Extra dimensions and
modified gravity
Cédric Deffayet
(APC, CNRS, Paris)
1. Introduction
IAGRG-26
2. DGP model
Allahabad
January 2011
3. The Vainshtein mechanism of
« massive gravity ».
4. New models for large distance
modifications of gravity (k-Mouflage,
Galileon, Kinetic Gravity Braiding …)
1. Introduction
Why to modify gravity ?
Why extra dimensions ?
Why to modify gravity ?
One way to get rid of dark energy ?
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 or dark
energy ?
Why extra dimensions ?
Idea borrowed
from string
theory
XA
RD
Taken to
« brane worlds »
Usual (4 dimensional )
space-time: a “brane”
4D
Localized Degrees of freedom:
typically those of the particle
physics standard model
n extra-dimensions
+ non localized fields
(e.g. gravitational field)
Several canonical brane world models
brane
brane
“Warped”
Compact
extra-dimensions
extra-dimension
Arkani-Hamed &
Randall & Sundrum (1999)
Dimopoulos & Dvali (1998)
Compact
Extra dimensions(s)
“Anti-de-Sitter”space-time
brane
But gravity equations of motion are
modified such that one has large
distance modification of gravity
DGP model
Dvali, Gabadadze,
Porrati (2000)
In a 5D flat space-time
infinite extra-dimension
String theory, M theory
Low energy effective
theories of some
superstring
Compact
constructions.
(ADD & Antoniadis)
extra-dimensions
Strings with a low
Arkani-Hamed &
“fundamental” Dimopoulos
scale: a & Dvali
new way to do
phenonomelogy with
superstrings
Warped
extra-dimension
Linked to the AdS-CFT
Randall & Sundrum
correspondance
and other constructions
Non stringy approaches
to beyond standard
model approaches:
a new way to tackle some
problems
(e.g. the “gauge hierarchy”)
An alternative to
compactification
Toy model to study
some brane world
characteristics
brane
Compact
In both cases gravity is standard
extra-dimensions
Compact extra-dimensions of size
for r >> R(n) or RAdS
Because of the “bulk” geometry
DGP model
Dvali, Gabadaze,
Porrati, 2000
R(n)
Arkani-Hamed &
Dimopoulos & Dvali
brane
Gravity is modified
at large distance!
Warped
extra-dimensions
Randall & Sundrum
“Anti-de-Sitter” space-time
of radius RAdS
In this talk
DGP model and ideas following
from it …. getting inspired by DGP
to build modification of gravity at
large distances
2. DGP model (or brane-induced gravity).
Dvali, Gabadadze, Porrati, 2000
Usual 5D brane
world action
Peculiar to
DGP model
A special hierarchy between
M(5) and MP is required
to make the model
phenomenologically interesting
Leads to the e.o.m.
• Brane localized kinetic
term for the graviton
• Will generically be induced
by quantum corrections
DGP model
Phenomenological interest
A new way to modify gravity at large distance,
with a new type of phenomenology …
(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 …
In the DGP model :
• Newtonian potential on the brane behaves as
V(r ) /
V(r ) /
1
r
1
4D behavior at small distances
5D behavior at large distances
r2
• 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 (KaluzaKlein) 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!
Homogeneous cosmology of DGP model
One obtains the following modified Friedmann equations (C.D. 2001)
Energy density of brane localized matter
Analogous to standard (4D) Friedmann equations
In the early Universe
(small Hubble radii
)
Late time cosmology
Depending on the
sign of
One virtue of DGP model:
can get accelerated
universe by large distance
modification of gravity
(C.D (2001); C.D., Dvali,
Gabadaze (2002)).
Brane cosmology in 5D
Minkowski
no R termfrom
on standard cosmology
Latebulk
timewith
deviation
Self accelerating solution
the brane (i.e.: solution to 5D
(asympotes de Sitter space
Einstein-Hilbert Action)
even with zero matter energy
density)
DGP self accelerating phase
The brane (first) Friedmann equation
Can be rewritten as
with
Phase diagram
with  = +1
Acts as a cosmological
constant if  = +1
Same number of
parameter as CDM
Maartens, Majerotto
Vs. CDM
DGP
Maartens, Majerotto 2006
(see also Fairbairn, Goobar 2005;
Rydbeck, Fairbairn, Goobar 2007)
DGP model and cosmology
Interesting (toy) model:
first model to show how to obtain
accelerating Universe from a large
distance modification of gravity
However a ghost was found in the
spectrum of the linearized theory in
the self-accelerating phase !
