1. Galileons in Cosmology Claudia de Rham Université de Genève Work with Clare Burrage, Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava, David Seery and Andrew Tolley

2. What is Dark Energy ? Is it a Cosmological Constant ???

3. What is Dark Energy ? Is it a Cosmological Constant ???  56 orders of magnitude discrepancy with physics from the Standard Model !

4. What is Dark Energy ? Is Dark Energy is Dynamical ??  new form of energy eg. Quintessence,…  or self-accelerating Universe eg. F(R), DGP Is it a Cosmological Constant ???  56 orders of magnitude discrepancy! OR

5. What is Dark Energy ? Is Dark Energy is Dynamical ??  new form of energy eg. Quintessence,…  or self-accelerating Universe eg. F(R), DGP Is it a Cosmological Constant ???  56 orders of magnitude discrepancy! These degrees of freedom must be extremely light! OR

6. Is Dark Energy is Dynamical ??  new form of energy eg. Quintessence,…  or self-accelerating Universe eg. F(R), DGP Is it a Cosmological Constant ???  56 orders of magnitude discrepancy! These degrees of freedom must be extremely light! OR What is Dark Energy ? How are they hidden ?

7. The Galileon Is a field weakly coupled near a source (kinetic Chameleon – or Vainhstein mechanism)

8. The Galileon cf. talks of Guido D’Amico, Sébastien Renaux-Petel, Alexander Vikman, George Zahariade. Originated from the decoupling limit of DGP new dof as Brane-bending mode gravity Dvali, Gabadadze & Porrati ’00

9. The Galileon The brane-bending mode behaves as It has 3 special features: 1. The theory is local 2. It possess a Shift and a Galileon symmetry 3. It has a well-defined Cauchy problem (eom remain 2 nd order) Luty, Porrati, Rattazzi hep-th/0303116

10. The Galileon The brane-bending mode behaves as The most general family of interactions with the same property is Nicolis, Rattazzi and Trincherini, 0811.2197 Corresponds to the Galileon family of interactions

11. The Galileon Is a field weakly coupled near a source (kinetic Chameleon – or Vainhstein mechanism) That arises in stable theories of massive Gravity

12. The Galileon Is a field weakly coupled near a source (kinetic Chameleon – or Vainhstein mechanism) That arises in stable theories of massive Gravity As an IR modified theory of gravity it can have important consequences for late-time cosmology

13. The Galileon Is a field weakly coupled near a source (kinetic Chameleon – or Vainhstein mechanism) That arises in stable theories of massive Gravity As an IR modified theory of gravity it can have important consequences for late-time cosmology Self-acceleration (Universe accelerates due to Gravity dof) Degravitation (Weakening of Gravity in IR could imply that a large Cosmological Constant can drive small acceleration)

14. The Galileon Is a field weakly coupled near a source (kinetic Chameleon – or Vainhstein mechanism) That arises in stable theories of massive Gravity As an IR modified theory of gravity it can have important consequences for late-time cosmology As well as at early time (Inflation)

15. 2 dof A massless spin-2 field in 4d, has 2 dof Massive Gravity

16. 2 dof A massless spin-2 field in 4d, has 2 dof 5 dof A massive spin-2 field, has 5 dof Massive Gravity

17. 5 th force constraints in the solar system imply that the extra degrees of freedom must be strongly coupled at a scale Strong Coupling rSrSrSrS

18. 5 th force constraints in the solar system imply that the extra degrees of freedom must be strongly coupled at a scale Strong Coupling rSrSrSrS r*r*r*r*

19. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time 2. Stückelberg language

20. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time 2. Stückelberg language Arkani-Hamed, Georgi, Schwartz, hep-th/0210184 Creminelli et. al. hep-th/0505147

21. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time 2. Stückelberg language ghost...

