BRANEBRANE COSMOLOGYCOSMOLOGY

Roy Maartens

Moriond 2004 University of Portsmouth whywhy branebrane--worlds?worlds?

„ GRGR breaksbreaks downdown need quantum gravity in the early universe

„ nono QGQG theorytheory asas yetyet but M theory is a promising candidate

„ MM theorytheory needsneeds extraextra dimensionsdimensions ++ branesbranes can lower the Planck scale „ standardstandard cosmologycosmology highlyhighly successfulsuccessful „ but – still a paradigm seeking a theory „ inflation? dark energy? (dark matter?) „ quantumquantum modificationsmodifications toto GRGR * solve puzzles - inflation, dark energy, low quadrupole?,... * predict new features „ slowslow progressprogress inin MM theorytheory cosmologycosmology use braneworld phenomenology GRGR phenomenologyphenomenology QGQG 22 keykey aspectsaspects

„ braneworld gravity brings new features KK modes, moduli fields, holography, shadow matter ….

„ precision cosmology can constrain braneworld models (and M Theory) * via dynamics – BBN, SNe * via CMB whywhy dondon’’tt wewe seesee thethe extraextra dimensions?dimensions? „ cconventionalonventional KaluzaKaluza--KleinKlein ideaidea:: iinternalnternal extraextra dimensiondimension tootoo smallsmall toto bebe seenseen

4D spacetime small extra dimension „ ddiscoveryiscovery ofof DD--branebrane „ matter fields restricted matter fields restricted large extra toto lowerlower dimensionaldimensional branebrane dimension „ eexternalxternal bulkbulk feltfelt onlyonly throughthrough gravitygravity

„ extraextra dimensiondimension biggerbigger gravity 66

M theory 1 time + 10 space dimensions visible 1+3

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

1 braneworld large extra dimension matter effective 5D braneworld ()2 1/3 << gravity M 5 ~ M 4 / L M 4 warpedwarped braneworldsbraneworlds t „ simple models (Randall-Sundrum) λ>0 Λ ‹ 5D Einstein gravity + vacuum energy 5<0

y „ use curvature to localize gravity ‹ brane self-gravity (tension) x ‹ tension balances bulk vacuum energy ‹ brane is Minkowski, bulk 5D AdS

„ two models

λλ|______||- λ ______compact non-compact massivemassive KKKK modesmodes background – 5D anti de Sitter

(5) 2 = 2 + − 2| y|/ l ()− 2 + r 2 Λ = − 2 ds dy e dt dx , 5 6 / l pµ

(5) 5D gravitons pA * massless in 5D * effective 4D mass nA 5D metric perturbation

(5) →(5) + << g AB g AB hAB , | hAB | 1 µ RS-gauge µ RS-gauge µν = = = ∂ ν hAy 0, h 0 h ⇒ 5 d.o. f .

5D5D spinspin-22 4D spin--2 + spin-1 + spinspin-00 → + Σ + β hµν (5) hij (2) i (2) (1) i = ∂ i = = ∂ i Σ hi hij 0 i

linearizedlinearized 5D field equation 2 δ (5)R = Λ h , []∂ h = 0 AB 3 5 AB y AB brane µν µ −ϕ ν (5) 2 = 2 + 2|y|/ lη µ − ⎡ 4 ⎤ ds dy e ∇ µ∇dx dx= 2y/l − ′′+ ′ wave equation h e ⎢ h h ⎥ ηµν ⎣ ⎦ µν l „ put a small particle on brane ϕ→η + µ µ hµν separate h → (xµ) f (y), ∇ µ∇ = m2ϕ m m µν m m „ TT-gauge (4D) h = 0 = ∂ νh