Luty, Porrati, Rattazzi; Nicolis, Rattazzi,
Koyama; Gorbunov, Koyama, Sibiryakov;
Charmousis, Kaloper, Gregory, Padilla;
C.D., Gabadadze, Iglesias
The meaning of the ghost is not
completely clear (because the theory also
has a low hidden cuttof)…
It could possibly be eliminated by putting
brane into branes (cascading DGP)
De Rahm, Dvali, Hofmann, Khoury, Pujolas,
Redi, Tolley, 2008-2010
It triggered investigations of other models
learning from DGP (the rest of this talk) based
on particular on the Vainshtein mechanism
3. The Vainshtein mechanism of « massive gravity »
3.1 Quadratic Pauli-Fierz theory and
the vDVZ discontinuity
3.2 Non linear Pauli-Fierz theory and
the Vainshtein mechanism
This mechanism is at the basis of DGP and
other more recent models featuring large
distance deviations from GR
3.1 Quadratic massive gravity (and the vDVZ discontinuity)
Pauli-Fierz action: second order action
for a massive spin two h ö ÷
second order in h  ´ g -  
Only Ghost-free (quadratic) action for a
massive spin two
Pauli, Fierz
(NB: breaks explicitly gauge invariance)
vDVZ discontinuity
The propagators read
propagator for m = 0
ö÷ëì
D 0 (p)
propagator for m 6= 0
ö ÷ë ì
D m (p)
=
=
ñöëñ÷ì + ñöëñ÷ë
ñö÷ñëì
à 2p2
2p2
+ O (p)
ñö÷ñëì
ñöëñ÷ì + ñöëñ÷ë
à 3( p2à m2)
2( p2à m 2)
+ O (p)
Coupling the graviton with a conserved energy-momentum tensor
R4 p
Sint = d x ghö÷Tö÷
h
ö÷
R ö÷ëì
= D
(x à x 0)Tëì (x 0)d4x 0
The amplitude between two conserved sources T and S
is given by
R 4 ö÷
A = d xS (x)hö÷(x)
ð
For a massless graviton: A 0 = êTö÷ à
ð
For a massive graviton: A m = êTö÷ à
ñ
1
ê
2ñö÷T
ñ
1
ê
3ñö÷T
ê
Sö÷
In Fourier
space
ê
Sö÷
e.g. amplitude between two non relativistic sources:
ö
ê
T÷ / diag(mê1; 0; 0; 0)
ö
ê
S÷ / diag(mê2; 0; 0; 0)
A ø 23m
ê 1m
ê2
Instead of
Rescaling of Newton constant
defined from Cavendish
experiment
A ø 12m
ê 1m
ê2
4
GNewt on = 3G(4)
appearing in
the action
but amplitude between an electromagnetic probe
and a non-relativistic source is the same as in the
massless case (the only difference between massive and massless
case is in the trace part)
wrong light bending! (factor ¾)
3.2 Non linear Pauli-Fierz theory (NB: not a realistic theory)
and the Vainshtein mechanism
Can be defined by an action of the form
Keep all order in h  ´ g -  
At quadratic order in h  this
reduces to Pauli-Fierz
Look at Spherically Symmetric
Solutions of this theory
Some
background
metric
2
÷(ú) 2
õ(ú)
2
ö(ú) 2
à
ds = e dt + e dú + e ú dÒ2
2
÷(r ) = à
õ(r ) = +
rS
7 …
+
r (1+ 32ï +
1
2
:::
rS
21
+ ï…
à
(1
r
8 +
with ï =
:::
rS
m4r 5
Vainshtein 1972
In some kind of
non linear PF
Wrong light bending!
This coefficient equals +1
Introduces
a new length
scale r
in Schwarzschild
solution
in the problem
below which the perturbation theory diverges!
V
For the sun: bigger than solar system!
with
r v = (r Smà 4)1=5
So, what is going on at smaller distances?
Vainshtein’s answer (1972):
There exists an other perturbative expansion at
smaller distances, reading:
º (r ) = ¡
rS
r
¸ (r ) = +
rS
r
n
n
³
1+ O
1+ O
³
5=2
r 5=2 =r v
5=2
r 5=2 =r v
´o
´o
No warranty that this
solution can be matched
with the other for large r!
Boulware, Deser 1972
with r v¡ 5=2 = m 2 r S¡ 1=2
• This goes smoothly toward
Schwarzschild as m goes to
zero
• This leads to corrections to
Schwarzschild which are non
analytic in the Newton
constant
To summarize: 2 regimes
÷(R) = à
RS
7
R (1 + 32ï
Valid for R À Rv
+ ::: wit h
ï =
RS
m 4R 5
with Rv = (RSmà 4)1=5
Standard
perturbation theory
around flat space
Crucial question: can one join the two
regimes in a single existing non singular
(asymptotically flat) solution? (Boulware Deser 1972)
Expansion around
Schwarzschild
solution
Valid for R ¿ Rv
Does it really work ?
