

9. 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

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

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

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

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

16. 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)

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

18. 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 + …

19. scalar perturbations
less power since
need to check curvature perturbations

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

21. 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

22. 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

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

27. background
Friedmann equations
dark radiation
radion

28. radion
low-energy solution

29. effective equations on +ve tension brane
scalar-tensor theory

30. cosmological perturbations
on large scales
Weyl anisotropic stress given by radion

31. define new variables
then

32. physical meaning of variables
brane displacements
bulk anisotropic perturbation

33. simple toy model Weyl anisotropic stress
completely compensates entropy perturbation

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