1. Gravity in brane world Takahiro Tanaka (Kyoto univ.) 2008/March @AIU2008

2. Higher dimensional models of particle physics Superstring theory (10dim), M-theory (11dim) Our universe is 4-dimensional. → Compactification is necessary ! 2 Compactifiation y Higher Dimensional bulk flat xx identify If the size of the extra-dimensions is very small, we will not notice its presence. Basic idea :

3. 3 Kaluza-Klein compactification Homogeneous in the direction of extra-dimensions. Branworld Only gravity propagates in Higher dimensional spacetime. Alternative compactification scheme (braneworld) Red ： gravity flux line F ∝1/ r D -2 (short ） 1/ r 2 (long ） Blue ： EM flux line F ∝1/ r 2 Since experimental constraints on gravity force are week, relatively large extradimensions are allowed. ～0.1mm Gravity naturally propagates in higher dimensional spacetime. Standard model fields are localized on the brane. -otherwise, contradiction with observation.

4. 4 Constraint on the deviation from 1/r 2 low Short rage force Capner et al hep-ph/0611184

5. 5 Large extra dimensions (ADD model) Effective 4-dim Newton’s constant ~1/16   G N 4-dim our universe ⊗2-dim torus ?? Arkani-Hamed, Dimopoulos and Dvali (199 ８ ) Assume homogeneity in the direction of extra-dimensions. Volume of extra-dimensions size of extra- dimensions n=2, M 6 : electroweak scale 1TeV= 10 3 GeV Size of the extra-dimensions d=1mm ≈ (10 -13 GeV) -1 (10 19 GeV) 2 ≈ (10 3 GeV) 4 (10 -13 GeV) -2 Hierarchy problem Example)

6. 6 Warped extra dimension 5-dim anti-de Sitter Randall Sundrum I model (1999) y AdS Bulk 55 ?? negative- tension brane positive- tension brane d Z 2 -symmetry  Another approach to the hierarchy problem AdS curvature length 5-dim negative cosmological constant brane tension

7. 7 d y Roughly speaking, 4-dim effective theory can be obtained by substituting Kintaro-candy configuration. Kintaro candy 代入 On the negative tension brane: (10 19 GeV) 2 (TeV) 2 Hierarchy is explained with d~40 {. On the positive tension brane: Hierarchy is not explained, but we have finite M pl even for d →∞. ⇒ new compactification （RSII） This part determines the effective gravitational coupling

8. What was the impact of RSII model on cosmology? Compactification leads to a 4-dim effective massless scalar field corresponding to the size of the extra-dimension. fifth-force (harmful ) Stabilization mechanism to kill this extra scalar d.o.f.. ⇒ The scalar d.o.f. becomes massive. Harmless, but the effect of extra-dimensions does not appear at all at the length scale larger than (mass scale) -1 ～(size of compactification). （￣▽￣）。ｏ０ ○ 8 Gravity in RSII braneworld ½ for 4-dim general relativity. Deviation from ½ is caused by extra scalar degree of freedom. current bound <10 -5 Yukawa potential

9. 9 In contrast, compactification is effectively realized due to the warped geometry in RS-II model, although the extra- dimension extends infinitely. As we do not need stabilization of the volume, gravity at large distance is non-trivial ! Brane ??

10. Static spherical symmetric case  Not exactly Schwarzschild ⇒ ℓ << 0.1mm Metric perturbations induced on the brane For static and spherically sym. configurations second order perturbation is well behaved. Correction to 4D GR=O (ℓ 2 /R 2 star ) »Giannakis & Ren (’00) exterior »Kudoh, T.T. (’01) interior »Wiseman (’01) numerical  No Schwarzshild-like BH solution? Gravity on the brane looks like 4D GR approximately, BUT 10 ( Randall Sundrum (‘99) Garriga & T.T. (’99))

11. z Kintaro candy solution Black string solution Metric induced on the brane is exactly Schwarzschild solution. However, this solution is singular. C  C  ∝ z 4 Moreover, this solution is unstable. Gregory Laflamme instability ( Chamblin, Hawking, Reall (’00) ) 11 “Black string longer than its radius is unstable.”

