Aspects of a supersymmetric Brans-Dicke theory

0.14 MB·English

by  Riccardo Catena

DESY 06 – 211
Aspects of a supersymmetric Brans-Dicke theory
Riccardo Catena
Deutsches Elektronen-Syncrotron (DESY),
22603 Hamburg, Germany
([email protected])
We consider a locally supersymmetric theory where the Planck mass is replaced by a dynam-
ical superfield. This model can be thought of as the Minimal Supersymmetric extension of the
Brans-Dicketheory(MSBD).Themotivation thatunderliesthisanalysis istheresearch ofpossible
connections between Dark Energy models based on Brans-Dicke-like theories and supersymmetric
Dark Matter scenarios. We find that the phenomenology associated with the MSBD model is very
different compared to the one of the original Brans-Dicke theory: the gravitational sector does
not couple to the matter sector in a universal metric way. This feature could make the minimal
supersymmetric extension of theBD idea phenomenologically inconsistent.
7
0
0 I. INTRODUCTION In the BD theory the Planck mass is replaced by a
2 dynamical scalar field. In this paper we consider the su-

n persymmetric analogous of this mechanism: we replace
During the last decade of cosmological observations,
a in the supergravity Lagrangian the Planck mass with a
thepictureofaUniversedominatedbyDarkMatterand
J chiralsuperfield,the“Plancksuperfield”. Suchareplace-
Dark Energy has emerged [1]. Although a real under-
2 ment defines the “natural” supersymmetric extension of
standing of the microscopic nature of such cosmologi-
2 the BD theory. Let us refer to it as the Minimal Super-
cal components is still missing, different theories are at
symmetric Brans-Dicke theory (MSBD) to distinguish it
2 present under analysis.
from other possible approaches. We find that, contrary
v Concerning Dark Matter particles, their indirectly ob-
4 totheoriginalBDtheory,intheMSBDthegravitational
served interaction properties naturally fit those of the
2 sectordoesnotcoupletothemattersectorinauniversal
lightest state of supersymmetric models with conserved
2 metric way. As a result, possible violations of the weak
1 R−parity [2]. If supersymmetry really exists, a simple equivalence principle couldmake the minimal supersym-
1 and well motivated cosmological model should include a metric extension of the BD idea phenomenologically in-
6 supersymmetric dark matter particle.
consistent.
0
Dark Energy interaction properties are even more ob-
In spite of this conclusion, we find the subject a good
/
h scurethantheDarkMatterones. Anexampleofthislack laboratory for studying realistic models of Dark Matter-
t of knowledge is the difficulty to explain the extremely
- Dark Energy unification. For instance, alternative ap-
p small value of its mass scale that, in a phenomenolog-
proaches to the problem could provide a consistent sce-
e ically consistent model, should be of the order of the
nariowhereDarkMatterandDarkEnergyareidentified
h
present value of the Hubble parameter [3]. As a conse-
: with different components of the Planck multiplet.
v quence, direct couplings of Dark Energy to matter fields
The plan of this paper is as follows. In section II and
i are strongly constrained by fifth force searches [4]. A
X III we introduce notation and review the BD model and
possible way to avoid such constraints is to work in the
r the concept of universal metric coupling. Section IV is
a framework of Scalar-Tensor theories [5]. In these theo- devotedto the MSBD theory;we willspeciallyunderline
ries the gravitationalinteraction is described in terms of
the differences between its phenomenology and the one
both a metric tensor and a scalar field. Moreover, the
of the original BD theory. Section V is concerned with
energy density of this extra scalar degree of freedom can
sometechnicaldetails,relatedwiththe componentfields
be easily identified with Dark Energy [6]. An interesting
formalism, that should make the arguments of section
feature of such models is that the gravitational sector
IV more precise. The results are discussedin section VI.
