Massive Gravity and the - PowerPoint Presentation

Slide1

Massive Gravity and the

Galileon

Claudia de

Rham

Universit

é de Genève

Miami 2010

Dec,

18th 2010

Work with

Gregory

Gabadadze,

Lavinia

Heisenberg,

David Pirtskhalava and

Andrew TolleySlide2

Why Massive Gravity ?

Phenomenology

Self-acceleration

C.C. ProblemSlide3

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 m

g

< 10-25 GeV

,how about the graviton?Slide4

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 m

g

< 10-25 GeV

,how about the graviton?Could dark energy be due to an IR modification of gravity?with no ghosts ... ? Deffayet, Dvali, Gabadadze, ‘01Koyama, ‘05Slide5

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 m

g

< 10-25 GeV,

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 gravitySlide6

A massless spin-2 field in 4d, has

2 dof

A massive spin-2 field, has 5 dof

Massive GravitySlide7

In GR,

Degrees of freedom

Gauge

invariance

ConstraintsSlide8

In GR,

In massive gravity,

Degrees of freedom

Gauge

invariance

Constraints

Shift

-

Shift does not propagate a constraint

remaining

degrees of

freedomSlide9

In GR,

In massive gravity,

Degrees of freedom

Gauge

invariance

Constraints

-

Shift does not propagate a constraint- Non-linearly, lapse no longer propagates the Hamiltonian Constraint…

remainingdegrees of freedomShiftlapseBoulware & Deser,1972Creminelli et. al. hep-

th

/0505147Slide10

Avoiding the Ghost

Relying on a

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

The Ghost can be avoided by

Massless spin-2 in 5d: 5

dof

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

The graviton acquires a soft massresonanceSlide11

Avoiding the Ghost

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

The Ghost can be avoided by

Typically, the ghost enters at the scaleSlide12

Avoiding the Ghost

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

The Ghost can be avoided by

Typically, the ghost enters at the scale

That scale can be

pushedSlide13

To give the graviton a mass, include the interactions

Mass for the fluctuations around flat space-time

Graviton massSlide14

To give the graviton a mass, include the interactions

Mass for the fluctuations around flat space-time

Graviton mass

Arkani-Hamed

,

Georgi

, Schwartz, hep-

th/0210184Creminelli et. al. hep-th/0505147 Slide15

To give the

graviton a mass, include the interactions

Mass for the fluctuations around flat space-timeGraviton massSlide16

To give the graviton a mass, include the interactions

Mass for the fluctuations around flat space-time

Graviton massSlide17

In the decoupling limit,

with fixed,

Decoupling limit

pl

Which can be formally

inverted such that

with Slide18

The ghost can usually be seen in the decoupling limit where the mass term is of the form

leading to higher order eomIt seems a formidable task to remove these terms to all order in the decoupling limit.

Decoupling limitSlide19

The ghost can usually be seen in the decoupling limit where the mass term is of the form

leading to higher order eom

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

Decoupling limitSlide20

The ghost can usually be seen in the decoupling limit where the mass term is of the form

leading to higher order eom

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

Decoupling limit

withSlide21

The ghost can usually be seen in the decoupling limit where the mass term is of the form

leading to higher order eom

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

Decoupling limit

with

CdR

, Gabadadze, Tolley, 1011.1232 Slide22

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

Decoupling limitSlide23

In the decoupling limit (keeping fixed)

with

Ghost-free

decoupling limitSlide24

In the decoupling limit (keeping fixed)

The Bianchi identity requires

Ghost-free decoupling limitSlide25

In the decoupling limit (keeping fixed)

The Bianchi identity requiresThe decoupling limit stops at 2

nd order.

Ghost-free decoupling limitSlide26

In the decoupling limit (keeping fixed)

The Bianchi identity requiresThe decoupling limit stops at 2

nd order. are at most 2nd order in derivativeThese mixings can be removed by a local field redefinition

Ghost-free decoupling limitSlide27

The Galileon

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

CdR

, Gabadadze, 1007.0443

The interactions have

3 special features:

They are localThey possess a Shift and a Galileon

symmetry

They have a

well-defined Cauchy problem

(

eom

remain 2

nd

order)

Nicolis

,

Rattazzi

and

Trincherini

, 0811.2197

Corresponds to the

Galileon

family

of interactions

The BD ghost can be pushed

beyond the scale

L

3

Coupling to matterSlide28

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/0505147Slide29

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/0505147Slide30

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 casesSlide31

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 4

th 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… Slide32

Consequences for CosmologySlide33

Degravitation

From naturalness considerations

, we expect a vacuum energy of the order of the cutoff scale (Planck scale).

But observations tell us

Slide34

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

Arkani-Hamed

et. al.,

‘02

Dvali

, Hofmann

&

Khoury,

‘07

k

: 4d momentumSlide35

The degravitation mechanism is a

causal process.

1/m

H

2

time

time

Phase transition

LDegravitationSlide36

The degravitation mechanism is a

causal process.

1/m

H

2

time

time

Phase transition

L

DegravitationSlide37

In Massive gravity,

DegravitationSlide38

Degravitation

Screening the CC

1/m

H

2

time

time

l

Relaxes towards a flat geometry even with a large CCSlide39

Dark Energy

Screening the CC

Relaxes towards a flat geometry even with a large CC

Self-acceleration

Source the late time accelerationSlide40

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 Slide41

Summary

Galileon

interactions arise naturally- in braneworlds with induced curvature (soft mass gravity) - in hard massive gravity with no ghosts in the dec

. limitThe 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)