22. To give the graviton a mass, include the interactions Mass for the fluctuations around flat space-time 2. Stückelberg language

23. In the decoupling limit, with fixed, Decoupling limit The ghost can be avoided in that limit, if is a total derivative

24. is a total derivative, for instance if Decoupling limit

25. is a total derivative, for instance if In the decoupling limit, Decoupling limit or with

26. The mass term Has no ghosts in the decoupling limit: Ghost-free theory with CdR, Gabadadze, Tolley, 1011.1232

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

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

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

30. 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 Ghost-free decoupling limit NO GHOSTS in the decoupling limit

31. 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

32. Galileon in disguise

33. For that 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 Coupling to matter

34. Consequences for Cosmology

35. 1.For late time acceleration 2.Inflation ?

36. C.C. Problem

37. Degravitation 1/m H2H2 time Could relax towards a flat geometry even with a large CC

38. Dark Energy Could relax towards a flat geometry even with a large CC Source the late time acceleration

39. 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

40. Consequences for Cosmology 1.For late time acceleration 2.Inflation ?

41. Higher derivative interactions are essential for the viability of this class of models. Within the solar system,  reaches the scale  *, yet, we are still within the regime of validity of the theory Vainshtein, Phys. Lett. B 39 (1972) 393 Babichev, Deffayet & Ziour, 0901.0393 EFT and relevant operators Luty, Porrati, Rattazzi hep-th/0303116 Nicolis & Rattazzi, hep-th/0404159

42. Higher derivative interactions are essential for the viability of this class of models. Within the solar system,  reaches the scale  *, yet, we are still within the regime of validity of the theory The breakdown of the EFT is not measured by but by itself gradients should be small So we can trust a regime where as long as EFT and relevant operators cf. Guido D’Amico’s talk

43. Dirac Born Infeld Kabat and Lifschytz, hep-th/9902073 One of the most attractive model of inflation is provided by the DBI action Which describes the dynamics of a probe-brane in a extra dimension

44. DBI - Galileon DBI is similar to the Galileon in that 1. It relies on higher derivative interactions, 2. While keeping quantum corrections under control 3. It exhibits a 5d Poincaré or AdS symmetry 4. It has a well-defined Cauchy problem

45. DBI - Inflation Silverstein, Tong, hep-th/0310221 Alishahiha, Silverstein, Tong, hep-th/0404084 The DBI brane model is one of the most important actors of potential inflation scenarios. 1. The brane can move relativistically, and yet exhibit slow-roll attenuate the usual slow-roll tuning issues of inflation 2. Leads to very characteristic observational features such as non-Gaussianities characteristic triangle shapes.

46. EFT and relevant operators Can we use these ideas to build a radiatively stable model of inflation with the Galileon ?

47. Galileon Inflation Model of Inflation grounded on the Galileon Symmetry

48. Galileon Inflation Model of Inflation grounded on the Galileon Symmetry

49. Galileon Inflation Model of Inflation grounded on the Galileon Symmetry Shift symmetry guarantees the conservation of ζ outside the horizon time independent power spectrum Non-renormalization theorem allows us to consider these interactions to be large, without breaking the regime of validity of EFT. Nicolis & Rattazzi, hep-th/0404159

50. Galileon Inflation Model of Inflation grounded on the Galileon Symmetry Shift symmetry guarantees the conservation of ζ outside the horizon time independent power spectrum Non-renormalization theorem allows us to consider these interactions to be large, without breaking the regime of validity of EFT. Remains true with a mass term

51. Strong coupling When the interactions are important, the inflationary phase satisfies And perturbations are given by

52. Strong coupling When the interactions are important, the inflationary phase satisfies And perturbations are given by dim - 6dim - 7

53. Non-Gaussianities The operator 7 can be of the same order as the others if the dim-6 operators are suppressed by additional H/ . This suppression is stable thanks to the Galileon symmetry. Potentially “large” nG, but with no specific shape. Burrage, CdR, Seery, Tolley, 1009.2497

54. New shapes of non-Gaussianity The situation is different, if the leading interactions are tuned to vanish Leading interactions then arise from terms nG can rely on dim-9 operators specific shapes nG: - 2 operators lead to ~ equilateral triangles - 1 to flattened isocele triangle. Creminelli, D’Amico, Musso, Norena and Trincherini, 1011.3004 cf. Guido D’Amico’s talk

55. 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...... Similar in spirit than DBI, but with different signatures