4D„ perturbed zero-mode 5D field – onlyequation tensor m=0: no normalizable scalar or vector 4 weak-fielde potential2 y / l []h&& + k 2 h = h′′ − h′ l 1 ⎛ 2 2 ⎞ Φ (r ) ∝ ⎜1 − l ⎟ + ... „ separate into modes ⎜ 2 ⎟ r ⎝ 3r ⎠ h(t, y) = ϕ (t) f ( y), h′(t,0) = 0 = h′(t, L) < 0.1mm ∑ m m l m m=0 m>0 λ > 4 > 5 ⇒ (1 TeV ) , M 5 10 TeV cosmologicalcosmological branebrane µν µν induced 4Dκ field equations κ 2 2 2 µν G = T + 6 ()T − Eµν ρ λ ρ high-energy ρ = ()+ λ eff 1 / 2 high or low energy (5) Eµν CABCD 5D gravitonµν - massive KK effects ρ → ()π E E E , µν µ = dark radiation Weyl anisotropic stress E µ 0 backgroundbackgroundbranebrane cosmologycosmology cosmologycosmology

„ MinkowskiFRW brane –brane moving – fixed in BH- in AdS AdS55 bulk

„ FRW brane – moving in Schw.- AdS5

CR=a(T) velocity H

C R=a(t)

velocity H C (5) ρ = CABCD E a4 dR 2 ⎛ dr 2 ⎞ (5)ds 2 = −F (R)dT 2 + + R 2 ⎜ E + r 2dΩ 2 ⎟ ⎜ π 2 = ⎟ F (R) ⎝1− Krµν 0 ⎠ 2 = + R − C F (R) K 2 2 l R κ „ generalized Friedmann equation 2 ρ ρ ⎛ ⎞ C Λ K H 2 = ⎜1+ ⎟ + + 4 − 3 ⎝ 2λ ⎠ a 4 3 a 2

high-energy term dark radiation ρ „ same conservation equation & + 3H (ρ + p) = 0 Λ „ solutions (C=0, K=0= 4) ρρ = = []+ 1/ 4 ⎡ = 1/ 2 ⎤ p ⇒ a a0 t(t tλ ) ⎢GR : a a0t ⎥ 3 ⎣ ⎦ κ ρ ρ = − = Ht = ⎛ + ⎞ > p ⇒ a a0e , H ⎜1 ⎟ HGR 3 ⎝ 2λ ⎠ ρ ρ λ ρ „ high energies ρ– new effects λ = >> << 1/ 4 p : ⇒ t tλρ⇒ a ~ t 3 = − >> ∝ ⎡ ∝ ρ ⎤ p : ⇒ H ⎢GR : H ⎥ ⎣ ⎦

‹ high-energy inflation ‹ high-energy reheating ‹ high-energy early radiation era ρ λ 4 << λ > ⎛ ⎞ „ low energies – recover GR , ⎜1 TeV ⎟ ⎝ ⎠

‹ below at least electroweak scale ‹ nucleosynthesis “safe” ‹ but perturbations will carry 5D effects into CMB standardstandard 4D4D pertperturbationurbation picturepicture

∆ T ≈ − 1ζ T 5

ζ = const Φ + Ψ = 0

− = ha∝ 1 hij const ij

ζ 2 2 〈 〉 〈h 〉 −1 ij H inflation t brane-world brane-world ∆T ≈ ζ + Ψ − Φ = ?? perturbationsperturbations T

κ Φ + Ψ = − 2 2π a E KK anisotropy bulk ∝ hij ??

0-mode + KK modes perturbationsperturbations fromfrom inflationinflation

de Sitter brane in AdS5 bulk ‹ like Minkowski brane: tensor zero mode no scalar/ vector 0-mode from 5D graviton ‹ unlike Minkowski: mass gap for KK modes

3 m > H 2

massive modes not excited during inflation tensor 0-mode frozen until horizon re-entry scalar perturbations

‹ no 5D graviton contribution to lowest order

‹ only from density perturbations - decoupleζ from 5D perturbations (large scales) δρ ‹ curvature perturbationδρ can be found without knowledge of bulkρ (large scales) ζ + δ 4 = Φ + ( E ) = + C / a tot 3( + p) m 3(ρ + p)

δρ matter curvature perturbation conserved δρ asρ in GR ρ 2 2 3 ⎛ ⎞ ≈ ⎛ ⎞ ⎛ Vλ⎞ ϕ << ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⇒ COBE M 4 ⎝ ⎠ ⎝ ⎠ GR ⎝ 2 ⎠ „ SachsSachs-Wolfe effect red shift ∆ T =+Ψ−Φζ m T photon Φ

