Review talk by Prof. Cedric Deffayet at the SuperJEDI Conference, July 2013

1. 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)

2. 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 ?

3. 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

4. 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 ?

5. 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))

6. 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

7. Einstein equations can be obtained from the action
With

8. 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)

9. 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.

10. 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, …)

11. 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

12. DGP
Vs. CDM
Maartens, Majerotto 2006
(see also Fairbairn, Goobar 2005;
Rydbeck, Fairbairn, Goobar 2007)

13. • 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;…

14. 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.

15. 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

16. A force mediated by a massive scalar would instead
obey the modified Helmholtz equation
  -  /C
2 / 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

18. Inserting this into the 5D massless field equation:
with mk = R
k
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 = R
1
m2 = R
2
m3 = R
3
m4 = R
4
Experiments at energies much below m1
only see the massless mode
Low energy effective theory is four-
dimensional
m
2
k k

19. 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 ?

23. 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]

24. 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. M2
P G¹º =
¡
T¹º + Tg
¹º(f; g)
¢
Effective energy-momentum
tensor ( f,g dependent)
Isham, Salam, Strathdee, 1971

25. 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
8
m2
M2
P
Z
d4
x
p
¡f H¹ºH¾¿ (f¹¾
fº¿
¡ f¹º
f¾¿
)
S
(3)
int = ¡
1
8
m2
M2
P
Z
d4
x
p
¡g H¹ºH¾¿ (g¹¾
gº¿
¡ g¹º
g¾¿
)

26. With the ansatz (not the most general one !)
gABdxA
dxB
= ¡J(r)dt2
+ K(r)dr2
+ L(r)r2
d-2
fABdxA
dxB
= ¡dt2
+ dr2
+ r2
d-2
Gauge transformation
g¹ºdx¹
dxº
= ¡eº(R)
dt2
+ e¸(R)
dR2
+ R2
d-2
f¹ºdx¹
dxº
= ¡dt2
+
µ
1 ¡
R¹0
(R)
2
¶2
e¡¹(R)
dR2
+ e¡¹(R)
R2
d-2
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

27. 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=5
For the sun: bigger than solar system!

28. 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) = +RS
R
n
1 + O
³
R5=2
=R
5=2
v
´o
º(R) = ¡RS
R
n
1 + O
³
R5=2
=R
5=2
v
´o
R
¡5=2
v = m2
R
¡1=2
S

29. To summarize: 2 regimes
÷(R) = à R
RS
(1 + 32
7
ï + ::: 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

30. 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)

31. 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 ?)

32. Analogous to the Stuckelberg « trick » used to introduce
gauge invariance into the Proca Lagrangian
(action for a massive photon)
F¹ºF¹º
+ m2
A¹A¹
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 Á
m2
A¹A¹
! m2
@¹Á@¹
Á
g 
f 

33. Do the same for non linear massive gravity
The theory considered has the usual diffeo invariance
g¹º(x) = @¹x0¾
(x)@ºx0¿
(x)g0
¾¿ (x0
(x))
f¹º(x) = @¹x0¾
(x)@ºx0¿
(x)f0
¾¿ (x0
(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)x
¹
´AB
X
A
f¹º(x)
XA
(x)
A
Á

34. One (trivial) example: our spherically symmetric ansatz
gABdxA
dxB
= ¡J(r)dt2
+ K(r)dr2
+ L(r)r2
d-2
fABdxA
dxB
= ¡dt2
+ dr2
+ r2
d-2
Gauge transformation
g¹ºdx¹
dxº
= ¡eº(R)
dt2
+ e¸(R)
dR2
+ R2
d-2
f¹ºdx¹
dxº
= ¡dt2
+
µ
1 ¡
R¹0
(R)
2
¶2
e¡¹(R)
dR2
+ e¡¹(R)
R2
d-2

35. Expand the theory around the unitary gauge as
XA
(x) = ±A
¹ x¹
+ ¼A
(x)
¼A
(x) = ±A
¹ (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
M2
P m2
8
Z
d4
x
£
h2
¡ h¹ºh¹º
¡ F¹ºF¹º
¡4(h@A ¡ h¹º@¹
Aº
) ¡ 4(h@¹
@¹Á ¡ h¹º@¹
@º
Á)]

36. One can demix  from h by defining
h¹º = ^h¹º ¡ m2
´¹ºÁ
And the interaction term reads then at quadratic order
S =
M2
P m2
8
Z
d4
x
n
^h2
¡ ^h¹º
^h¹º
¡ F¹ºF¹º
¡ 4(^h@A ¡ ^h¹º@¹
Aº
)
+6m2
h
Á(@¹@¹
+ 2m2
)Á ¡ ^hÁ + 2Á@A
io
The canonically normalized  is given by ~Á = MP m2
Á
Taking then the
« Decoupling Limit »
One is left with …
MP ! 1
m ! 0
¤ = (m4
MP )1=5
» const
T¹º=MP » const;

37. 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 ?)

38. 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

39. 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

40. 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 ?

41. The end of part 1