Gravity in brane world

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

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

3. Alternative compactification scheme (braneworld) Gravity naturally propagates in higher dimensional spacetime. Blue：EM flux line F∝1/r2 Since experimental constraints on gravity force are week, relatively large extradimensions are allowed. ～0.1mm Red：gravity flux line F∝1/rD-2(short） 1/r2(long） Branworld Only gravity propagates in Higher dimensional spacetime. Kaluza-Klein compactification Homogeneous in the direction of extra-dimensions. Standard model fields are localized on the brane. -otherwise, contradiction with observation.

4. Constraint on the deviation from 1/r2 low • Short rage force Capner et al hep-ph/0611184

5. Large extra dimensions (ADD model) Arkani-Hamed, Dimopoulos and Dvali (199８) Example) ?? Assume homogeneity in the direction of extra-dimensions. Volume of extra-dimensions 4-dim our universe ⊗2-dim torus Hierarchy problem n=2 , M6: electroweak scale 1TeV= 103GeV Size of the extra-dimensions d=1mm≈(10-13GeV)-1 (1019GeV)2≈(103GeV)4 (10-13GeV)-2 Effective 4-dim Newton’s constant ~1/16pGN size of extra-dimensions

6. Warped extra dimension Randall Sundrum I model (1999) Another approach to the hierarchy problem 5-dimanti-de Sitter Z2-symmetry ?? +s -s AdScurvature length 5-dim negative cosmological constant L5 y brane tension negative- tension brane positive- tension brane AdS Bulk d

7. Roughly speaking, 4-dim effective theory can be obtained by substituting Kintaro-candy configuration. y 代入 Kintaro candy This part determines the effective gravitational coupling d On the positive tension brane: On the negative tension brane: Hierarchy is not explained, but we have finite Mpleven for d →∞. ⇒ new compactification（RSII） (1019GeV)2 (TeV)2 Hierarchy is explained with d~40{.

8. Gravity in RSII braneworld • What was the impact ofRSII model on cosmology? Compactification leads to a 4-dim effective masslessscalar field corresponding to the size of the extra-dimension. fifth-force (harmful ) Stabilization mechanismto 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○ ½for 4-dim general relativity. Deviation from ½is caused by extra scalar degree of freedom. current bound <10-5 Yukawa potential

9. In contrast, compactification is effectively realized due to the warped geometryin 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. Metric perturbations induced on the brane (Randall Sundrum(‘99) Garriga& T.T. (’99)) Static spherical symmetric case • Not exactly Schwarzschild ⇒ ℓ << 0.1mm • For static and spherically sym. configurations second order perturbation is well behaved. • Correction to 4D GR=O(ℓ 2/R2star) • Giannakis & Ren (’00) exterior • Kudoh, T.T. (’01) interior • Wiseman (’01) numerical Gravity on the brane looks like 4D GR approximately, BUT • No Schwarzshild-like BH solution?

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

12. AdS/CFT correspondence WCFT[q]=SEH+ SGH- S1- S2- S3 ( Maldacena (’98) ) ( Hawking, Hertog, Reall (’00) ) z0→ 0limit is well defined with the counter terms SRS= 2(SEH+ SGH)- 2S1+Smatter = 2S2+Smatter+ 2(WCFT+ S3) Boundary metric Counter terms Brane position brane tension z0 ⇔ cutoff scale parameter 4D Einstein-Hilbert action

13. Generalized AdS/CFT correspondence Classical 5D dynamics in RS II model 4D Einstein gravity +CFT quantum correction equivalent 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 )

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

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

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

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

18. Dvali-Gabadadze-Porrati model (Phys. Lett. B485, 208 (2000)) • Action: induced gravity term Critical length scale • Forr < rc, 4-diminduced gravity term dominates? • Extension is infinite, but4-dim GR seems to be recovered forr < rc. ?? y Brane Minkowski Bulk y=constant

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

20. Self-acceleration has a Ghost normal ghost • 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.

21. Ghost in self-accelerating branch in DGP model • A massive graviton in de Sitter space with 0 <m2 < 2H2 contains a spin-2 ghost mode in general.(Higuchi Nucl. Phys. B282, 397 (1987)) Spin2 ghost: helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation 1 2 2 This mode becomes a ghost • The mass of the lowest KK graviton in self-accelerating branch • m2 = 2H2 for r = 0, 0 < m2 < 2H2 for r > 0. Marginal, but there is a ghost mode in DGP.(e.g. Gorbunov, Koyama and Sibiryakov, Phys. Rev. D73, 044016 (2006))

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. m2 > 2H2 The ghost will disappear. Can we erase the ghost? t y=y+ y=y- r Bulk is Rindler wedge of Minkowski space

23. However, at the point where the spin 2 ghost disappears, spin 0 (brane bending mode) ghost appears instead. • In fact, there is no ghost in spin-2 sector once the second brane (or negative energy density) is introduced. t r+ < 0: H+<1/rc y=y+ y=y- r self-acceleration H+=1/rc spin-2 ghost exists r+ > 0: H+>1/rc far limit close limit (K. Izumi, K. Koyama & T.T, 2007) (single brane: Charmousis et al. 2006) 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. InH→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.

25. Strong coupling scale may introduce a natural cutoff scale?? • If a mode has a small quadratic kinetic term a→ 0 limit strong coupling. loop integral propagator x x vertex x x strong coupling scale = L/a1/2 When spin-2 ghost marginally appears (m2= 2H2), all the scales are necessarily in the strong coupling regime!

26. We may justify 3-momentum cutoff? Action for spin 2, helicity-0 mode (K. Izumi & T.T, 2007) Spin2 ghost: helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation 1 2 2 ghost 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

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

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. Correction to gravity in DGPnormal branch Perturbation equation in weakly non-linear regime: ★ Since this coefficient is extremely small,non-linear termsbecomes important even for weakly gravitating system. :brane bending d.o.f. relatively large strong coupling scale 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 BHsolutions？ 29

30. Gravity in higher co-dimension brane world very shortly In general gravitational potential becomes singular at the brane. 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) 30

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/r3 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.