1. Cascading gravity and de gravitation Claudia de Rham Perimeter Institute/McMaster Miami 2008 Dec, 18 th 2008

2. Based on work on collaboration with “Cascading Gravity and Degravitation”, JCAP02(2008)011 “Cascading DGP”, PRL 100 (251603), 2008 “Tensing the ghost in 6D cascading gravity”, to appear “Towards Cosmology in theories of massive gravity”, to appear Stefan Hofmann, Nordita, Stockholm Justin Khoury, Perimeter, Waterloo Andrew Tolley, Perimeter, Waterloo Oriol Pujolas, CERN Gia Dvali, NYU, New York &CERN Michele Redi, EPFL, Lausanne

3. The c.c. problem The current acceleration of the Universe is well described by a c.c.,  /M pl 2, with  (10 -2 eV) 4 while m e 4 /  ~ 10 36 and M pl 4 /  ~ 10 120 The current acceleration of the Universe is well described by a c.c.,  /M pl 2, with  (10 -2 eV) 4 while m e 4 /  ~ 10 36 and M pl 4 /  ~ 10 120 Why is the vacuum energy so small when quantum effects lead to much bigger corrections? Why is the vacuum energy so small when quantum effects lead to much bigger corrections?

4. The c.c. problem The current acceleration of the Universe is well described by a c.c.,  /M pl 2, with  (10 -2 eV) 4 while m e 4 /  ~ 10 36 and M pl 4 /  ~ 10 120 The current acceleration of the Universe is well described by a c.c.,  /M pl 2, with  (10 -2 eV) 4 while m e 4 /  ~ 10 36 and M pl 4 /  ~ 10 120 Why is the vacuum energy so small when quantum effects lead to much bigger corrections? Why is the vacuum energy so small when quantum effects lead to much bigger corrections? Is the vacuum energy actually small or does it simply gravitate very little ? Is the vacuum energy actually small or does it simply gravitate very little ? idea behind degravitation Dvali, Hofmann&Khoury, hep-th/0703027

5. Small c.c. / weakly gravitating In GR, gravity is mediated by a massless spin-2 particle and gauge invariance makes both questions equivalent. (universality of graviton coupling) In GR, gravity is mediated by a massless spin-2 particle and gauge invariance makes both questions equivalent. (universality of graviton coupling) If gravity was mediated by an effectively massive graviton, gravity would be weaker in the IR the vacuum energy (and other IR sources) would gravitate differently If gravity was mediated by an effectively massive graviton, gravity would be weaker in the IR the vacuum energy (and other IR sources) would gravitate differently Dvali, Hofmann&Khoury, hep-th/0703027

6. Filtering gravity In Einstein’s gravity, the c.c. is bound to gravitate as any other source In Einstein’s gravity, the c.c. is bound to gravitate as any other source The idea behind degravitation is to promote the Newton’s constant G N to a filter operator, The idea behind degravitation is to promote the Newton’s constant G N to a filter operator,

7. Filtering gravity At short wavelengths compared to L, if  >0 G N G 0 N there is no filter and sources gravitate normally, At short wavelengths compared to L, if  >0 G N G 0 N there is no filter and sources gravitate normally, While at long distances, G N 0, so sources with large wavelengths, (such as the c.c.) are filtered out and effectively gravitate very weakly. While at long distances, G N 0, so sources with large wavelengths, (such as the c.c.) are filtered out and effectively gravitate very weakly. with

8. Filtering and graviton mass As such, the theory would not satisfy the Bianchi identity, As such, the theory would not satisfy the Bianchi identity, This cannot represent a consistent theory of massless spin-2 gravitons (with only 2 degrees of freedom) This cannot represent a consistent theory of massless spin-2 gravitons (with only 2 degrees of freedom) Instead the theory should be understood as the limit of a theory of massive gravity, with mass ~1/L. Instead the theory should be understood as the limit of a theory of massive gravity, with mass ~1/L.

9. Filtering and graviton mass Any degravitating (filter) theory must reduce at the linearized level to a theory of massive gravity Any degravitating (filter) theory must reduce at the linearized level to a theory of massive gravity Corresponding to the filter theory Corresponding to the filter theory

10. Filtering and graviton mass To be a satisfying ghost-free degravitating theory, the mass should satisfy with 0 d  < 1. To be a satisfying ghost-free degravitating theory, the mass should satisfy with 0 d  < 1.  = 1 corresponds to the effective 4d theory arising from the 5d DGP model.  = 1 corresponds to the effective 4d theory arising from the 5d DGP model. R (4) Dvali, Gabadadze & Porrati, hep-th/0005016 R (5)

11. DGP – eg. of massive gravity Extra dof arise from 5d nature of theory. Extra dof arise from 5d nature of theory. We live in a (3+1)-brane embedded in an infinite flat extra dimension We live in a (3+1)-brane embedded in an infinite flat extra dimension Dvali, Gabadadze & Porrati, hep-th/0005016 R (5)

