SUPERJEDI

Pointe aux Piments

July, the 4th 2013

A short introduction

to « massive gravity » …

or …

Can one give a mass

to the graviton

1. The 3 sins of massive

gravity (or why it is hard ?)

2. Cures

(or why it may be possible ?)

(to be discussed later)

Cédric Deffayet

(APC, CNRS Paris)

Part 1. the 3 sins of massive gravity

1.1. Introduction: why « massive gravity » ?

and some properties of « massless gravity »

1.2. The DGP model (as an invitation to take the trip)

1.3. Kaluza-Klein gravitons

1.4. Quadratic massive gravity: the Pauli-Fierz theory and the vDVZ

discontinuity

1.5. Non linear Pauli-Fierz theory and the Vainshtein Mechanism

1.6 The Goldstone picture (and « decoupling limit ») of non linear

massive gravity, and what can one get from it ?

1.1 Introduction: Why « massive gravity » ?

• Dark matter, to explain in

particular the dynamics of

galaxies.

• Dark energy (to explain in

particular the observed

acceleration of the expansion of

the Universe) … in a form close to

that of a cosmological constant ½ » constant

Friedmann (Einstein) equation

Standard model of cosmology

requires the presence in the

matter content of the Universe of

Why « massive gravity » ?

One way to modify gravity at « large distances »

… and get rid of dark energy (or dark matter) ?

Changing the dynamics

of gravity ?

Historical example the

success/failure of both

approaches: Le Verrier and

• The discovery of Neptune

• The non discovery of Vulcan…

but that of General Relativity

Dark matter

dark energy ?

for this idea to work…

One obviously needs

a very light graviton

(of Compton length

of order of the size of

the Universe)

I.e. to « replace » the cosmological constant by a

non vanishing graviton mass…

NB: It seems one of the

Einstein’s motivations to

introduce the cosmological

constant was to try to « give a

mass to the graviton »

(see « Einstein’s mistake and the

cosmological constant »

by A. Harvey and E. Schucking,

Am. J. of Phys. Vol. 68, Issue 8 (2000))

Some properties of « massless gravity »

(i.e. General Relativity – GR )

In GR, the field equations (Einstein equations) take

the same form in all coordinate system (« general

covariance »)

Einstein tensor:

Second order non

linear differential

operator on the

metric g¹ º

Energy momentum

tensor: describes the

sources

Newton

constant

The Einstein tensor obeys the identities

In agreement with the conservation relations

Einstein equations can be obtained from the action

With

If one linearizes Einstein equations around e.g. a flat metric ´¹º

one obtains the field equations for a « graviton »

given by

Kinetic operator of the graviton h¹ º : does not contain any mass term / (undifferentiated h¹ º)

The masslessness of the graviton is guaranteed by the gauge

invariance (general covariance)

Which also results in the graviton having 2 = (10 – 4 £ 2)

physical polarisations (cf. the « photon » A¹)

1.2 The DGP model

(as an invitation to take the trip)

Peculiar to

DGP model

Usual 5D brane

world action

• Brane localized kinetic

term for the graviton

• Will generically be induced

by quantum corrections

A special hierarchy between

M(5) and MP is required

to make the model

phenomenologically interesting

Dvali, Gabadadze, Porrati, 2000

Leads to the e.o.m.

Phenomenological interest

A new way to modify gravity at large distance, with a new type

of phenomenology … The first framework where cosmic

acceleration was proposed to be linked to a large distance

modification of gravity (C.D. 2001; C.D., Dvali, Gababadze 2002)

(Important to have such models, if only to disentangle what

does and does not depend on the large distance dynamics of

gravity in what we know about the Universe)

Theoretical interest

Consistent (?) non linear massive gravity …

DGP model

Intellectual interest

Lead to many subsequent developments (massive gravity,

Galileons, …)

Energy density of brane localized matter

Homogeneous cosmology of DGP model

One obtains the following modified Friedmann equations (C.D. 2001)

• Analogous to standard (4D) Friedmann equations

In the early Universe

(small Hubble radii )

• Deviations at late time (self-acceleration)

Two

branches of

solutions

Light cone

Brane

Cosmic time

Equal cosmic time

Big Bang

DGP

Vs. CDM

Maartens, Majerotto 2006

(see also Fairbairn, Goobar 2005;

Rydbeck, Fairbairn, Goobar 2007)

• Newtonian potential on the brane behaves as

4D behavior at small distances

5D behavior at large distances

• The crossover distance between the two regimes is given by

This enables to get a “4D looking” theory of gravity out of

one which is not, without having to assume a compact

(Kaluza-Klein) or “curved” (Randall-Sundrum) bulk.