„ metricmetric perturbationsperturbations Φ ζ =Φ−H ⎛⎞& −Ψ tot ⎜⎟, HH& ⎝⎠ Φ+Ψ=− 2122−− l ka ε κπ cannotcannot predict CMB anisotropiesanisotropies unlessunless WeylWeyl anisotropicanisotropic stressstress isis known π E -- lowlow energyenergy approximationapproximation structurestructure formationformation -- veryvery lowlow energyenergy „ bulk curvature scale < 1mm << brane’sbrane’s „ use gradient expansion to solve 5D field equationsequations ∂ << ∇ | y F | | µ F | λ+

λ− < 0 „ need 2 boundary T+ need 2 boundary T− conditions ϕ radion effectiveeffective equationsequations onon ++veve tensiontension branebrane scalar-tensor theory µ κ ν =ϕ 2 1 []+ − 2 + 1 ∇ ∇ − ∇ 2 G µ T+ (1 ) T− ( ) ν ϕ ω ϕ + ( ) ⎛∇ ∇ µ − 1 ∇ 2 ⎞ ϕ 2 ⎜ ν ϕ ( ) ⎟ ⎝ ϕ 2 µ ϕ⎠ µ ϕ µ ϕ δ δ ν µ ϕ ∇ ∇ µ = f (ϕ,T± ) µ ω µ ϕ ν ϕ ϕ ν 3 T+ where ( ) = T_ 2(1−ϕ) scalarscalar perturbationsperturbations

ζ ζ κ δ

2 −4 = − Ca± ± m± ϕ 6 H& ± ϕ + ⎛ + & ⎞ = ()ϕ ϕ S S&& ⎜3H + ⎟S& f + , − ⎝ ⎠ ϕ ϕ + ϕ − π = ϕ E f ( ± , S) branebrane--worldworld CMBCMB anisotropiesanisotropies

modelmodel withwith mostmost simplesimple backgroundbackground WMAP „ radion fixed „ no dark radiation in background

δρ ρ

/ Temperature fluctuation πcdark = ζE r 4 m = ζ E f (cdark, m ) angular scale 0.032 0.18 0.03 0.16 0.028 0.14 2 2 h

h 0.026

b 0.12 DM Ω 0.024 Ω 0.1 0.022 0.08 0.02 −0.2 −0.1 0 0.1 −0.2 −0.1 0 0.1 cdark cdark

30 95 90 25 85 20 0

80 re H z 75 15 70 10 65 60 5 −0.2 −0.1 0 0.1 −0.2 −0.1 0 0.1 cdark cdark furtherfurther workwork

„ simplesimple RSRS model OK so far „ computecompute CMBCMB forfor „ more realistic background „ one-brane model

„ models with bulk dilaton/ 6 6 moduli field „ models with quantum corrections 1+3 „ M theory models? 1+3

1 moremore developeddeveloped modelsmodels

bulkbulk scalarscalar fieldfield ((dilatondilaton//modulimoduli))

κ1 S = d 5 x − (5)g [](5) R − 2Λ (Φ) − 2 ()∂Φ 2 grav 2 ∫ 5 5 2 5 − 4 − λ Φ ∫ d x g ± ( ) κ

*dilaton can drive brane inflation Λ (Φ) 5 *dilaton + shadow matter λ+ dilaton ? dark matter λ− T+ ? dark energy T− *brane-brane collision radion ϕ ? big bang (ekpyrotic) quantumquantum curvaturecurvature correctionscorrections

Gauss-Bonnet (early universe) κ 1 S = d 5 x − (5)g [](5) R − 2Λ + {}(5) R 2 + ... grav 2 ∫ 5 2 5 κ− ∫ d 4 x − gλ α

induced gravity (late universe)

1 λ S = d 5 x −(5)g [](5) R − 2Λ − d 4 x − g []− µ 2 R grav 2 ∫ 5 ∫ 2 5