12. AdS/CFT correspondence W CFT [q]=S EH + S GH  S 1  S 2  S 3 ( Maldacena (’98) ) ( Hawking, Hertog, Reall (’00) ) z 0 →  limit is well defined with the counter terms S RS = 2(S EH + S GH )  2S 1  S matter = 2S 2  S matter  2(W CFT + S 3 ) Boundary metricCounter terms Brane position z 0 ⇔ cutoff scale parameter brane tension 4D Einstein-Hilbert action 12

13. Evidences for AdS/CFT correspondence Linear perturbation around flat background (Duff & Liu (’00)) Friedmann cosmology ( Shiromizu & Ida (’01) ) Localized Black hole solution in 3+1 dimensions ( Emparan, Horowitz, Myers (’00) ) Tensor perturbation around Friedmann ( Tanaka ) 13 4D Einstein gravity +CFT quantum correction equivalent Classical 5D dynamics in RS II model Generalized AdS/CFT correspondence

14. 4D Einstein+CFT with the lowest order quantum correction Classical black hole evaporation conjecture 5D BH on brane 4D BH with CFT equivalent Classical 5D dynamics in RS II model number of field of CFT Hawking radiation in 4D Einstein+CFT picture equivalent Classical evaporation of 5D BH AdS/CFT correspondence (T.T. (’02), Emparan et al (’02)) Time scale of BH evaporation 14

15. Metric induced on the brane looks like Schwarzschild solution, but Black Hole solution in 3+1 braneworld This static solution is not a counter example of the conjecture. Casimir energy of CFT fields on with is given by “At the lowest order there is no black hole. Hence, absence of Hawking radiation is consistent.” ( Emparan, Horowitz, Myers (’00) ) The above metric is a solution with this effective energy momentum tensor. Emparan et al (’02) 15 Exact solution exists in 3+1-dim.

16. Numerical construction of brane BH Static and spherical symmetric configuration T, R and C are functions of z and r. Kudoh, Nakamura & Tanaka (‘03) Kudoh (’04) Comparison of 4D areas with 4D and 5D Schwarzschild sols. 4D Sch. 5D Sch.  is surface gravity It becomes more and more difficult to construct brane BH solutions numerically for larger BHs. Small BH case (  –1 < ℓ ) is beyond the range of validity of the AdS/CFT correspondence. 16

17. We need to solve only the Hamiltonian constraint to obtain a time-symmetric initial data: easier! Time-symmetric initial data for brane BH Tanahashi & Tanaka (to appear in JHEP) 1) Even an initial data might be difficult to construct for large AH area. 1) It was possible to construct an initial data with large AH area. 2) We failed to obtain an initial data with M BH < M BS for the same AH area, Initial data is not unique, but 2) If there is a stable static BH, we expect M BH < M BS for the same horizon area. Results: Next step is its time evolution! which is consistent with “classical BH evaporation conjecture”. 17

18. Dvali-Gabadadze-Porrati model Action: (Phys. Lett. B485, 208 (2000)) For r  r c, 4- dim induced gravity term dominates? Extension is infinite, but 4- dim GR seems to be recovered for r  r c. Brane ?? y Minkowski Bulk Critical length scale y  constant induced gravity term 18

19. Flat Friedmann equation In early universe, cosmic expansion is normal. Late time behavior for  = +1 Cosmology in DGP model self-acceleration in the limit  → 0 figure taken from Chamousis et.al. hep-th/0604086 (Deffayet (2006)) 19

20. Self-acceleration has a Ghost normalghost Spontaneous pair production of ghost and normal particles unstable vacuum. Once a channel opens, Lorentz invariance leads to divergence. Maybe we need non-Lorentz invariant cutoff. 20

21. Ghost in self-accelerating branch in DGP model The mass of the lowest KK graviton in self-accelerating branch m 2 = 2H 2 for  = 0, 0 0. A massive graviton in de Sitter space with 0 < m 2 < 2H 2 contains a spin-2 ghost mode in general. (Higuchi Nucl. Phys. B282, 397 (1987)) Marginal, but there is a ghost mode in DGP. (e.g. Gorbunov, Koyama and Sibiryakov, Phys. Rev. D73, 044016 (2006)) Spin2 ghost: helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation 1 2 2 This mode becomes a ghost 21