• But the tensorial structure of the graviton propagator is that of a massive

graviton (gravity is mediated by a continuum of massive modes)

Leads to the « van Dam-Veltman-Zakharov discontinuity » on

Minkowski background (i.e. the fact that the linearized theory differs

drastically – e.g. in light bending - from linearized GR at all scales)!

In the DGP model :

the vDVZ discontinuity, is believed to disappear via the « Vainshtein

mechanism » (taking into account of non linearities) C.D.,Gabadadze, Dvali,

Vainshtein, Gruzinov; Porrati; Lue; Lue & Starkman; Tanaka; Gabadadze, Iglesias;…

Many (open) questions about DGP model…

… but for the purpose of this talk,

just take it as an example of a

theory with some flavour of

« massive gravity »…

… and extra space dimensions.

1.3. Kaluza-Klein gravitons

Massive gravitons (from the standpoint of a 4D observer) are

ubiquitous in models with extra space-time dimensions in the

form of « Kaluza-Klein » modes.

Consider first a massless scalar-mediated force in 4D.

E = - /0

N = 4 GN m

It is obtained from the Poisson equation

(e.g. for an electrostatic or a gravitationnal potential)

Yielding a force between two bodies / 1/r2

A force mediated by a massive scalar would instead

obey the modified Helmholtz equation

- /C2 / source

Compton length C = ~ / m c

And results in the finite range Yukawa

potential

(r) / exp (-r/C) / r

This comes from a quadratic (m2 ©2)

mass terms in the Lagrangian

Inserting this into the 5D massless field equation:

with mk = Rk

Field equation for a 4D scalar

field of mass mk

A 5D massless scalar appears as a

collection of 4D massive scalars

(Tower of “Kaluza-Klein” modes):

m0 = 0

m1 = R1

m2 = R2

m3 = R3

m4 = R4

Experiments at energies much below m1

only see the massless mode

Low energy effective theory is four-

dimensional

m2

k k

The same reasoning holds for the graviton:

with g (x,y) = + h (x

,y)

Metric describing the

4+1D space-time

Flat metric describing

the reference cylinder

Small perturbation in the

vicinity of a reference

“cylinder” :

Decomposed in terms of a

Fourier serie : ² One massless

graviton ² A tower of massive

“Kaluza-Klein” graviton

Can one consider consistently a single

massive graviton ?

N.B., the PF mass term reads

h00 enters linearly both in the kinetic

part and the mass term, and is thus a

Lagrange multiplier of the theory…

… which equation of motion enables to eliminate

one of the a priori 6 dynamical d.o.f. hij

By contrast the h0i are not Lagrange multipliers

5 propagating d.o.f. in the quadratic PF h is transverse traceless in vacuum.

[cf. a massive photon (Proca field), which has 3

polarisation, vs a massless photon, which has 2]

1.5 Non linear Pauli-Fierz theory and the Vainshtein Mechanism

Can be defined by an action of the form

The interaction term is chosen such that

• It is invariant under diffeomorphisms

• It has flat space-time as a vacuum

• When expanded around a flat metric

(g = + h , f = )

It gives the Pauli-Fierz mass term

Einstein-Hilbert action

for the g metric

Matter action (coupled

to metric g)

Interaction term coupling

the metric g and the non

dynamical metric f

Matter energy-momentum tensor

Leads to the e.o.m. M2PG¹º =

¡T¹º + Tg

¹º(f; g)¢

Effective energy-momentum

tensor ( f,g dependent)

Isham, Salam, Strathdee, 1971

Some working examples

Look for static spherically symmetric solutions

with

H¹º = g¹º ¡ f¹º

(infinite number of models with similar properties)

