1. Claudia de Rham July 30 th 2012 1206.3482 with S. Renaux-Petel YITP International Molecule-type Workshop "Nonlinear massive gravity theory and its observational test"

2. GR has been a successful theory from mm length scales to Cosmological scales Then why Modify Gravity ?

3. Why Modify Gravity ? There is little doubt that GR breaks down at high energy (Mpl ? TeV ???)

4. Why Modify Gravity ? There is little doubt that GR breaks down at high energy (Mpl ? TeV ???) Does gravity break down in the IR ?) Late time acceleration & CC problem First signs of the breakdown of GR on cosmological scales ?

5. What is Dark Energy? Graviton is spin-2, massless field Graviton is not spin-2, massless field DE is not dynamical (no new dof) DE is dynamical (new dof) Minimally coupled to matter Non-minimally coupled to matter Quintessence Backreaction Brans Dicke eg. f(R) Unique Landscape of vacua Tunneling dynamics Anthropic arguments Eternal inflation Graviton has mass (resonance) Lorentz invariance is broken Nonlocality Consistent IR modification of gravity Ghost condensate Infinite extra dimensions Self- acceleration Filtering / degravitation mechanism DGP Cascading Gravity de Rham, Tolley 2008

6. Tackling Dark Energy Either ignore the vacuum energy from particle physics and find another source for the acceleration candidate for Dark Energy Additional component in the Universe Modification of General Relativity

7. Tackling Dark Energy Either ignore the vacuum energy from particle physics and find another source for the acceleration candidate for Dark Energy Additional component in the Universe Modification of General Relativity New degrees of freedom

8. Tackling Dark Energy Either ignore the vacuum energy from particle physics and find another source for the acceleration Or try to reconciliate the amount of vacuum energy with the observed acceleration Change natural scaleFind a way to hide a large vacuum energy

9. Tackling Dark Energy Either ignore the vacuum energy from particle physics and find another source for the acceleration Or try to reconciliate the amount of vacuum energy with the observed acceleration Change natural scaleFind a way to hide a large vacuum energy Can we change Gravity to tackle this issue ?

10. Massive Gravity In GR (or Newtonian gravity), Gravity has an infinite range If gravity was mediated by a massive particle, the Newtons law would shut down at some distance ~ m -1

11. Massive Gravity effectively filters out a CC Massive Gravity

12. Massive Gravity effectively filters out a CC Massive Gravity k2k2 m2m2

13. Massive Gravity effectively filters out a CC Massive Gravity k2k2 m2m2 Degravitation Arkani-Hamed, Dimopoulos, Dvali &Gabadadze, 02 Dvali, Hofmann & Khoury, 07

14. Worries 1.Tuning issue…

15. To be consistent with observations, the graviton mass ought to be tuned, Which is the same tuning as the vacuum energy, Tuning / Fine-tuning

16. To be consistent with observations, the graviton mass ought to be tuned, Which is the same tuning as the vacuum energy, But… we expect the graviton mass to be protected against large quantum corrections Tuning / Fine-tuning

17. Worries 1.Tuning issue… 2. If gravity does not see the cosmological constant, why is the Universe accelerating … k2k2 m2m2 CC

18. The degravitation mechanism is a causal process. Relaxation mechanism 1/m H2H2 time Phase transition

19. The degravitation mechanism is a causal process. Relaxation mechanism 1/m H2H2 time Phase transition

20. Worries 1.Tuning issue… 2. If gravity does not see the cosmological constant, why is the Universe accelerating … 3. How do we describe a theory of massive gravity & hide the new dofs ??? Stirring up a hornet's nest…

21. Ghost-free Massive Gravity In 4d, there is a 2-parameter family of ghost free theories of massive gravity CdR, Gabadadze, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232

22. Ghost-free Massive Gravity In 4d, there is a 2-parameter family of ghost free theories of massive gravity Absence of ghost has now been proved fully non- perturbatively in many different languages CdR, Gabadadze, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232 Hassan & Rosen, 1106.3344 CdR, Gabadadze, Tolley, 1107.3820 CdR, Gabadadze, Tolley, 1108.4521 Hassan & Rosen, 1111.2070 M. Mirbabayi, 1112.1435 Hassan, Schmidt-May & von Strauss, 1203.5283

