braneworld cosmology YITP Kyoto U Misao Sasaki Plan

曲がる余剰次元と宇宙項 - braneworld cosmology - YITP, Kyoto U Misao Sasaki

Plan of this talk 1. Introduction --- relativist/cosmologist’s view of braneworld --- Randall-Sundrum’s (single) braneworld 2. Brane cosmology from the bulk point of view 3. Alternative models --- DGP model, Bulk inflaton model, … 4. Einstein Gauss-Bonnet braneworld 5. Summary

1. Introduction n Braneworld domain wall with matter fields 3 -brane = 4 D time like (singular) hypersurface in 5 D spacetime (bulk). Brane tension (s) = vacuum energy 0 → self-gravitating domain wall Causality on the brane causality in the bulk Ishihara ‘ 01 A new picture of the Universe!

Two different views on gravity in the bulk string (~particle) picture geometrical (~wave) picture emission of closed loops spacetime fluctuations This talk is based on a view from this standpoint Need to UNIFY the two pictures!

One fact worth keeping in mind …. In General Relativity or any theory with 1/rn-2 force law in n-dim spacetime, embedding a self-gravitating brane with co-dim > 2 ie, object with Tmn~ dp(x) where p>2 in n-dim will generically form a black hole (brane)! Example: n=4 • domain wall (p=1) • string (p=2) • monopole (p=3) thin (singular) limit → OK → marginally OK → BH! need to consider a thick brane for co-dim>2 (future issue)

l Randall-Sundrum’s (single-) braneworld RS ‘ 99 ··· 5 D Einstein theory with L 5<0 5 D Anti-de Sitter (Ad. S 5) • Warp factor b(y) e-|y|/ decreases exponentially as |y|→. • Extra dim is effectively compact with size ~ • gravity is confined within |y| Z 2 -symmetry

n n L 4 on the brane vacuum energy s On scales > , gravity reduces to 4 D Einstein. On scales < , 5 D Einstein gravity is recovered: Randall & Sundrum ’ 99 Garriga & Tanaka ’ 99 n Only gravity propagates in the bulk Experimental bound: <0. 1 mm (‘large’ extra-dimension) cosmology of the early universe is modified significantly ( <0. 1 mm t <10 -13 sec)

2. Brane cosmology from the bulk point of view Binetruy et al. (’ 99), Krauss (’ 99), Ida (’ 99), . . . 5 D Ad. S-Schwarzschild as a bulk Induced metric on the brane

Junction condition • Cosmological evolution deviates from 4 D at H 1 • m/R 4 term due to BH in Ad. S bulk (5 D Weyl tensor ‘Emn’) ~ ‘radiation’ on the brane Shiromizu, Maeda & MS ‘ 99 dark radiation / Weyl fluid ~ conformal field Ad. S/CFT

3. Alternative models RS braneworld 5 D Einstein gravity • Simplest non-trivial realization of braneworlds. • There are many other possibilities. • RS two-brane model → existence of ‘radion’ = distance between 2 branes radion is tachyonic in cosmology! m 2~ -4 H 2 Gen & MS (’ 01) Frolov & Kofman (’ 03) may have interesting implications e. g. , Kanno, Soda & Wands (’ 05)

• DGP braneworld Dvali, Gabadadze & Porrati (’ 00) induced gravity on the brane 5 D gravity on large scales: r > r 0 4 D gravity on small scales: r < r 0 IR modification of gravity: our universe will NOT be described by 4 D Friedmann if r 0~H 0 -1 ‘Dark energy’ due to 5 D gravity, leading to H=2/r 0

• 5 D theory may not be pure Einstein: • scalar fields f (moduli, dilaton, …) ··· bulk inflaton model Kobayashi & Soda, Himemoto & MS, Maeda Wands (’ 00), … L 5, eff determined by 5 D dynamics of f H 2=L 4, eff/3 ··· Hubble parameter on the brane 1/ 2=-L 5, eff/6 ··· Effective Ad. S curvature in the bulk Low energy (H 1) effective theory on the brane = 4 D Einstein-scalar system Minamitsuji, Himemoto & MS, Kanno & Soda (’ 03) Brane can inflate without inflaton on the brane, while the bulk is collapsing!