(that also includes the Dark Energy scalar) couples to
Finally, we list in the appendix useful expressions that
the matter sector in a universal metric way so that fifth
we used during the computations.
forceboundsaresatisfiedbyconstruction[4]. IfDarkEn-
ergy has a scalar nature, Scalar-Tensor theories provide
a natural framework to discuss its properties.
II. NOTATION
TheinterestingpossibilitytorelateaScalar-Tensorin-
terpretationofDarkEnergytoasupersymmetricdescrip-
tion of Dark Matter leads to study supersymmetric ex- We will use in the following the same notation and
tensionsofScalar-Tensortheories. Thisisthetopicofthe conventions of [8]. We list here for clarity some of them.
present paper. As we will see, the results of this analy- The superspace is described in terms of the coordi-
sis do not rely on the particular choice of the underlying nates (ym,θ ). Greek indexes label two components
α
Scalar-Tensortheory;forthis reasonwewillconsiderthe Weylspinorswhile latinindexesthe componentsoffour-
simplest one, i.e. the Brans-Dicke (BD) theory [7]. vectors. Indexes transforming under local coordinates
2
transformationsinsuperspacearecalledEinsteinindexes matter-gravitycoupling is also called universal and met-
and are taken from the end of the alphabet, for example ric. Thisisanontrivialpropertyandhasveryimportant
(m, n, ...). Instead, indexes transforming under local phenomenological implications. It can be shown, for in-
Lorentz transformations are called Lorentz indexes and stance, that in a theory where matter couples to gravity
aretakenfromthebeginningofthealphabet,forexample in a universalmetric way the weak equivalence principle
(a, b, ...). The power series expansion in θ of a chiral is satisfied by construction [4].
α
superfield Φ is given by A typical example of non universal metric coupling is
thefollowing. Letusintroduceinthegravitationalsector
Φ(ym,θα)=A(ym)+√2θαχα(ym)+θαθαF(ym) (1) a long range scalar field φ that couples like a dilaton to
the field strength F of some (for simplicity) abelian
whereA(ym)andF(ym)arecomplexscalarsandχ (ym) µν
α gauge group with gauge coupling g¯
a Weyl spinor. We will couple matter superfields to the
minimal supergravity multiplet. This contains the viel-
1
bein eam, the gravitino ψαa and two auxiliary fields: a SφFF =−4g¯2 d4x√−ggαµgβνφFαβFµν (5)
vector ba and a scalar M. Finally, covariant derivatives Z
withrespecttosupergravitytransformationsaredenoted
by Dα, D¯α˙ and Dm. winhfeoruergdµiνmiesntshioenms tehtreiccotmenbsionratainond√g ≡gdgeαtµ(ggµβνν)i.sSWinecyel
−
invariant, the scalar field φ can not be reabsorbed by
III. THE BRANS-DICKE THEORY AND THE means of a rescalingof the metric. Therefore, in this ex-
UNIVERSAL METRIC COUPLING ample the gauge field strength Fµν feels gravity through
the metric g andthe scalarφ. Inotherwords,sinceno
µν
Weyl rescaling of g can “remove” φ from the matter
InGeneralRelativitythecouplingbetweengravityand µν
sector (S ), it is not possible in this case to define a
matter is described by the following Lagrangian φFF
Jordanframe. Such a non metric and universalcoupling
1 can be easily interpreted in terms of an effective, scalar-
LEH =−2eMP2lR+LM[eam,Ψ], (2) field dependent, gauge coupling, i.e. g¯e−ff2(φ) ≡ g¯−2φ.
