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(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

(Partially) Massless Gravity

Claudia de Rham

July 5th 2012

Work in collaboration with Sébastien Renaux-Petel

1206.3482

massive gravity1
Massive Gravity
  • The notion of mass requires a reference !
massive gravity2
Massive Gravity
  • The notion of mass requires a reference !
massive gravity3
Massive Gravity
  • The notion of mass requires a reference !

Flat Metric

Metric

massive gravity4
Massive Gravity
  • The notion of mass requires a reference !
  • Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)
massive gravity5
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 gravity6
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
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 gravity1
Fierz-Pauli Massive Gravity
  • Mass term for the fluctuations around flat space-time
  • Transforms under a change of coordinate
fierz pauli massive gravity2
Fierz-Pauli Massive Gravity
  • Mass term for the ‘covariantfluctuations’
  • Does not transform under that change of coordinate
fierz pauli massive gravity3
Fierz-Pauli Massive Gravity
  • Mass term for the ‘covariantfluctuations’
  • The potential has higher derivatives...

Total derivative

fierz pauli massive gravity4
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 gravity2
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 gravity3
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 gravity4
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 gravity5
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 gravity6
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)

slide21
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

slide22
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
slide23
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
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

massi v e gravity in de sitter
Massive Gravity in de Sitter
  • Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metric
massi v e gravity in de sitter1
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)

massi v e gravity in de sitter2
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)

massi v e gravity in de sitter3
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

massi v e gravity in de sitter4
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
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 massless1
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 massless2
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
(Partially) massless limit
  • Massless limit

GR + mass term

Recover 4d diff invariance

GR

in 4d: 2 dof (helicity 2)

partially massless limit1

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 massless1
Non-linear partially massless
  • Let’s start with ghost-free theory of MG,
  • But around dS

dS ref metric

non linear partially massless2
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....

helicity 0 on ds

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:
helicity 0 on ds1

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:
helicity 0 on ds2

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
helicity 0 on ds3

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...

decoupling limit on ds

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,
decoupling limit on ds1

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...
dl on ds

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)

dl on ds1

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)

massless limit
Masslesslimit

In the massless limit, the helicity-0 mode still couples to matter

The Vainshtein mechanism is active to decouple this mode

partially massless limit2
Partially massless limit

Coupling to matter eg.

partially massless limit3
Partially massless limit

The symmetry cancels the coupling to matter

There is no Vainshtein mechanism, but there is no vDVZ discontinuity...

partially massless limit4
Partially massless limit

Unless we take the limit without considering the PM parameters a.

In this case the standard Vainshtein mechanism applies.

partially massless3
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