Massive gravity and the galileon
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Miami 2010 Dec, 18 th 2010. Massive Gravity and the Galileon. Work with Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava and Andrew Tolley. Claudia de Rham Universit é de Genève. Why Massive Gravity ?. Phenomenology Self-acceleration C.C. Problem.

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Massive gravity and the galileon

Miami 2010

Dec, 18th2010

Massive Gravity and the Galileon

Work with

Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava and Andrew Tolley

Claudia de Rham

Université de Genève


Why massive gravity

Why Massive Gravity ?

  • Phenomenology

  • Self-acceleration

  • C.C. Problem


Why massive gravity1

Why Massive Gravity ?

  • Phenomenology

  • Self-acceleration

  • C.C. Problem

what are the theoretical and observational bounds on gravity in the IR ?

mass of the photon is bounded to mg < 10-25GeV,

how about the graviton?


Why massive gravity2

Why Massive Gravity ?

  • Phenomenology

  • Self-acceleration

  • C.C. Problem

what are the theoretical and observational bounds on gravity in the IR ?

mass of the photon is bounded to mg < 10-25GeV,

how about the graviton?

Could dark energy be due to an IR modification of gravity?

with no ghosts ... ?

Deffayet, Dvali, Gabadadze, ‘01

Koyama, ‘05


Why massive gravity3

Why Massive Gravity ?

  • Phenomenology

  • Self-acceleration

  • C.C. Problem

what are the theoretical and observational bounds on gravity in the IR ?

mass of the photon is bounded to mg < 10-25GeV,

how about the graviton?

Could dark energy be due to an IR modification of gravity?

with no ghosts ... ?

Is the cosmological constant small ? ORdoes it have a small effect on the geometry ?

Gravity modified in IR

Massive gravity


Massive gravity and the galileon

Massive Gravity

  • A massless spin-2 field in 4d, has 2 dof

  • A massive spin-2 field, has 5 dof


Massive gravity and the galileon

Degrees of freedom

  • In GR,

Gauge

invariance

Constraints


Massive gravity and the galileon

Degrees of freedom

  • In GR,

  • In massive gravity,

Gauge

invariance

Constraints

- Shift does not propagate a constraint

remainingdegrees of

freedom

Shift


Massive gravity and the galileon

Degrees of freedom

  • In GR,

  • In massive gravity,

Gauge

invariance

Constraints

- Shift does not propagate a constraint

- Non-linearly, lapse no longer propagates the Hamiltonian Constraint…

remainingdegrees of

freedom

Boulware & Deser,1972

Creminelli et. al. hep-th/0505147

Shift

lapse


Avoiding the ghost

Avoiding the Ghost

The Ghost can be avoided by

  • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... )

Massive spin-2 in 4d: 5 dof (+ ghost…)

Massless spin-2 in 5d: 5 dof

The graviton acquires a soft mass

resonance


Avoiding the ghost1

Avoiding the Ghost

The Ghost can be avoided by

  • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... )

  • Pushing the ghost above an acceptable cutoff scale

Typically, the ghost enters at the scale


Avoiding the ghost2

Avoiding the Ghost

The Ghost can be avoided by

  • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... )

  • Pushing the ghost above an acceptable cutoff scale

Typically, the ghost enters at the scale

That scale can be pushed


Massive gravity and the galileon

Graviton mass

  • To give the graviton a mass, include the interactions

  • Mass for the fluctuations around flat space-time


Massive gravity and the galileon

Graviton mass

  • To give the graviton a mass, include the interactions

  • Mass for the fluctuations around flat space-time

Arkani-Hamed, Georgi, Schwartz, hep-th/0210184

Creminelli et. al. hep-th/0505147


Massive gravity and the galileon

Graviton mass

  • To give the graviton a mass, include the interactions

  • Mass for the fluctuations around flat space-time


Massive gravity and the galileon

Graviton mass

  • To give the graviton a mass, include the interactions

  • Mass for the fluctuations around flat space-time


Massive gravity and the galileon

Decoupling limit

pl

  • In the decoupling limit,with fixed,

  • Which can be formally inverted such thatwith


Massive gravity and the galileon

Decoupling limit

  • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom

  • It seems a formidable task to remove these terms to all order in the decoupling limit.


