Roy Maartens- Brane Cosmology

8/3/2019 Roy Maartens- Brane Cosmology

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BRANE COSMOLOGY BRANE COSMOLOGY

Moriond 2004 University of

Portsmouth

Roy Maartens



GR breaks downGR breaks down

need quantum gravityneed quantum gravity

in the early universein the early universe

whywhy branebrane--worlds?worlds?

no QG theory as yetno QG theory as yet

but M theory is abut M theory is apromising candidatepromising candidate

M theory needs extraM theory needs extra

dimensions +dimensions + branesbranes

can lower the Planck scalecan lower the Planck scale



GR GR phenomenologyphenomenology QGQG

standard cosmology highlystandard cosmology highlysuccessfulsuccessful

butbut – – still a paradigm seeking a theorystill a paradigm seeking a theory

inflationinflation

??

dark energydark energy

??

(dark matter(dark matter

??))

quantum modifications to GR quantum modifications to GR

** solve puzzlessolve puzzles -- inflation,inflation,

dark energy, lowdark energy, low quadrupolequadrupole?,...?,...** predict new featurespredict new features

slow progress in M theoryslow progress in M theory

cosmologycosmologyuseuse braneworldbraneworld phenomenologyphenomenology



2 key aspects2 key aspects

braneworldbraneworld gravity bringsgravity brings newnew featuresfeatures

KK modes,KK modes, modulimoduli fields, holography, shadow matter … .fields, holography, shadow matter … .

precision cosmology canprecision cosmology can constrainconstrain braneworldbraneworldmodels (and M Theory)models (and M Theory)

* via dynamics* via dynamics – – BBN,BBN, SNeSNe

* via CMB* via CMB



why donwhy don’ ’ t we see the extrat we see the extra dimensions?dimensions? cconventionalonventional KaluzaKaluza--KleinKlein ideaidea::

iinternalnternal extra dimension too small to be seenextra dimension too small to be seen

ddiscoveryiscovery of Dof D--branebrane

mmatteratter fieldsfields restrictedrestricted

toto lower dimensionallower dimensional branebrane

eexternalxternal bulk felt onlybulk felt only

throughthrough gravitygravity

extra dimension biggerextra dimension bigger

4D spacetimesmall extradimension

large extradimension

gravity



M theoryM theory1 time + 10 space dimensions

1+10 1+3+1+(6)

braneworldlarge extra dimension

effective 5D braneworld

matter

gravity

shadowshadow

visiblevisible

6 6

1

1+3

1+3

( ) 4

3 / 12

45 / ~ M L M M <<



x

simple models (Randall-Sundrum)

5D Einstein gravity + vacuum energy

use curvature to localize gravity

brane self-gravity (tension)

tension balances bulk vacuum energy

brane is Minkowski, bulk 5D AdS

two models

| ______ | | ____ _ _ compact non-compact

warped braneworldswarped braneworlds

y

t

Λ5<0>0

- λ λ



background – 5D anti de Sitter

5D gravitons

* massless in 5D* effective 4D mass

5D metric perturbation

massive KK modesmassive KK modes

A p)5(

An

µ p

( ) 2

5

22 / ||222)5( / 6, l

rl−=Λ+−+=

− xd dt edyds

y

1||,)5()5(

<<+→ AB AB AB AB hhgg



RSRS--gaugegauge

5D5D spinspin--22 4D4D spinspin--22 ++ spinspin--11 ++ spinspin--00

linearizedlinearized 5D field equation5D field equation

[ ] 0,3

25

)5(=∂Λ=

brane AB y AB AB hh Rδ

...50,0 f od hhh Ay ⇒∂===µν

ν µ µ

ii

ijii

i

iij

hh

hh

Σ∂==∂=

+Σ+→

0

)1()2()2()5( β µν



put a small particle on brane

TT-gauge (4D)