Boulware Deser, 1972, BD in the following

Arkani-Hamed, Georgi, Schwarz, 2003

AGS in the following

(Damour, Kogan, 2003)

S(2)

int = ¡1

8m2M2

P

Zd4x

p¡f H¹ºH¾¿ (f

¹¾fº¿ ¡ f¹ºf¾¿ )

S(3)int = ¡

1

8m2M2

P

Zd4x

p¡g H¹ºH¾¿ (g

¹¾gº¿ ¡ g¹ºg¾¿ )

With the ansatz (not the most general one !)

gABdxAdxB = ¡J(r)dt2 +K(r)dr2 +L(r)r2d­2

fABdxAdxB = ¡dt2 + dr2 + r2d­2

Gauge transformation

g¹ºdx¹dxº = ¡eº(R)dt2 + e¸(R)dR2 + R2d­2

f¹ºdx¹dxº = ¡dt2 +

µ1¡

R¹0(R)

2

¶2

e¡¹(R)dR2 + e¡¹(R)R2d­2

Then look for an expansion in GN (or in RS / GN M) of the would be solution

Interest: in this form the g metric can easily be

compared to standard Schwarzschild form

This coefficient equals +1

in Schwarzschild solution

Wrong light bending!

Vainshtein 1972

In « some kind »

[Damour et al. 2003]

of non linear PF

…

…

O(1) ²

O(1) ²

Introduces a new length scale R in the problem

below which the perturbation theory diverges! V

with Rv = (RSmà 4)1=5For the sun: bigger than solar system!

So, what is going on at smaller distances?

Vainshtein’72

There exists an other perturbative expansion at smaller distances,

defined around (ordinary) Schwarzschild and reading:

with

• This goes smoothly toward Schwarzschild as m goes to zero

• This leads to corrections to Schwarzschild which are non

analytic in the Newton constant

¸(R) = +RSR

n1 +O

³R5=2=R

5=2v

´oº(R) = ¡RS

R

n1 +O

³R5=2=R

5=2v

´oR¡5=2v = m2R

¡1=2

S

To summarize: 2 regimes

÷(R) = àR

RS(1 + 32

7ï + ::: with ï = m4R5

RS

Valid for R À Rv with Rv = (RSmà 4)1=5

Valid for R ¿ Rv

Expansion around

Schwarzschild

solution

Crucial question: can one join the two

regimes in a single existing non singular

(asymptotically flat) solution? (Boulware Deser 72)

Standard

perturbation theory

around flat space

This was investigated (by numerical integration) by

Damour, Kogan and Papazoglou (2003)

No non-singular solution found

matching the two behaviours (always

singularities appearing at finite radius)

(see also Jun, Kang 1986)

In the 2nd part of this talk:

A new look on this problem using in

particular the « Goldstone picture » of

massive gravity in the « Decoupling limit. »

(in collaboration with E. Babichev and R.Ziour

2009-2010)

1.6 The Goldstone picture (and « decoupling limit »)

of non linear massive gravity,

and what can one get from it ?

Originally proposed in the analysis of Arkani-Hamed,

Georgi and Schwartz using « Stückelberg » fields …

and leads to the cubic action in the scalar sector

(helicity 0) of the model

Other cubic terms omitted

With = (m4 MP)1/5

« Strong coupling scale »

(hidden cutoff of the model ?)

Analogous to the Stuckelberg « trick » used to introduce

gauge invariance into the Proca Lagrangian

(action for a massive photon)

F¹ºF¹º +m2A¹A¹

Unitary gauge

The obtained theory has the gauge invariance

A¹ ! A¹ + @¹®

Á ! Á ¡ ®

A¹ ! A¹ + @¹ÁDo then the replacement

with the new field Á

The Proca action is just the same theory

written in the gauge, while gets a kinetic term

via the Proca mass term ( )

Á = 0 Ám2A¹A¹ ! m2@¹Á@¹Á

g

f

Do the same for non linear massive gravity

The theory considered has the usual diffeo invariance

g¹º(x) = @¹x0¾(x)@ºx0¿ (x)g0¾¿ (x

0(x))

f¹º(x) = @¹x0¾(x)@ºx0¿ (x)f 0¾¿ (x

0(x))

