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(Partially) Massless Gravity

(Partially) Massless Gravity . Claudia de Rham July 5 th 2012 Work in collaboration with Sébastien Renaux-Petel 1206.3482 . Massive Gravity. Massive Gravity. The notion of mass requires a reference !. Massive Gravity. The notion of mass requires a reference !. Massive Gravity.

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(Partially) Massless Gravity

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  1. (Partially) Massless Gravity Claudia de Rham July 5th 2012 Work in collaboration with Sébastien Renaux-Petel 1206.3482

  2. Massive Gravity

  3. Massive Gravity • The notion of mass requires a reference !

  4. Massive Gravity • The notion of mass requires a reference !

  5. Massive Gravity • The notion of mass requires a reference ! Flat Metric Metric

  6. Massive Gravity • The notion of mass requires a reference ! • Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)

  7. Massive Gravity • The notion of mass requires a reference ! • Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance) • The loss in symmetry generates new dof GR Loss of 4 sym

  8. Massive Gravity • The notion of mass requires a reference ! • Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance) • The loss in symmetry generates new dof Boulware & Deser, PRD6, 3368 (1972)

  9. Fierz-Pauli Massive Gravity • Mass term for the fluctuations around flat space-time Fierz & Pauli, Proc.Roy.Soc.Lond.A173, 211 (1939)

  10. Fierz-Pauli Massive Gravity • Mass term for the fluctuations around flat space-time • Transforms under a change of coordinate

  11. Fierz-Pauli Massive Gravity • Mass term for the ‘covariantfluctuations’ • Does not transform under that change of coordinate

  12. Fierz-Pauli Massive Gravity • Mass term for the ‘covariantfluctuations’ • The potential has higher derivatives... Total derivative

  13. Fierz-Pauli Massive Gravity • Mass term for the ‘covariantfluctuations’ • The potential has higher derivatives... Ghost reappears atthe non-linear level Total derivative

  14. Ghost-free Massive Gravity

  15. Ghost-free Massive Gravity

  16. Ghost-free Massive Gravity • With • Has no ghosts at leading order in the decoupling limit CdR, Gabadadze, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232

  17. 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

  18. 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-perturbativelyin 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 Hassan, Schmidt-May & von Strauss, 1203.5283

  19. 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-perturbativelyin many different languages • As well as around any reference metric, be it dynamical or not BiGravity !!! Hassan, Rosen & Schmidt-May, 1109.3230 Hassan & Rosen, 1109.3515

  20. Ghost-free Massive Gravity One can construct a consistent theory of massive gravity around any reference metric which- propagates 5 dofin 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)

  21. But... • The Vainshtein mechanism always comes hand in hand with superluminalities...This doesn’t 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

  22. 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

  23. 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 inparameter space where thehelicity-0 mode could also beabsent ???

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

  25. Massive Gravity in de Sitter • Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metric

  26. 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)

  27. 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)

  28. 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

  29. Massive Gravity in de Sitter Deser & Waldron, 2001 • 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

  30. Partially massless • Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still present

  31. Partially massless • Is different from the minimal modelfor 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 disappearsin this case the fundamental theory has a helicity-0 mode but it cancels on a specific background

  32. Partially massless • Is different from the minimal modelfor 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 disappearsin this case the fundamental theory has a helicity-0 mode but it cancels on a specific background • Is different from Lorentz violating MGno Lorentz symmetry around dS, but still have same amount of symmetry.

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

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

  35. Non-linear partially massless

  36. Non-linear partially massless • Let’s start with ghost-free theory of MG, • But around dS dS ref metric

  37. Non-linear partially massless • Let’s 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....

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

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

  40. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 Helicity-0 on dS • 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

  41. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 Helicity-0 on dS • 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 in any dimensions...

  42. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 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,

  43. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 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...

  44. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 DL on dS + non-diagonalizable terms mixing h and p. d terms + d-3 terms (d-1) free parameters (m2 and a3,...,d)

  45. CdR & Sébastien Renaux-Petel, arXiv:1206.3482 DL on dS • The kinetic term vanishes if • All the other interactions vanish simultaneously if + non-diagonalizable terms mixing h and p. d terms + d-3 terms (d-1) free parameters (m2 and a3,...,d)

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

  47. Partially massless limit Coupling to matter eg.

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

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

  50. 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... See Deser&Waldron Zinoviev Work in progress with Kurt Hinterbichler, Rachel Rosen & Andrew Tolley

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