Moreover, it can be shown that in this picture also the
where e ≡ det(eam), R is the Ricci scalar and Ψ symbol- masses of the particles become φ-dependent. However
ically represents all matter fields involved in the theory. the protonandneutronmasses,forinstance,acquiredif-
In the BD approach to the gravitational interaction the ferent dependences from φ. This is a consequence of the
Planck mass appearing in eq. (2) becomes dynamical by fact that a gauge interaction contributes differently to
means of the substitution the proton and neutron binding energies. As a result,
thetheorymanifestlyviolatestheweakequivalenceprin-
M2 = ϕ2(ym), (3)
Pl ⇒ ciple [4].
where ϕ(ym) is a real scalar field. As a consequence
eq. (2) is replaced by
IV. THE MINIMAL SUPERSYMMETRIC
LBD = Lϕ[eam,ϕ]+LM[eam,Ψ] BRANS-DICKE THEORY
1
= e ϕ2 +ω∂ ϕ∂mϕ + [ea ,Ψ], (4)
−2 R m LM m Eq. (3) gives a prescription to construct the BD La-
(cid:0) (cid:1) grangian starting from the Einstein-Hilbert one. In this
where the factor ω that multiplies the kinetic term of
section we apply an analogousprescriptionto the super-
ϕ has to be tuned to fit the post-newtonian bounds [9].
gravity Lagrangian
Eq.(4)givesthesocalled“Jordanframe”formulationof
the theory. In this frame the BD scalar does not appear
in the matter Lagrangianand particle physics is just the = 3M2 d2θ2 R+ [H,Ψ]+h.c., (6)
standard one. The theory can be formulated in other Lsg − Pl E LM
Z
frames related to the Jordan one by a Weyl rescaling of
the vielbein such as ea ea el(ϕ), where l(ϕ) is some whereH isthesupergravitymultiplet, isthechiralden-
ϕ-dependent function.mIn→themse alternative formulations sityandRrepresentsthecurvaturesupEerfield,definedas
thematterLagrangianacquiresanexplicitfunctionalde- the covariantderivative of the spin connection.
pendence from ϕ, i.e. = [ea el(ϕ),Ψ]. However, Let us start introducing a chiral superfield Φ with com-
the inverse Weyl rescaliLngMea LMeame−l(ϕ) always brings ponents givenin the power series expansion(1). We will
back the theory to its originmal→vermsion in which particle call Φ the Planck superfield. This dynamical object al-
physics is just the standard one. lowsthe naturalsupersymmetricextensionofthe substi-
Eq. (4) shows that in the BD theory all matter fields tution (3)
feel the gravitational interaction through the same viel-
bein, the Jordan frame vielbein. For this reason such a M2 = Φ2(ym,θ ). (7)
Pl ⇒ α
3
Applying the substitution (7) to eq. (6) one finds theory the weak equivalence principle is not satisfied by
constructionandtimevariationsofmassesandcouplings
MSBD = Φ[H,Φ]+ M[H,Ψ] are not under control. In the next section we will give
L L L
the explicit expressions for eqs. (9) and (10).
= 3 d2θΦ22 R
− E −
Z
1
d2θ2 ¯ ¯α˙ 8R Φ†Φ+ V. COMPONENT FIELDS
α˙
− 8 E D D −
Z
+ [H,Ψ] +(cid:0) h.c., (cid:1) (8)
LM The Lagrangiansgivenin this sectionare obtainedus-
ingthe results summarizedinthe appendix. Letus start
whereinthe thirdline,inanalogywith eq.(4), weintro-
with the first term of eq. (8). Its component fields ex-
duced a kinetic term for Φ. To be as general as possible
pansion reads
we do not assume any particular form for .