perturbed 5D field equation

separate into modes

µν µν µν η η h+→

[ ] hhhk he y ′−′′=+l

&&l 42 / 2

),(0)0,(),()(),( Lt ht h y f t yt hm

mm′==′= ∑ ϕ

µν

ν

µ

µ hh ∂== 0

ν µ µν η dxdxedyds

y l / ||222)5( −+=

wave equation

separate

4D zero-mode – only tensor

m=0: no normalizable scalar or vector

weak-field potential

⎥⎦

⎤

⎢⎣

⎡′+′′−=∇∇

−hheh

y

l

l 4 / 2 µ µ

mmmm m y f xh ϕ ϕ ϕ µ µ

2),()( =∇∇→

...3

21

1)(

2

2

+⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −∝Φ

r r r

l

m=0 m>0

TeV M TeV

mm

5

5

410,)1(

1.0

>>⇒

<

λ

l



induced 4D field equations

high-energy

high or low energy

5D graviton - massive KK effects

cosmological branecosmological brane

( ) µν µν µν µν λ

κ κ E T T G −+=2

22

6

( )λ ρ ρ ρ 2 / 1 +=eff

( ) E

E E µν µν π ρ ,→

µν E ABCDC )5(

0=

µ

µ E dark radiation Weyl anisotropic stress



branebrane cosmologycosmology FRW brane – moving in BH- AdS5 bulk

C R=a(T)

velocity H

22

2

22

2

22

222)5(

)(

1)()(

R

C RK RF

d r Kr

dr R RF

dRdT RF ds

−+=

⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛ Ω+

−++−=

l

background cosmologybackground cosmology Minkowski brane – fixed in AdS5

FRW brane – moving in Schw.- AdS5

C R=a(t)

velocity H

4a

C E = ρ

ABCDC )5(

0= E

µν π



generalized Friedmann equation

high-energy term dark radiation

same conservation equation

solutions (C=0, K=0= Λ4)

2

4

4

22

321

3 a

K

a

C H −

Λ++⎟

⎠

⎞⎜⎝

⎛ +=

λ

ρ ρ

κ

[ ] ⎥⎦

⎤

⎢⎣

⎡=+=

⇒=

2 / 1

0

4 / 1

0 :)(3 t aaGRt t t aa p λ

ρ

GR

Ht H H eaa p >

⎟ ⎠

⎞

⎜⎝

⎛ +==⇒−=

λ

ρ ρ

κ ρ 21

3,

0

0)(3 =++ p H ρ ρ &



high energies – new effects

high-energy inflation

high-energy reheating

high-energy early radiation era

low energies – recover GR

below at least electroweak scale

nucleosynthesis “safe”

but perturbations w ill carry 5D effects into CMB

⎥⎦

⎤

⎢⎣

⎡∝∝⇒>>−=

⇒<<⇒>>=

ρ ρ λ ρ ρ

λ ρ ρ λ

H GR H p

t at t p

::

~:3

4 / 1

4

1, ⎟ ⎠

⎞⎜⎝

⎛ ><< TeV λ λ ρ



standard 4D perturbation picturestandard 4D perturbation picture

t

1

ijh a−

∝

inflation

1 H

−

⟩⟨2

ζ

0=Ψ+Φ

= const ζ

ζ 5

1−≈

∆

T

T

const hij =

⟩⟨2

ijh



branebrane--worldworld

perturbationsperturbations

??ijh ∝

bulk

E a π κ 22−=Ψ+Φ

0-mode +

KK modes

KK anisotropy

??=Φ−Ψ+≈

∆

ζ T

T



de Sitter brane in AdS5 bulk

l ike M inkowski brane:

tensor zero modeno scalar/ vector 0-mode from 5D graviton

unlike Minkowski:

mass gap for KK modes

massive modes not excited during inflation

tensor 0-mode frozen until horizon re-entry

perturbations from inflationperturbations from inflation

H m2

3>



scalar perturbations

no 5D graviton contribution to lowest order only from density perturbations -

decouple from 5D perturbations (large scales)

curvature perturbation can be found withoutknow ledge of bulk (large scales)

