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## (Partially) Massless Gravity

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**(Partially) Massless Gravity**Claudia de Rham July 5th 2012 Work in collaboration with Sébastien Renaux-Petel 1206.3482**Massive Gravity**• The notion of mass requires a reference !**Massive Gravity**• The notion of mass requires a reference !**Massive Gravity**• The notion of mass requires a reference ! Flat Metric Metric**Massive Gravity**• The notion of mass requires a reference ! • Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)**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**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)**Fierz-Pauli Massive Gravity**• Mass term for the fluctuations around flat space-time Fierz & Pauli, Proc.Roy.Soc.Lond.A173, 211 (1939)**Fierz-Pauli Massive Gravity**• Mass term for the fluctuations around flat space-time • Transforms under a change of coordinate**Fierz-Pauli Massive Gravity**• Mass term for the ‘covariantfluctuations’ • Does not transform under that change of coordinate**Fierz-Pauli Massive Gravity**• Mass term for the ‘covariantfluctuations’ • The potential has higher derivatives... Total derivative**Fierz-Pauli Massive Gravity**• Mass term for the ‘covariantfluctuations’ • The potential has higher derivatives... Ghost reappears atthe non-linear level Total derivative**Ghost-free Massive Gravity**• With • Has no ghosts at leading order in the decoupling limit CdR, Gabadadze, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232**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**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**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**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)**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**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**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 ???**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**Massive Gravity in de Sitter**• Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metric**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)**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)**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**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**Partially massless**• Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still present**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**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.**(Partially) massless limit**• Massless limit GR + mass term Recover 4d diff invariance GR in 4d: 2 dof (helicity 2)**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)**Non-linear partially massless**• Let’s start with ghost-free theory of MG, • But around dS dS ref metric**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....**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:**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:**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**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...**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,**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...**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)**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)**Masslesslimit**In the massless limit, the helicity-0 mode still couples to matter The Vainshtein mechanism is active to decouple this mode**Partially massless limit**Coupling to matter eg.**Partially massless limit**The symmetry cancels the coupling to matter There is no Vainshtein mechanism, but there is no vDVZ discontinuity...**Partially massless limit**Unless we take the limit without considering the PM parameters a. In this case the standard Vainshtein mechanism applies.**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