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)
and hence failure of the « Vainshtein
mechanism »
(see also Jun, Kang 1986)
We provided 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
• JHEP 0905:098,2009. (arXiv:0901.0393)
• PRL 103.201102(arXiv:0907.4103)
• PRD 82:104008,2010 (arXiv:1007.4506)
To obtain our solutions, we used the Decoupling Limit,
analytic insights (based on resurgence theory extending
Borel resummation) …. ……. and numerically… we first…
« shooted »
Then « relaxed »
With the ansatz
(for non linear massive gravity, with two metrics)
physical, dynamical, metric
g¹ º dx ¹ dx º
=
f ¹ º dx ¹ dx º
=
¡ eº ( R ) dt 2 + e¸ ( R ) dR 2 + R 2 d- 2
µ
¶2
0
R¹ (R)
2
¡ dt + 1 ¡
e¡ ¹ ( R ) dR 2 + e¡
2
Background flat metric
¹ (R)
R 2 d-
2
Numerical solutions of the full non linear system
«GR regime »
« linear regime »
Yukawa decay
source
The vDVZ discontinuity gets erased for
distances smaller than RV as expected
4. Getting inspired by DGP and massive gravity
4.1 The decoupling limit
4.2 k-Mouflage
4.3 Galileons
4.4 Kinetic Gravity Braiding
4.1 The decoupling limit
The vDVZ discontinuity can be understood as being due to
the scalar polarization of the (massive) graviton being
coupled to the trace of the source energy momentum
tensor…
While the Vainshtein mecanism can be related
to this polarization having strong self interaction
C.D.,Gabadadze, Dvali, Vainshtein, 2002
A good (at least qualitatively speaking)
description of this is given by the action
(obtained by taking a « decoupling limit »)
Where for the scalar sector of massive (bi-)gravity, one gets
®(
~3+ ¯(
Á)
~Á
~;¹ º Á
~;¹ º )
Á
With  = (m4 MP)1/5 and ® and ¯ model dependent coefficients
In the decoupling limit, the Vainshtein radius is kept fixed, and
one can understand the Vainshtein mechanism as
E.g. around a heavy source:
+
Interaction M/M P of
the external source
with þ
à
of mass M
+ ….
The cubic interaction above generates
O(1) coorrection at R = Rv ñ (RSmà 4)1=5
Question:
Can one use this do build interesting new
models with large distance modifications of
gravity ?
4.2 k-Mouflage
Receipe : take the DL action
N.L. completion (and extension)
k-Mouflage
Nice (toy model) arena to explore
to modify gravity in the IR
(Nicolis, Rattazzi and Trincherini; Chow, Khoury; Silva,
Koyama… for Galileon)
What about cosmology ?
4.3 Galileon
4.4 Kinetic Gravity Braiding
Starting from the Decoupling Limit of the
DGP model:
Scalar sector of the model
Energy
scale
One obtains the following field equations
NB: Those are purely second order
Luty, Porrati, Rattazzi
The field equations
can be obtained from the (5D) « Hamiltonian » constraint
Where one substitutes the Israel junction condition
To obtain
A last substitution
Yield the e.o.m. for 
The self accelerating brane « K = constant » and T =0
Corresponds in the DL
(using the substitution
To the solution
)
(and T=0)
of the field equations
Similar solutions exist in any scalar field theory with
field equations which are pure second order
Galileon
Nicolis, Rattazzi, Trincherini 2009
In 4D: only 3 non trivial models
(With the notations ¼¹ = ¹ ¼ and ¼¹ º = ¹ º ¼ )
Which can be expressed using product of epsilon
tensors … e.g.
A lot of interesting aspects an phenomenology
still being explored
Covariantization
C.D., Deser, Esposito-Farese, Vikman
Inflation and non Gaussianity
Creminelli, Nicolis, Trincherini, de Felici, Tsujikawa,
D’Amico, Musso, Norena, Nesseris, Mota, Sandstad,
Zlosnik,Burrage, de Rham, Mota, Tolley, Kobayashi,
Silva, Koyama, Yamagushi, Yokoyama …
Multi Galileon and p-form Galileon
C.D., Deser, Esposito-Farese, Padilla, Saffin,
Zhou, Hinterbichler, Trodden, Wesley …
4.4 Kinetic Gravity Braiding
C.D., Pujolas, Sawicki, Vikman
with
The scalar field equations read
« Kinetic braiding »
Lead to various interesting properties
Among which crossing of the phantom
divide without ghosts or instabilities
To conclude
DGP model
• First model to get an accelerating Universe via a
large distance modification of gravity
• Lack of a good UV completion
• Gave new insights on « massive gravity »
(use of the Vainshtein mechanism)
• Inspired a lot of work
Lead to a new family of scalar-tensor models
(k-Mouflage, Galileon, Kinetic Gravity Braiding)
still to be fully explored
(only other competing mechanisms
« Chameleon » / « Symmetron »)
Thank you for your invitation and attention