M
L
Eq. (8) defines the Minimal Supersymmetric Brans
Dicke theory (MSBD). Its invariance under supergrav- 3 d2θΦ22 R+h.c.=
− E
ity transformations follows from the properties of chiral Z
1
densities. By definitions, chiral densities transform like e(A2+A2∗)
totalderivativesinthespace(ym,θ )andtheproductof −4 R
α
1
a chiral density and a chiral superfield is again a chiral + eεabcd ψ¯ σ¯ ψ A2 ψ σ ψ¯ A2∗
density [8]. Moreover, the superfields ¯¯ 8R Φ†Φ 2 a bDc d − a bDc d
and Φ2 are chiral if Φ is chiral. This proDvDes−the invari- + 1 e(A2(cid:0) A2∗)εabcd ψ¯ σ¯ ψ +ψ σ(cid:1) ψ¯
(cid:0) (cid:1) a b c d a b c d
ance of the Lagrangian (8) under supergravity transfor- 16 − D D
mations. 1 eAχσaσ¯bψ 1(cid:0) eA∗χ¯σ¯aσbψ¯ (cid:1)
ab ab
Let us focus now on its phenomenology. As we will −√2 − √2
see explicitly in the next section, the component fields 1 1
e(A2+A2∗)MM∗+ e(A2+A2∗)bab
expansion of eq. (8) gives rise to a Lagrangian with the −6 6 a
following structure i
eem ba(A2 A2∗)
−2 a Dm −
= [ea ,ψa,ba,M,A,χ ,F]
LMSBD + LΦ[ema ,ψαa,ba,M,Ψ],α (9) 1eψaσaψ¯bbb(A2 A2∗) 1eψ¯aσ¯aψbbb(A2 A2∗)
LM m α −4 − − 4 −
i i
where eachfields was alreadyintroduced during the pre- eAχψaba+ eA∗χ¯ψ¯aba
−√2 √2
vious sections. Eq. (9) is the supersymmetric version
1 1
of eq. (4). The crucial difference between the two La- eχχM eχ¯χ¯M∗+eAFM +eA∗F∗M∗, (11)
grangiansis that in the supersymmetric one and −2 − 2
M Φ
communicatealsothroughthe auxiliaryfieldsLba andML .
where
This has deep phenomenological consequences when the
auxiliary fields are removed by means of their equations ψα = ψα ψα,
of motion. To show this point, let us write the general nm Dn m−Dm n
ψα =∂ ψα +ψβω α,
solution of the equations of motion for M and ba as fol- Dn m n m m nβ
lows and ω α is the algebra-valued spin connection.
nβ
Nowwe focus onthe kinetic termofthe Plancksuper-
ba = h (...,A,χ ),
1 α field. Its component fields expansion is given by
M = h (...,A,χ ), (10)
2 α
1
d2θ2 ¯ ¯α˙ 8R Φ†Φ=
where h1 and h2 are two appropriate functions of the −8 E Dα˙D −
fields involved in the theory. In eq. (10) we underlined 1Z (cid:0) (cid:1)
the crucial dependence of h1 and h2 from A and χα. +6e|A|2R−e∂mA∂mA∗
Now, replacing the solutions (10) in the Lagrangian(9),
i
the degrees of freedom of the Planck multiplet explicitly e(χσm mχ¯+χ¯σ¯m mχ)
−2 D D
appear in the matter Lagrangian. Since no Weyl rescal-
1
ing of the vielbein can remove the auxiliary fields from eA2εabcd ψ¯ σ¯ ψ ψ σ ψ¯
a b c d a b c d
−6 | | D − D
, it followsthat the Planckmultiplet couples intrinsi-
M
Lcally to matter. Therefore, there is no way to write the +√2e A∗χσa(cid:0)bψ +Aχ¯σ¯abψ¯ (cid:1)
matter Lagrangianas [ea ,ψa,Ψ] by means of a suit- 3 ab ab
