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

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

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

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

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

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

7.  4 on the brane  vacuum energy   4 on the brane  vacuum energy  On scales >, gravity reduces to 4D Einstein. On scales >, gravity reduces to 4D Einstein. On scales <, 5D Einstein gravity is recovered: On scales <, 5D Einstein gravity is recovered: Only gravity propagates in the bulk Only gravity propagates in the bulk Randall & Sundrum ’99 Garriga & Tanaka ’99 cosmology of the early universe is modified significantly Experimental bound: <0.1 mm (‘large’ extra-dimension) ( <0.1 mm  t <10 -13 sec)

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

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

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

11. 5D gravity on large scales: r > r 0 4D gravity on small scales: r < r 0 IR modification of gravity: our universe will NOT be described by 4D Friedmann if r 0 ~ H 0 -1 Dvali, Gabadadze & Porrati (’00) DGP braneworld DGP braneworld induced gravity on the brane ‘Dark energy’ due to 5D gravity, leading to H =2/ r 0

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

13. Signatures of 5D bulk appear only if H  1 bulk degrees of freedom = Kaluza-Klein (KK) modes bulk degrees of freedom = Kaluza-Klein (KK) modes ~ massive fields on the brane KK modes are non-negligible when H  1 ‘zero’ mode KK modes important difference from KK cosmology: 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 & Langlois ‘05

14. 5D action with higher derivatives 5D action with higher derivatives Gauss-Bonnet term ··· ghost-free; inspired by string theory Einstein Gauss-Bonnet (EGB) gravity: Deruelle & Delezel ‘00, Charmousis & Dufaux ‘02 Davis ’02, Gravanis & Willison ’02, …….. Lovelock ’71 GB term is topological in 4D. EGB is non-trivial only for D  5. GB term is topological in 4D. EGB is non-trivial only for D  5. Natural extension of Einstein theory in the sense that Natural extension of Einstein theory in the sense that no derivatives higher than second appear in the field eqns. no derivatives higher than second appear in the field eqns. A brane can be consistently embedded in the EGB bulk A brane can be consistently embedded in the EGB bulk despite the presence of R 2 terms. despite the presence of R 2 terms. e.g., Mavromatos & Papantonopoulos ‘05

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

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

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

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

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

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

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

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

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