22. Can we erase the ghost simply by putting the second regulator brane in the bulk? The idea is: if the distance between two branes becomes closer, the KK mass will increase. m 2 > 2H 2 The ghost will disappear. Can we erase the ghost? Bulk is Rindler wedge of Minkowski space t r y=y  y=y  22

23. In fact, there is no ghost in spin-2 sector once the second brane (or negative energy density) is introduced. t r y=y  y=y  self-acceleration H   r c    : H   r c far limitclose limit However, at the point where the spin 2 ghost disappears, spin 0 (brane bending mode) ghost appears instead. (K. Izumi, K. Koyama & T.T, 2007) (single brane: Charmousis et al. 2006)    : H   r c spin-2 ghost exists 23 Stubborn ghost

24. Spontaneous pair production of ghost and normal particles Vacuum becomes unstable. particle production rate diverges due to UV contribution, but it is a bit strange that UV behavior is affected by the value of cosmological constant. In H → 0 limit there is no ghost. If there is a non-covariant cutoff scale, the pair production rate becomes finite. Then, the model might be saved. Do we really need to be afraid of spin 2 ghost in de Sitter space? Here we discuss general massive gravity theory in de Sitter background. 24

25. Strong coupling scale If a mode has a small quadratic kinetic term  → 0 limit strong coupling. x x xx When spin-2 ghost marginally appears (m  = 2H  ), all the scales are necessarily in the strong coupling regime ! strong coupling scale = L/  1/2 may introduce a natural cutoff scale?? 25 vertex propagator loop integral

26. We may justify 3-momentum cutoff? Spin2 ghost: helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation 1 2 2 ghost Action for spin 2, helicity-0 mode Action for helicity-0 mode, depending on 3-momentum k, is not covariant, but spin 2 graviton in total is covariant classical mechanically. ≡background covariance ≡de Sitter invariance (K. Izumi & T.T, 2007) 26

27. However, quantum mechanical state will lose covariance. Quantization of a ghost)  ⇒ -ve (ghost case)  ⇒ +ve (normal case) To make the ground state wave function normalizable, & negative energy states Quantization which avoids negative norm distinguishes ghost from normal case. 27

28. 28 Self-acceleration branch of DGP model has a ghost. Ghost is composed of helicity 0-mode in spin-2 sector. Quantization of this ghost breaks Lorentz invariance. The strong coupling energy scale is low. It’s not completely clear if violent particle production occurs because the relevant modes are in the strong coupling regime. Why did the ghost appear?

29. 29 Correction to gravity in DGP normal branch Perturbation equation in weakly non-linear regime: Substituting the linearised version of the above equation into ★, Once non-linear term becomes important, one can neglect in Eq.★. Then, 4D GR is reproduced. ★ Is 4D GR a good approximation even for strongly gravitating system？ How about BH solutions？ Since this coefficient is extremely small, non-linear terms becomes important even for weakly gravitating system. :brane bending d.o.f. relatively large strong coupling scale

30. 30 Gravity in higher co-dimension brane world In general gravitational potential becomes singular at the brane. very shortly Some regularization is necessary. co-dimension 1 brane + KK compactification. ~similar to 5-dim cases Gauss-Bonnet term in the bulk (6-dim). ~does not seem to work as is initially proposed. Nested brane world with induced gravity terms. Ghost appears but it is claimed that the ghost can be erased by putting sufficiently large 4-dim tension. Not as stubborn as the ghost in self-accelerating branch? (Bostock, Gregory, Navarro, Santiago(2003)) (de Rham, Dvali, Hofmann, Khoury, Pujolas, Redi, Tolley, arXiv:0711.2072)

31. 31 Summary Gravity is quite non-trivial in several brane world models. RS-II model Extension is infinite, but effectively 4-dim gravity is realized. 1/ r 3 potential: correction is not exponentially suppressed. Stationary black hole solution may not exist. (classical BH evaporation conjecture) Induced gravity (DGP model) Self-acceleration branch has a ghost (helicity 0 mode of massive graviton), which might be less harmful than the usual ghost. Normal branch is also abnormal. Non-linear terms are important for the recovery of 4-dim GR. Superluminal motion around the gravitating body. Black hole solution is not found. Higher derivative, Higher co-dimensions, etc.