23. Ghost-free Massive Gravity In 4d, there is a 2-parameter family of ghost free theories of massive gravity Absence of ghost has now been proved fully non- perturbatively in many different languages As well as around any reference metric, be it dynamical or notBiGravity !!! Hassan, Rosen & Schmidt-May, 1109.3230 Hassan & Rosen, 1109.3515

24. Degrees of Freedom Massive Gravity 1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0 5 dofs

25. Degrees of Freedom Massive Gravity 1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0 - 2 dof in metric (after gauge fixing) - 3 Stückelberg fields 5 dofs Restore diff invariance

26. Degrees of Freedom Massive Gravity Bi-Gravity 1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0 - 2 dof in metric (after gauge fixing) - 3 Stückelberg fields 1 massive & 1 massless spin-2 - 2x2 helicity-2 - 2 helicity-1 - 1 helicity-0 - 2x2 dof in both metrics (after gauge fixing) - 3 Stückelberg fields 5 dofs7 dofs Restore diff invarianceRestore 2 nd copy of diff invariance

27. Ghost-free Massive Gravity One can construct a consistent theory of massive gravity around any reference metric which - propagates 5 dof in the graviton (free of the BD ghost) - one of which is a helicity-0 mode which behaves as a scalar field couples to matter - hides itself via a Vainshtein mechanism Vainshtein, PLB39, 393 (1972)

28. But... The Vainshtein mechanism always comes hand in hand with superluminalities... This doesnt necessarily mean CTCs, but - there is a family of preferred frames - there is no absolute notion of light-cone. Burrage, CdR, Heisenberg & Tolley, 1111.5549

29. But... The Vainshtein mechanism always comes hand in hand with superluminalities... The presence of the helicity-0 mode puts strong bounds on the graviton mass

30. But... The Vainshtein mechanism always comes hand in hand with superluminalities... The presence of the helicity-0 mode puts strong bounds on the graviton mass Is there a different region in parameter space where the helicity-0 mode could also be absent ???

31. Change of Ref. metric Hassan & Rosen, 2011 Consider massive gravity around dS as a reference ! dS Metric Metric dS is still a maximally symmetric ST Same amount of symmetry as massive gravity around Minkowski !

32. Massive Gravity in de Sitter Only the helicity-0 mode gets seriously affected by the dS reference metric

33. Massive Gravity in de Sitter Only the helicity-0 mode gets seriously affected by the dS reference metric Healthy scalar field (Higuchi bound) Higuchi, NPB282, 397 (1987)

34. Massive Gravity in de Sitter Only the helicity-0 mode gets seriously affected by the dS reference metric Fasiello & Tolley, 1206.3852 Healthy scalar field (Higuchi bound) Higuchi, NPB282, 397 (1987)

35. Massive Gravity in de Sitter Only the helicity-0 mode gets seriously affected by the dS reference metric The helicity-0 mode disappears at the linear level when

36. Massive Gravity in de Sitter Only the helicity-0 mode gets seriously affected by the dS reference metric The helicity-0 mode disappears at the linear level when Recover a symmetry Deser & Waldron, 2001

37. Partially massless Is different from the minimal model for which all the interactions cancel in the usual DL, but the kinetic term is still present

38. Partially massless Is different from the minimal model for which all the interactions cancel in the usual DL, but the kinetic term is still present Is different from FRW models where the kinetic term disappears in this case the fundamental theory has a helicity-0 mode but it cancels on a specific background

39. Partially massless Is different from the minimal model for which all the interactions cancel in the usual DL, but the kinetic term is still present Is different from FRW models where the kinetic term disappears in this case the fundamental theory has a helicity-0 mode but it cancels on a specific background Is different from Lorentz violating MG no Lorentz symmetry around dS, but still have same amount of symmetry.