matter curvature perturbation conserved

as in GR

4

322

2 M

V COBE

GR

<<⇒⎟ ⎠

⎞⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛ ϕ

λ ρ

δρ

ρ

δρ

)(3

/

)(3

)(4

p

aC

p m

E

tot +

+=

+

++Φ=

ρ

δ ζ

ρ

δρ δρ ζ



SachsSachs

--Wolfe effectWolfe effect

metric perturbationsmetric perturbations

cannot predict CMB anisotropies unlesscannot predict CMB anisotropies unless

Weyl anisotropicWeyl anisotropic stress is knownstress is known

m

T

T ζ

∆= + Ψ − Φ

2 1 2 2

,tot H H H

k a ε

ζ

κ π − −

⎛ ⎞Φ

= Φ − − Ψ⎜ ⎟⎝ ⎠

Φ + Ψ = −

&

&

l

Φ

red shift

photon



-- low energy approximationlow energy approximation

structure formationstructure formation -- very low energyvery low energy

bulk curvature scale < 1mm <<bulk curvature scale < 1mm << brane’sbrane’s

use gradient expansion to solveuse gradient expansion to solve

5D field equations5D field equations

need 2 boundaryneed 2 boundary

conditionsconditions

Eπ

+

+

T

λ

−

−

<

T 0λ

ϕ

radion

|||| F F y µ ∇<<∂



effective equations oneffective equations on

++veve tensiontension branebrane

[ ]

),(

)(2

1)(

)(1

)1(1

2

2

222

±

−+

=∇∇

⎟ ⎠ ⎞⎜

⎝ ⎛

∇−∇∇+

∇−∇∇+−+=

T f

T T G

ϕ ϕ

ϕ δ ϕ ϕ ϕ

ϕ ω

ϕ δ ϕ ϕ

ϕ ϕ

κ

µ

µ

ν µ

µ µ

ν µ

ν µ

ν µ

ν µ

ν µ

scalar-tensor theory

)1(2

3)(

ϕ

ϕ ϕ ω

−=where +

T _T



scalar perturbationsscalar perturbations

),( S f ±

= ϕ π E

+ϕ ϕ −ϕ

S ( )−++

±

−

±

±±

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ++

−=

ϕ ϕ ϕ

ϕ

δ κ ζ ζ

,3

6

42

f S H S

H

Cam

&&

&&

&



),(

4

/

m E

m

r E

cdark f

cdark

ζ π

ζ

ρ δρ

=

=

branebrane--world CMB anisotropiesworld CMB anisotropies

model w ith mostmodel w ith most

simple backgroundsimple background

radionradion fixedfixed

no dark radiation inno dark radiation inbackgroundbackground

WMAP

angular scale

T e m p e

r a t u r e f l u c t u

a t i o n



cdark

Ω b

h 2

−0.2 −0.1 0 0.1

0.02

0.022

0.024

0.026

0.028

0.03

0.032

cdark

Ω D M h

2

−0.2 −0.1 0 0.1

0.08

0.1

0.12

0.14

0.16

0.18

cdark

H 0

−0.2 −0.1 0 0.1

60

65

70

75

80

85

90

95

cdark

z r e

−0.2 −0.1 0 0.1

5

10

15

20

25

30



further work further work

simple RS model OK so farsimple RS model OK so far

compute CMB forcompute CMB for more realistic backgroundmore realistic background

oneone--branebrane modelmodel

models w ith bulk models w ith bulk dilatondilaton / /modulimoduli fieldfield

models w ith quantummodels w ith quantum

correctionscorrections M theory models?M theory models?

66 66

11

1+31+3

1+31+3



more developed modelsmore developed models

( )[ ]

∫ ∫

Φ−−

Φ∂−ΦΛ−−=

±)(

)(22

1

4

2255)5()5(5

2

5

λ

κ κ

g xd

Rg xd S grav

bulk scalar field (bulk scalar field (dilatondilaton / / modulimoduli ))

**dilatondilaton can drivecan drive branebrane inflationinflation

**dilatondilaton + shadow matter+ shadow matter

?? dark matterdark matter?? dark energydark energy

**branebrane--branebrane collisioncollision

?? big bang (big bang (ekpyroticekpyrotic ))

ϕ radion

dilaton

)(5 ΦΛ

+

+

T

λ

−

−

T

λ



quantum curvature correctionsquantum curvature corrections

[ ]

∫

∫

−−

++Λ−−=

λ

α κ

g xd

R Rg xd S grav

4

2)5(

5

)5()5(5

2

5

...22

1

GaussGauss--Bonnet (early universe)Bonnet (early universe)

induced gravity (late universe)induced gravity (late universe)

[ ] [ ]∫ ∫ −−−Λ−−= Rg xd Rg xd S grav

24

5

)5()5(5

2

5

22

1 µ λ

κ

Roy Maartens- Brane Cosmology

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