1. Massive Gravity and the Galileon Claudia de Rham Université de Genève Work with Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava and Andrew Tolley

2. Why Massive Gravity ? Phenomenology Self-acceleration C.C. Problem

3. Why Massive Gravity ? Phenomenology Self-acceleration C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to m  < 10 -25 GeV, how about the graviton?

4. Why Massive Gravity ? Phenomenology Self-acceleration C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to m  < 10 -25 GeV, how about the graviton? Could dark energy be due to an IR modification of gravity? with no ghosts... ? Deffayet, Dvali, Gabadadze, ‘01 Koyama, ‘05

5. Why Massive Gravity ? Phenomenology Self-acceleration C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to m  < 10 -25 GeV, how about the graviton? Could dark energy be due to an IR modification of gravity? with no ghosts... ? Is the cosmological constant small ? OR does it have a small effect on the geometry ? Gravity modified in IR Massive gravity

6. 2 dof A massless spin-2 field in 4d, has 2 dof 5 dof A massive spin-2 field, has 5 dof Massive Gravity

7. In GR, Degrees of freedom Gauge invariance Constraints

8. In GR, In massive gravity, Degrees of freedom Gauge invariance Constraints Shift - Shift does not propagate a constraint remaining degrees of freedom

9. In GR, In massive gravity, Degrees of freedom Gauge invariance Constraints - Shift does not propagate a constraint - Non-linearly, lapse no longer propagates the Hamiltonian Constraint… remaining degrees of freedom Shift lapse Boulware & Deser,1972 Creminelli et. al. hep-th/0505147

10. Avoiding the Ghost  Relying on a larger symmetry group, eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading,... ) The Ghost can be avoided by Massless spin-2 in 5d: 5 dof Massive spin-2 in 4d: 5 dof (+ ghost…) The graviton acquires a soft mass resonance

11. Avoiding the Ghost  Relying on a larger symmetry group, eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading,... )  Pushing the ghost above an acceptable cutoff scale The Ghost can be avoided by Typically, the ghost enters at the scale

12. Avoiding the Ghost  Relying on a larger symmetry group, eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading,... )  Pushing the ghost above an acceptable cutoff scale The Ghost can be avoided by Typically, the ghost enters at the scale That scale can be pushed

13. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time Graviton mass

14. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time Graviton mass Arkani-Hamed, Georgi, Schwartz, hep-th/0210184 Creminelli et. al. hep-th/0505147

15. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time Graviton mass

16. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time Graviton mass

17. In the decoupling limit, with fixed, Decoupling limit pl Which can be formally inverted such that with

18. The ghost can usually be seen in the decoupling limit where the mass term is of the form leading to higher order eom It seems a formidable task to remove these terms to all order in the decoupling limit. Decoupling limit

19. The ghost can usually be seen in the decoupling limit where the mass term is of the form leading to higher order eom But we can attack the problem by the other end: starting with what we want in the decoupling limit Decoupling limit

20. The ghost can usually be seen in the decoupling limit where the mass term is of the form leading to higher order eom But we can attack the problem by the other end: starting with what we want in the decoupling limit Decoupling limit with

21. The ghost can usually be seen in the decoupling limit where the mass term is of the form leading to higher order eom But we can attack the problem by the other end: starting with what we want in the decoupling limit Decoupling limit with CdR, Gabadadze, Tolley, 1011.1232

22. That potential ensures that the problematic terms cancel in the decoupling limit Decoupling limit

23. In the decoupling limit (keeping fixed) with Ghost-free decoupling limit

24. In the decoupling limit (keeping fixed) The Bianchi identity requires Ghost-free decoupling limit

25. In the decoupling limit (keeping fixed) The Bianchi identity requires The decoupling limit stops at 2 nd order. Ghost-free decoupling limit

26. In the decoupling limit (keeping fixed) The Bianchi identity requires The decoupling limit stops at 2 nd order. are at most 2 nd order in derivative These mixings can be removed by a local field redefinition Ghost-free decoupling limit

27. The Galileon For a stable theory of massive gravity, the decoupling limit is CdR, Gabadadze, 1007.0443 The interactions have 3 special features: 1. They are local 2. They possess a Shift and a Galileon symmetry 3. They have a well-defined Cauchy problem (eom remain 2 nd order) Nicolis, Rattazzi and Trincherini, 0811.2197 Corresponds to the Galileon family of interactions The BD ghost can be pushed beyond the scale  3 Coupling to matter

28. Back to the BD ghost… In the ADM decomposition, with The lapse enters quadratically in the Hamiltonian, Boulware & Deser,1972 Creminelli et. al. hep-th/0505147

29. Back to the BD ghost… In the ADM decomposition, with The lapse enters quadratically in the Hamiltonian, Does it really mean that the constraint is lost ? Boulware & Deser,1972 Creminelli et. al. hep-th/0505147

30. Back to the BD ghost… In the ADM decomposition, with The lapse enters quadratically in the Hamiltonian, Does it really mean that the constraint is lost ? The constraint is manifest after integrating over the shift This can be shown - at least up to 4 th order in perturbations - completely non-linearly in simplified cases

31. Massive gravity - Summary We can construct an explicit theory of massive gravity which: 1. Exhibits the Galileon interactions in the decoupling limit ( has no ghost in the decoupling limit) 2. Propagates a constraint at least up to 4 th order in perturbations ( does not excite the 6th BD mode to that order) and indicates that the same remains true to all orders 3. Whether or not the constraint propagates is yet unknown. secondary constraint ? 4.Symmetry ??? CdR, Gabadadze, Tolley, in progress…

32. Consequences for Cosmology

33. Degravitation From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale). But observations tell us

34. Degravitation From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale). But observations tell us Idea behind degravitation: Gravity modified on large distances such that the vacuum energy gravitates more weakly Arkani-Hamed et. al., ‘02 Dvali, Hofmann & Khoury, ‘07 k: 4d momentum

35. The degravitation mechanism is a causal process. 1/m H2H2 time Phase transition  Degravitation

36. The degravitation mechanism is a causal process. 1/m H2H2 time Phase transition  Degravitation

37. In Massive gravity, Degravitation

38. 1/m H2H2 time Relaxes towards a flat geometry even with a large CC

39. Dark Energy Relaxes towards a flat geometry even with a large CC Source the late time acceleration

40. Dark Energy Which branch is possible depends on parameters Branches are stable and ghost-free (unlike self-accelerating branch of DGP) In the screening case, solar system tests involve a max CC to be screened. CdR, Gabadadze, Heisenberg, Pirtskhalava, 1010.1780

41. Summary Galileon interactions arise naturally - in braneworlds with induced curvature (soft mass gravity) - in hard massive gravity with no ghosts in the dec. limit The Galileon can play a crucial role in (stable) models of self- acceleration… …or provide a framework for the study of degravitation On different scales, it can provide a radiatively stable model of inflation leading to potentially large nG... (Cf. Andrew’s talk - Sunday)