This can be used to go from a « unitary gauge » where

fAB = ´AB

To a « non unitary gauge » where some of the d.o.f.

of the g metric are put into f thanks to a gauge

transformation of the form

f¹º(x) = @¹XA(x)@ºXB(x)´AB (X(x))

g¹º(x) = @¹XA(x)@ºXB(x)gAB (X(x))

g¹º(x)

x¹

´AB

XA

f¹º(x)

XA(x)

A

Á

One (trivial) example: our spherically symmetric ansatz

gABdxAdxB = ¡J(r)dt2 +K(r)dr2 +L(r)r2d­2

fABdxAdxB = ¡dt2 + dr2 + r2d­2

Gauge transformation

g¹ºdx¹dxº = ¡eº(R)dt2 + e¸(R)dR2 + R2d­2

f¹ºdx¹dxº = ¡dt2 +

µ1¡

R¹0(R)

2

¶2

e¡¹(R)dR2 + e¡¹(R)R2d­2

Expand the theory around the unitary gauge as

XA(x) = ±A¹ x¹ + ¼A(x)

¼A(x) = ±A¹ (A¹(x) + ´¹º@ºÁ) :

Unitary gauge

coordinates

« pion » fields

The interaction term expanded

at quadratic order in the new fields A and reads

A gets a kinetic term via the mass term

only gets one via a mixing term

M2Pm2

8

Zd4x

£h2 ¡ h¹ºh

¹º ¡ F¹ºF¹º

¡4(h@A ¡ h¹º@¹Aº)¡ 4(h@¹@¹Á ¡ h¹º@

¹@ºÁ)]

One can demix from h by defining

h¹º = h¹º ¡ m2´¹ºÁ

And the interaction term reads then at quadratic order

S =M2

Pm2

8

Zd4x

nh2 ¡ h¹º h

¹º ¡ F¹ºF¹º ¡ 4(h@A ¡ h¹º@¹Aº)

+6m2hÁ(@¹@¹ + 2m2)Á ¡ hÁ + 2Á@A

io

The canonically normalized is given by ~Á = MPm2Á

Taking then the

« Decoupling Limit »

One is left with …

MP ! 1

m ! 0

¤ = (m4MP )1=5 » const

T¹º=MP » const;

With = (m4 MP)1/5

E.g. around a heavy source: of mass M

+ + ….

Interaction M/M of

the external source

with þà

P The cubic interaction above generates

O(1) coorrection at R = Rv ñ (RSmà 4)1=5

In the decoupling limit, the Vainshtein radius is kept fixed, and

one can understand the Vainshtein mechanism as

® ( ~Á)3 + ¯ ( ~Á ~Á;¹º ~Á;¹º)

and ® and ¯ model dependent coefficients

« Strong coupling scale »

(hidden cutoff of the model ?)

An other non trivial property of non-linear Pauli-Fierz: at non

linear level, it propagates 6 instead of 5 degrees of freedom,

the energy of the sixth d.o.f. having no lower bound!

Using the usual ADM decomposition of the metric, the

non-linear PF Lagrangian reads (for flat)

With Neither Ni , nor N are

Lagrange multipliers

6 propagating d.o.f., corresponding to the gij

The e.o.m. of Ni and N determine those as

functions of the other variables Boulware, Deser ‘72

Moreover, the reduced Lagrangian for

those propagating d.o.f. read

) Unbounded from below Hamiltonian

Boulware, Deser 1972

This can be understood in the « Goldstone » description

C.D., Rombouts 2005

(See also Creminelli, Nicolis, Papucci, Trincherini 2005)

Indeed the action for the scalar polarization

Leads to order 4 E.O.M. ), it describes two

scalars fields, one being ghost-like

Summary of the first part: the 3 sins of massive gravity

They can all be seen at the Decoupling Limit level

• 1. vDVZ discontinuity

Cured by the Vainshtein mechanism ?

• 2. Boulware Deser ghost

Can one get rid of it ?

• 3. Low Strong Coupling scale

Can one have a higher cutoff ?

The end of part 1