Massive gravity and the galileon

Decoupling limit

  • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom

  • But we can attack the problem by the other end: starting with what we want in the decoupling limit


Massive gravity and the galileon

Decoupling limit

  • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom

  • But we can attack the problem by the other end: starting with what we want in the decoupling limit

with


Massive gravity and the galileon

Decoupling limit

  • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom

  • But we can attack the problem by the other end: starting with what we want in the decoupling limit

with

CdR, Gabadadze, Tolley, 1011.1232


Massive gravity and the galileon

Decoupling limit

  • That potential ensures that the problematic terms cancel in the decoupling limit


Massive gravity and the galileon

Ghost-free decoupling limit

  • In the decoupling limit (keeping fixed)with


Massive gravity and the galileon

Ghost-free decoupling limit

  • In the decoupling limit (keeping fixed)

  • The Bianchi identity requires


Massive gravity and the galileon

Ghost-free decoupling limit

  • In the decoupling limit (keeping fixed)

  • The Bianchi identity requires

  • The decoupling limit stops at 2nd order.


Massive gravity and the galileon

Ghost-free decoupling limit

  • In the decoupling limit (keeping fixed)

  • The Bianchi identity requires

  • The decoupling limit stops at 2nd order.

  • are at most 2nd order in derivative

  • These mixings can be removed by a local field redefinition


The galileon

The Galileon

  • For a stable theory of massive gravity, the decoupling limit is

  • The interactions have 3 special features:

The BD ghost can be pushedbeyond the scale L3

They are local

They possess a Shift and a Galileonsymmetry

They have a well-defined Cauchy problem(eom remain 2nd order)

  • Corresponds to the Galileon family of interactions

Coupling to matter

CdR, Gabadadze, 1007.0443

Nicolis, Rattazzi and Trincherini, 0811.2197


Back to the bd ghost

Back to the BD ghost…

  • In the ADM decomposition,

  • with

  • The lapse enters quadratically in the Hamiltonian,

Boulware & Deser,1972

Creminelli et. al. hep-th/0505147


Back to the bd ghost1

Back to the BD ghost…

  • In the ADM decomposition,

  • with

  • The lapse enters quadratically in the Hamiltonian,

  • Does it really mean that the constraint is lost ?

Boulware & Deser,1972

Creminelli et. al. hep-th/0505147


Back to the bd ghost2

Back to the BD ghost…

  • In the ADM decomposition,

  • with

  • The lapse enters quadratically in the Hamiltonian,

  • Does it really mean that the constraint is lost ?

  • The constraint is manifest after integrating over the shift

  • This can be shown - at least up to 4th order in perturbations- completely non-linearly in simplified cases


Massive gravity summary

Massive gravity - Summary

We can construct an explicit theory of massive gravity which:

Exhibits the Galileon interactions in the decoupling limit (has no ghost in the decoupling limit)

Propagates a constraint at least up to 4th order in perturbations (does not excite the 6th BD mode to that order) and indicates that the same remains true to all orders

Whether or not the constraint propagates is yet unknown. secondary constraint ?

Symmetry ???

CdR, Gabadadze, Tolley, in progress…


Consequences for cosmology

Consequences for Cosmology


Degravitation

Degravitation

  • From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale).

  • But observations tell us


Degravitation1

Degravitation

  • From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale).

  • But observations tell us

  • Idea behind degravitation: Gravity modified on large distances such that the vacuum energy gravitates more weakly

k: 4d momentum

Arkani-Hamed et. al., ‘02

Dvali, Hofmann & Khoury, ‘07


Degravitation2

Degravitation

L

Phase transition

  • The degravitation mechanism is a causal process.

time

H2

1/m

time


Degravitation3

Degravitation

L

Phase transition

  • The degravitation mechanism is a causal process.

time

H2

1/m

time


Degravitation4

Degravitation

In Massive gravity,


Degravitation5

Degravitation

l

time

H2

Screening the CC

1/m

time

Relaxes towards a flat geometry even with a large CC


Dark energy

Dark Energy

Screening the CC

Self-acceleration

Source the late time acceleration

Relaxes towards a flat geometry even with a large CC


Dark energy1

Dark Energy

Screening the CC

Self-acceleration

  • 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


Summary

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 stablemodel of inflation leading to potentially large nG... (Cf. Andrew’s talk - Sunday)


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