12. DGP – eg. of massive gravity Extra dof arise from 5d nature of theory. Extra dof arise from 5d nature of theory. We live in a (3+1)-brane embedded in an infinite flat extra dimension We live in a (3+1)-brane embedded in an infinite flat extra dimension In the UV, the 4d curvature term dominates, gravity looks 4d In the UV, the 4d curvature term dominates, gravity looks 4d In the IR, gravity is 5d. In the IR, gravity is 5d. Dvali, Gabadadze & Porrati, hep-th/0005016 R (4) R (5)

13. DGP – eg. of massive gravity Effective 4d propagator for DGP Effective 4d propagator for DGP This corresponds to a degravitating theory with  =1/2 with induced Friedmann eq. This corresponds to a degravitating theory with  =1/2 with induced Friedmann eq.  =1/2 is too large ! Is there an extension with  <1/2 ???  =1/2 is too large ! Is there an extension with  <1/2 ??? k: 4d momentum m 5 =M 5 3 /M 4 2 Cf. Ghazal Geshnizjani ’s talk

14. Gravity in higher dimensions For a given spectral representation, w e have the “Newtonian potential” For a given spectral representation, w e have the “Newtonian potential” In a (4+n)-dimensional spacetime, the gravitational potential goes as ie. In a (4+n)-dimensional spacetime, the gravitational potential goes as ie. If n=1 (DGP), in the IR G ~p -1  =1/2 If n=1 (DGP), in the IR G ~p -1  =1/2 If n=2, in the IR G ~ log p  =0 If n=2, in the IR G ~ log p  =0 Any higher dim DGP model corresponds to  =0. Any higher dim DGP model corresponds to  =0.  (s)~s n/2-1

15. Higher-codimension sources Cod-1 or pure tension cod-2 are the only meaningful distributional sources. (Geroch&Traschen) Cod-1 or pure tension cod-2 are the only meaningful distributional sources. (Geroch&Traschen) Arbitrary matter on cod-2 and higher distributions lead to metric divergences on the defect. The defect should be regularized. Arbitrary matter on cod-2 and higher distributions lead to metric divergences on the defect. The defect should be regularized. Geroch & Traschen, 1987

16. Cod-2 sources Cod-1 example Cod-1 example Cod-2 divergences Cod-2 divergences

17. Regularizing Cod-2 sources If we had instead the solution is regular (easier to see in momentum space) If we had instead the solution is regular (easier to see in momentum space) The new kinetic term plays the role of a regulator. Effectively represents a brane localized kinetic term. The new kinetic term plays the role of a regulator. Effectively represents a brane localized kinetic term.

18. Cascading gravity

19. Cod-2 cascading Consider the 6d action with couplings Consider the 6d action with couplings L1L1 L2L2 z y

20. Momentum space In momentum space, this corresponds to brane localized couplings  =-M 5 3 (q 5 +k 2 ), and 2 =-M 4 2 k 2. with 2 mass scales m 5 =M 5 3 /M 4 2 and m 6 = M 6 4 /M 5 3. In momentum space, this corresponds to brane localized couplings  =-M 5 3 (q 5 +k 2 ), and 2 =-M 4 2 k 2. with 2 mass scales m 5 =M 5 3 /M 4 2 and m 6 = M 6 4 /M 5 3. L1L1 L2L2 z y 

21. Cod-2 propagator Including both couplings, the propagator on the brane is Including both couplings, the propagator on the brane is As m 6 p k, the propagator behaves as in 6d (  =0) As m 6 p k, the propagator behaves as in 6d (  =0) As m 5 p k p m 6 it takes a 5d behavior As m 5 p k p m 6 it takes a 5d behavior At small scales, k p m 5, we recover 4d. At small scales, k p m 5, we recover 4d. log k log k 2 G -1

22. Cascading Gravity : A Naïve approach The generalization to gravity is straightforward The generalization to gravity is straightforward The tensor mode behaves precisely as the scalar field toy-model, The tensor mode behaves precisely as the scalar field toy-model, However one of the scalar modes propagates a ghost. However one of the scalar modes propagates a ghost.

23. Propagating modes Working around flat space-time, Working around flat space-time, where the tensor mode behaves as expected where the tensor mode behaves as expected and the scalar field  is also regularized by the cod-1 brane and the scalar field  is also regularized by the cod-1 brane source term

24. Ghost mode  is finite on the cod-2 brane,  is finite on the cod-2 brane, However in the UV,  ~ + T However in the UV,  ~ + T While in the IR,  ~ - T. While in the IR,  ~ - T. The kinetic term changes sign, signaling the presence of a ghost. The kinetic term changes sign, signaling the presence of a ghost. In the UV, the gravitational amplitude is In the UV, the gravitational amplitude is  

25. Ghost mode  is finite on the cod-2 brane,  is finite on the cod-2 brane, However in the UV,  ~ + T However in the UV,  ~ + T While in the IR,  ~ - T. While in the IR,  ~ - T. The kinetic term changes sign, signaling the presence of a ghost. The kinetic term changes sign, signaling the presence of a ghost. In the UV, the gravitational amplitude is In the UV, the gravitational amplitude is   = -1/3-1/6

26. Ghost mode This ghost is completely independent to the ghost present in the self-accelerating branch of DGP. This ghost is completely independent to the ghost present in the self-accelerating branch of DGP. However, it is generic to any cod-2 and higher framework with localized kinetic terms. However, it is generic to any cod-2 and higher framework with localized kinetic terms. In particular it is present when considering a pure cod-2 scenario (no cascading). In particular it is present when considering a pure cod-2 scenario (no cascading). L2L2 Gabadadze&Shifman hep-th/0312289

27. Curing the ghost There are two ways to cure the ghost: 1. Adding a tension on the brane 2. Regularizing the brane. There are two ways to cure the ghost: 1. Adding a tension on the brane 2. Regularizing the brane.

