(Partially) - PowerPoint Presentation

Slide1

(Partially)

Massless

Gravity

Claudia de Rham

July 5

th

2012

Work in collaboration with Sébastien Renaux-Petel 1206.3482 Slide2

Massive GravitySlide3

Massive Gravity

The notion of mass requires a

reference !Slide4

Massive Gravity

The notion of mass requires a

reference !Slide5

Massive Gravity

The notion of mass requires a

reference !

Flat Metric

MetricSlide6

Massive Gravity

The notion of mass requires a

reference ! Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)Slide7

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 symSlide8

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)Slide9

Fierz

-Pauli Massive Gravity

Mass term for the fluctuations around flat space-time

Fierz & Pauli, Proc.Roy.Soc.Lond.A

173, 211 (1939)Slide10

Fierz

-Pauli Massive Gravity

Mass term for the fluctuations around flat space-timeTransforms under a change of coordinateSlide11

Fierz

-Pauli Massive Gravity

Mass term for the ‘covariant fluctuations’

Does not transform under that change of coordinateSlide12

Fierz

-Pauli Massive Gravity

Mass term for the ‘covariant fluctuations’

The potential has higher derivatives...

Total derivativeSlide13

Fierz

-Pauli Massive Gravity

Mass term for the ‘covariant fluctuations’

The potential has higher derivatives...

Total derivative

Ghost reappears at

the non-linear levelSlide14

Ghost-free Massive GravitySlide15

Ghost-free Massive GravitySlide16

Ghost-free Massive Gravity

With

Has no ghosts at leading order in the

decoupling limit

CdR

, Gabadadze, 1007.0443

CdR

, Gabadadze, Tolley, 1011.1232 Slide17

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 Slide18

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-perturbatively in many different languages

CdR

, Gabadadze, 1007.0443CdR

, Gabadadze, Tolley, 1011.1232Hassan & Rosen, 1106.3344

CdR, Gabadadze, Tolley, 1107.3820CdR, Gabadadze, Tolley, 1108.4521Hassan & Rosen, 1111.2070 Hassan, Schmidt-May & von Strauss, 1203.5283 Slide19

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-perturbatively in many different languagesAs well as around

any reference metric, be it dynamical or not BiGravity !!!

Hassan, Rosen & Schmidt-May, 1109.3230

Hassan & Rosen,

1109.3515Slide20

Ghost-free Massive Gravity

One 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)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 massSlide23

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 in

parameter space where thehelicity-0 mode could also beabsent ???Slide24

Change of Ref. metric

Hassan & Rosen, 2011

Consider massive gravity around

dS

as a

reference

!

dS

Metric

Metric

dS

is still a maximally symmetric ST

Same amount of symmetry as massive gravity around

Minkowski

!Slide25

Massi

ve Gravity in de Sitter

Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metricSlide26

Massi

v

e Gravity in de SitterOnly the helicity-0 mode

gets ‘seriously’ affected by the dS reference metric

Healthy scalar field(Higuchi bound)

Higuchi, NPB

282, 397 (1987)Slide27

Massi

v

e Gravity in de SitterOnly the helicity-0 mode

gets ‘seriously’ affected by the dS reference metric

Higuchi, NPB

282, 397 (1987)

Healthy scalar field

(Higuchi bound)Slide28

Massi

ve 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

whenSlide29

Massi

ve 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

whenRecover a symmetry

Deser & Waldron, 2001Slide30

Partially

massless

Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still presentSlide31

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 backgroundSlide32

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 MG

no Lorentz symmetry around dS, but still have same amount of symmetry.Slide33

(Partially)

massless limitMassless

limit GR + mass term

Recover 4d diff invariance

GR

in 4d:

2

dof (helicity 2)Slide34

(Partially)

massless limitMassless

limit Partially Massless limit

GR + mass term

Recover 4d diff invariance

GR

GR + mass term

Recover 1 symmetryMassive GR4 dof

(

helicity

2 &1)

Deser & Waldron, 2001

in 4d:

2

dof

(

helicity

2)Slide35

Non-linear partially

masslessSlide36

Non-linear partially

masslessLet’s start with ghost-free theory of MG,

But around dS

dS

ref metricSlide37

Non-linear partially

masslessLet’s start with ghost-free theory of MG,

But around dSAnd 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....Slide38

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 Slide39

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 Slide40

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 Slide41

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

CdR & Sébastien Renaux-Petel, arXiv:1206.3482

in any dimensions...Slide42

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 Slide43

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 Slide44

DL on

dS

CdR & Sébastien Renaux-Petel, arXiv:1206.3482

+ non-diagonalizable terms

mixing h and p.

d terms + d-3 terms

(d-1) free parameters (m

2 and

a

3,...,d)Slide45

DL on

dS

The kinetic term vanishes ifAll the other interactions vanish simultaneously if

CdR & Sébastien Renaux-Petel, arXiv:1206.3482

+ non-diagonalizable terms

mixing h and p.

d terms + d-3 terms

(d-1) free parameters (m

2

and

a

3,...,d)Slide46

Massless

limit

In the

massless

limit, the helicity-0 mode still couples to matter

The Vainshtein mechanism is active to decouple this modeSlide47

Partially

massless limit

Coupling to matter

eg

. Slide48

Partially

massless limit

The symmetry cancels the coupling to matter

There is no Vainshtein mechanism, but there is no

vDVZ

discontinuity...Slide49

Partially

massless limit

Unless we take the limit without considering

the PM parameters

a

.

In this case the standard Vainshtein mechanism applies.Slide50

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 dimensionsWhat 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 Tolley

See Deser&Waldron Zinoviev