Signatures of 5 D bulk appear only if H 1 • bulk degrees of freedom = Kaluza-Klein (KK) modes ‘zero’ mode KK modes ~ massive fields on the brane KK modes are non-negligible when H 1 • important difference from KK cosmology: KK effects on the brane KK modes in KK cosmology (classical) KK modes act as negative energy dust on the brane! Minamitsuji, MS & Langlois ‘ 05

• 5 D action with higher derivatives Gauss-Bonnet term ··· ghost-free; inspired by string theory e. g. , Mavromatos & Papantonopoulos ‘ 05 Einstein Gauss-Bonnet (EGB) gravity: GB term is topological in 4 D. EGB is non-trivial only for D 5. • • • Natural extension of Einstein theory in the sense that Lovelock ’ 71. no derivatives higher than second appear in the field eqns A brane can be consistently embedded in the EGB bulk despite the presence of R 2 & terms. Deruelle Delezel ‘ 00, Charmousis & Dufaux ‘ 02 Davis ’ 02, Gravanis & Willison ’ 02, ……. .

4. Einstein Gauss-Bonnet (EGB) braneworld • EGB also admits Ad. S bulk, but with curvature radius given by convenient parameter (We assume a>0 & L 5<0) ‘-’ branch reduces to Ad. S in Einstein in the limit a→ 0: 1/ 2 → 1/ 02 ≡-L 5/6 ‘+’ branch is known to be unstable (but may have interesting cosmological implications)

• (Z 2 -symmetric) de Sitter brane in Ad. S bulk de Sitter with radius H-1 tension: t =const. hypersurface in 5 D (two identical H 4 cut and pasted at radius y=y 0 , b(y 0)=1)

• Flat (Minkowski) brane if s is tuned: • Gravity on the brane reduces to 4 D Einstein on scales > , as in the case of RS braneworld. (because the extra-dimension is essentially compact) However, ……. .

Gravity never becomes 5 D even at short distances! Deruelle & MS ‘ 03 scalar-tensor (~Brans-Dicke) theory at r < Davis ’ 04 Experimental bound on is relaxed substantially: 1 km ~ < ~< 100 km is also possible if a ~ 1

Thus, 5 D EGB leads to intrigueing possibilities that • Size (curvature radius) of the extra-dim may be macroscopic. ~10 km is an interesting scale for SN/NS/GW physics • Scalar-tensor gravity on the brane at short distances, while Einstein (tensor) gravity at long distances. Emergence of an (effectively massive) 4 D scalar field Impact on the early universe cosmology? ~10 km t ~10 -4 sec

Linearized gravity on de Sitter brane Minamitsuji & MS ‘ 04 fluctuation of brane position negligible at r 5 D (KK) effect → BD gravity ‘KK propagator’ spin 2 part of matter stress tensor Integro-differential equations

5. Summary l Braneworld gives a new picture of the universe l Different models can lead to a variety of predictions difference can appear at short or long distances l. In DGP braneworld, 4 D at r r 0 , 5 D at r r 0 Infinitely large extra dimension “dark energy” without dark energy l. In EGB braneworld, BD at r , Einstein at r KK modes summed up to give 4 D-like behavior Extra dim can be as large as ~ 10 km

Effect of KK modes (5 D gravity) in the braneworlds is highly non-trivial: brane geometry effective 4 D geometry in conventional KK cosmology KK modes lead to • cosmological constant in DGP • massive scalar in EGB • negative energy dust in RS • ….

There are surely much more to be done in braneworld cosmology. string-inspired models, cosmological perturbations, ‘thick’ braneworld, higher co-dimensions, black holes, quantum effects, …. and observational predictions! any hint of deviation from Einstein? Models with instabilities should not be rejected at once. After all, instabilities are needed in cosmology!