28. Curing the ghost There are two ways to cure the ghost: 1. Adding a tension on the brane 2. Regularizing the brane. There are two ways to cure the ghost: 1. Adding a tension on the brane 2. Regularizing the brane. Both approaches lead to a well-defined 4d effective theory, with gravitational amplitude Both approaches lead to a well-defined 4d effective theory, with gravitational amplitude = 1/3-1/12 = 1/2-1/6-1/12

29. Cosmology Cf. Ghazal Geshnizjani ’s talk

30. de Sitter solutions To find some de Sitter solution, can slice the 6d Minkowski bulk as To find some de Sitter solution, can slice the 6d Minkowski bulk as and take the cod-1 brane located at the cod-2 at. and take the cod-1 brane located at the cod-2 at.

31. dS solutions in 6d The Cod-1 is not flat The Cod-1 is not flat But the brane adapts its position to balance the extrinsic curvature and the Einstein tensor on the brane for y > 0 But the brane adapts its position to balance the extrinsic curvature and the Einstein tensor on the brane for y > 0 R5R5R5R5for this configuration can only support a minimal H

32. dS solutions in 6d The Friedmann eq. on the brane is then The Friedmann eq. on the brane is then from brane EH R 4

33. dS solutions in 6d The Friedmann eq. on the brane is then The Friedmann eq. on the brane is then Solution only makes sense for minimal tension Solution only makes sense for minimal tension from brane EH R 4

34. dS solutions in 6d The Friedmann eq. on the brane is then The Friedmann eq. on the brane is then Solution only makes sense for minimal tension Solution only makes sense for minimal tension which is the same bound as the no-ghost condition in the deficit angle solution. which is the same bound as the no-ghost condition in the deficit angle solution. from brane EH R 4

35. Properties of the solution Away for the source, the cod-1 brane asymptotes to a constant position Away for the source, the cod-1 brane asymptotes to a constant position The 6d bulk is Minkowski (in non trivial coordinates) volume of the extra dimensions is infinite, there are no separate massless zero mode. The 6d bulk is Minkowski (in non trivial coordinates) volume of the extra dimensions is infinite, there are no separate massless zero mode. Asymptotically, the 5d brane is flat

36. Properties of the Friedmann eq. Does correspond to a IR modification of gravity Does correspond to a IR modification of gravity Could in principle have a large  with a small H Could in principle have a large  with a small H BUT still a local expression… BUT still a local expression…

37. Properties of the Friedmann eq. Does correspond to a IR modification of gravity Does correspond to a IR modification of gravity Could in principle have a large  with a small H Could in principle have a large  with a small H BUT still a local expression… BUT still a local expression… In the absence of brane EH term, there is a self- accelerating solution ghost?? In the absence of brane EH term, there is a self- accelerating solution ghost??

38. Properties of the Friedmann eq. Does correspond to a IR modification of gravity Does correspond to a IR modification of gravity Could in principle have a large  with a small H Could in principle have a large  with a small H BUT still a local expression… BUT still a local expression… In the absence of brane EH term, there is a self- accelerating solution ghost?? In the absence of brane EH term, there is a self- accelerating solution ghost?? although different from the “standard self-acceleration’’ although different from the “standard self-acceleration’’

39. Properties of the Friedmann eq. Does correspond to a IR modification of gravity Does correspond to a IR modification of gravity Could in principle have a large  with a small H Could in principle have a large  with a small H BUT still a local expression… BUT still a local expression… In the absence of brane EH term, there is a self- accelerating solution ghost?? In the absence of brane EH term, there is a self- accelerating solution ghost?? If the solution was unstable, would be interesting to see where it decays to… If the solution was unstable, would be interesting to see where it decays to…

40. Conclusions Models of massive gravity represent a novel framework to understand the c.c. problem Models of massive gravity represent a novel framework to understand the c.c. problem There is to date only one known ghost-free non- perturbative theory capable of exhibiting a model of massive gravity that does not violate Lorentz invariance: that is DGP and its Cascading extension. There is to date only one known ghost-free non- perturbative theory capable of exhibiting a model of massive gravity that does not violate Lorentz invariance: that is DGP and its Cascading extension.

41. Conclusion In 6d cascading gravity, there are at least 2 kind of different solutions for a pure tension source: In 6d cascading gravity, there are at least 2 kind of different solutions for a pure tension source: static, “wedge solution”  de Sitter solution 