40. (Partially) massless limit Massless limit GR + mass term Recover 4d diff invariance GR in 4d: 2 dof (helicity 2)

41. (Partially) massless limit Massless limit Partially Massless limit GR + mass term Recover 4d diff invariance GR GR + mass term Recover 1 symmetry Massive GR 4 dof (helicity 2 &1) Deser & Waldron, 2001 in 4d: 2 dof (helicity 2)

42. Implications of the Symmetry GR 4d diff invariance -Kills 3 dofs -Imposes matter to be conserved !

43. Implications of the Symmetry GR Partially Massless 4d diff invariance new symmetry -Kills 3 dofs -Imposes matter to be conserved ! -Kills 1 dof -Does NOT impose matter to be conserved !

44. Implications of the Symmetry GR Partially Massless 4d diff invariance new symmetry -Kills 3 dofs -Imposes matter to be conserved ! -Kills 1 dof -Does NOT impose matter to be conserved ! - Instead PM symmetry fixes the vacuum energy to 0 ! First symmetry which could explain the CC problem ! Deser & Waldron

45. Non-linear partially massless

46. Lets start with ghost-free theory of MG, But around dS dS ref metric

47. Non-linear partially massless Lets start with ghost-free theory of MG, But around dS And derive the decoupling limit (ie leading interactions for the helicity-0 mode) dS ref metric But we need to properly identify the helicity-0 mode first....

48. Helicity-0 on dS on de Sitter on Minkowski To identify the helicity-0 mode on de Sitter, we copy the procedure on Minkowski. Can embed d-dS into (d+1)-Minkowski: CdR & Sébastien Renaux-Petel, arXiv:1206.3482

49. Helicity-0 on dS on de Sitter on Minkowski To identify the helicity-0 mode on de Sitter, we copy the procedure on Minkowski. Can embed d-dS into (d+1)-Minkowski: CdR & Sébastien Renaux-Petel, arXiv:1206.3482

50. Helicity-0 on dS on de Sitter on Minkowski To identify the helicity-0 mode on de Sitter, we copy the procedure on Minkowski. Can embed d-dS into (d+1)-Minkowski: behaves as a scalar in the dec. limit and captures the physics of the helicity-0 mode CdR & Sébastien Renaux-Petel, arXiv:1206.3482

51. Helicity-0 on dS on de Sitter on Minkowski To identify the helicity-0 mode on de Sitter, we copy the procedure on Minkowski. The covariantized metric fluctuation is expressed in terms of the helicity-0 mode as CdR & Sébastien Renaux-Petel, arXiv:1206.3482 in any dimensions...

52. Decoupling limit on dS Using the properly identified helicity-0 mode, we can derive the decoupling limit on dS Since we need to satisfy the Higuchi bound, CdR & Sébastien Renaux-Petel, arXiv:1206.3482

53. Decoupling limit on dS Using the properly identified helicity-0 mode, we can derive the decoupling limit on dS Since we need to satisfy the Higuchi bound, The resulting DL resembles that in Minkowski (Galileons), but with specific coefficients... CdR & Sébastien Renaux-Petel, arXiv:1206.3482

54. DL on dS CdR & Sébastien Renaux-Petel, arXiv:1206.3482 + non-diagonalizable terms mixing h and. d terms + d-3 terms (d-1) free parameters (m 2 and 3,...,d)

55. DL on dS The kinetic term vanishes if All the other interactions vanish simultaneously if CdR & Sébastien Renaux-Petel, arXiv:1206.3482 + non-diagonalizable terms mixing h and. d terms + d-3 terms (d-1) free parameters (m 2 and 3,...,d)

56. Massless limit In the massless limit, the helicity-0 mode still couples to matter The Vainshtein mechanism is active to decouple this mode

57. Partially massless limit Coupling to matter eg.

58. Partially massless limit The symmetry cancels the coupling to matter There is no Vainshtein mechanism, but there is no vDVZ discontinuity...

59. Partially massless limit Unless we take the limit without considering the PM parameters. In this case the standard Vainshtein mechanism applies.

60. Partially massless We uniquely identify the non-linear candidate for the Partially Massless theory to all orders. In the DL, the helicity-0 mode entirely disappear in any dimensions What happens beyond the DL is still to be worked out As well as the non-linear realization of the symmetry... Work in progress with Kurt Hinterbichler, Rachel Rosen & Andrew Tolley See Deser&Waldron Zinoviev