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Massive Gravity on de Sitter

(Partially) Massless Gravity Claudia de RhamJuly 5th 2012Work in collaboration with Sbastien Renaux-Petel 1206.3482

Massive Gravity

Massive GravityThe notion of mass requires a reference !



Flat MetricMetricMassive GravityThe notion of mass requires a reference ! Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)

Massive GravityThe 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

GRLoss of 4 symMassive GravityThe 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 GravityMass term for the fluctuations around flat space-timeFierz & Pauli, Proc.Roy.Soc.Lond.A 173, 211 (1939)

Fierz-Pauli Massive GravityMass term for the fluctuations around flat space-time

Transforms under a change of coordinate

Fierz-Pauli Massive GravityMass term for the covariant fluctuations

Does not transform under that change of coordinate


The potential has higher derivatives...

Total derivative


The potential has higher derivatives...

Total derivative

Ghost reappears atthe non-linear levelGhost-free Massive Gravity

Ghost-free Massive Gravity

Ghost-free Massive GravityWith

Has no ghosts at leading order in the decoupling limit

CdR, Gabadadze, 1007.0443CdR, Gabadadze, Tolley, 1011.1232

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

Ghost-free Massive GravityIn 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 languagesCdR, Gabadadze, 1007.0443CdR, Gabadadze, Tolley, 1011.1232Hassan & Rosen, 1106.3344CdR, Gabadadze, Tolley, 1107.3820CdR, Gabadadze, Tolley, 1108.4521Hassan & Rosen, 1111.2070 Hassan, Schmidt-May & von Strauss, 1203.5283

Ghost-free Massive GravityIn 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 languagesAs well as around any reference metric, be it dynamical or notBiGravity !!!Hassan, Rosen & Schmidt-May, 1109.3230Hassan & Rosen, 1109.3515Ghost-free Massive GravityOne 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)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 But...


The presence of the helicity-0 mode puts strong bounds on the graviton mass

But...


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. metricHassan & Rosen, 2011Consider massive gravity around dS as a reference !

dS MetricMetricdS is still a maximally symmetric STSame amount of symmetry as massive gravity around Minkowski !Massive Gravity in de SitterOnly the helicity-0 mode gets seriously affected by the dS reference metric

Massive Gravity in de SitterOnly the helicity-0 mode gets seriously affected by the dS reference metricHealthy scalar field(Higuchi bound)Higuchi, NPB282, 397 (1987)

Massive Gravity in de SitterOnly the helicity-0 mode gets seriously affected by the dS reference metricHiguchi, NPB282, 397 (1987)

Healthy scalar field(Higuchi bound)

Massive Gravity in de SitterOnly the helicity-0 mode gets seriously affected by the dS reference metricThe helicity-0 mode disappears at the linear levelwhen

Massive Gravity in de SitterOnly the helicity-0 mode gets seriously affected by the dS reference metricThe helicity-0 mode disappears at the linear levelwhen

Recover a symmetryDeser & Waldron, 2001

Partially masslessIs different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still present30To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

Partially masslessIs different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still presentIs 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 background31To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

Partially masslessIs different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still presentIs 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 backgroundIs different from Lorentz violating MGno Lorentz symmetry around dS, but still have same amount of symmetry.32To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

(Partially) massless limitMassless limit GR + mass term

Recover 4d diff invariance

GRin 4d: 2 dof (helicity 2)(Partially) massless limitMassless limit Partially Massless limitGR + mass term

Recover 4d diff invariance

GRGR + mass term Recover 1 symmetryMassive GR4 dof (helicity 2 &1)Deser & Waldron, 2001

in 4d: 2 dof (helicity 2)Non-linear partially masslessNon-linear partially masslessLets start with ghost-free theory of MG,

But around dS

dS ref metricNon-linear partially masslessLets 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 metricBut we need to properly identify the helicity-0 mode first....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 & Sbastien Renaux-Petel, arXiv:1206.3482 38To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

Helicity-0 on dS To identify the helicity-0 mode on de Sitter, we copy the procedure on Minkowski.





behaves as a scalar in the dec. limit and captures the physics of the helicity-0 modeCdR & Sbastien Renaux-Petel, arXiv:1206.3482 40To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like


The covariantized metric fluctuation is expressed in terms of the helicity-0 mode as

CdR & Sbastien Renaux-Petel, arXiv:1206.3482

in any dimensions...41To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

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,


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...


DL on dS CdR & Sbastien Renaux-Petel, arXiv:1206.3482

+ non-diagonalizable terms mixing h and p.

d terms + d-3 terms(d-1) free parameters (m2 and a3,...,d)44To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

DL on dS The kinetic term vanishes ifAll the other interactions vanish simultaneously if CdR & Sbastien Renaux-Petel, arXiv:1206.3482

+ non-diagonalizable terms mixing h and p.

d terms + d-3 terms(d-1) free parameters (m2 and a3,...,d)

45To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

Massless limit

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.



The symmetry cancels the coupling to matter

There is no Vainshtein mechanism, but there is no vDVZ discontinuity...48To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like


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


Partially masslessWe 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 TolleySee Deser&Waldron Zinoviev50To see that, we can simply look at the linearized Einstein equation.If I omit the tensor structure for a second, the Einstein equation looks like

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