ablevielbeinredefinitioLnMofmthefαormea ea el(A,χα,F), √2 (cid:0) (cid:1)
m → m e ψ¯ σ¯bσaχ¯∂ A+χσaσ¯bψ ∂ A∗
where l is an appropriate function of the components of − 2 a b a b
Φa.thIneoortyh.erTwhoerdmsa,ianJcoorndsaenqfureanmceedisoetshnatotinextishtefoMrSsuBcDh +1eεab(cid:0)cd(A∗∂aA A∂aA∗)ψbσcψ¯d (cid:1)
4 −
4
1 i
eA2b ba+ eba(A∂ A∗ A∗∂ A) and finally,
a a a
−9 | | 3 −
−16eχσabaχ¯−i√62eba Aψ¯aχ¯−A∗ψaχ Laux = 61ef(A,A∗)baba+ 3i eba(A∂aA∗−A∗∂aA)
+eFF∗+ 1eA2 M 2 (cid:0) 1eMA∗F 1(cid:1)eM∗F∗A 1eχσab χ¯ i√2eba Aψ¯ χ¯ A∗ψ χ
9 | | | | − 3 − 3 −6 a − 6 a − a
+L4, (12) −2i eema Dmba(A2−A2∗)(cid:0) (cid:1)
where includes only 4 fermions interactions and it 1 1
L4 − eψ σaψ¯ bb(A2 A2∗) eψ¯ σ¯aψ bb(A2 A2∗)
will be given afterwords. −4 a b − − 4 a b −
Using eqs. (11) and (12) one can write the explicit i i
eAχψ ba+ eA∗χ¯ψ¯ ba
component fields expansion of eq. (8). For simplicity we −√2 a √2 a
decompose the final Lagrangianas follows
1 1 1
ef(A,A∗)M 2+eFF∗ eMA∗F eM∗F∗A
−6 | | − 3 − 3
= + + + , (13)
LMSBD LK Lint L4 Laux 1eχχM 1eχ¯χ¯M∗+eAFM +eA∗F∗M∗
−2 − 2
where is the Lagrangian for the kinetic terms of the
fields cLoKntained in the Planck and supergravity multi- +LM[eam,ψαa,ba,M,Ψ]. (18)
plets, describes the interactions between the Planck
Lint To recover a complete analogy with eq. (4) one has
and supergravitymultiplets not included in and
L4 Laux to perform in eq. (13) a Weyl rescaling of the vielbein
is the Lagrangian for the auxiliary fields where we also
in order to have a kinetic term for the graviton of the
absorbed . We list in the following their explicit ex-
LM form 1/2eA2 . However, also includes a con-
pressions. LK reads tribut−ion pro|po|rRtional to ; aLsMa consequence such a
R
rescaling should be performed only after having speci-
1
LK = −4ef(A,A∗)R−e∂mA∂mA∗ fiAed=LAM∗.=AMddinangdLFM=toχe=q.0(,1o1n)eagnedtsttahkeinegxptrheessliiomnit
i Pl
e(χσm χ¯+χ¯σ¯m χ)
m m
− 2 D D 1
+ eεabcd g1(A,A∗)ψ¯aσ¯b cψd −2eMP2lR
D
+ g2(A,A(cid:2)∗)ψaσbDcψ¯d , (14) +21eMP2lεabcd ψ¯aσ¯bDcψd−ψaσbDcψ¯d
where the functions f, g1 and g2 are(cid:3)defined as follows 1eM2MM(cid:0)∗+ 1eM2bab (cid:1)
−3 Pl 3 Pl a
f(A,A∗) = A2+A2∗ 2 A2, +LM[eam,ψαa,ba,M,Ψ]. (19)
− 3| |
9 1 1 that, in agreement with [8], gives the component fields
g (A,A∗) = A2 A2∗ A2,
1 16 − 16 − 6| | expansion of the Lagrangian(6).
1 9 1 As usual, auxiliary fields can be expressed in terms
g (A,A∗) = A2 A2∗+ A2. (15)
2 16 − 16 6| | of other fields involved in the theory by means of their
equationsofmotion. UsingtheLagrangian(18)onefinds
The Lagrangian is given by
int
L
1 1
b = i (A∗∂ A A∗∂ A)+ χσ χ¯
√2 a − f a − a 2f a
= e A∗χσabψ +Aχ¯σ¯abψ¯
int ab ab
L 3 √2 3
+i (Aψ¯ χ¯ A∗ψ χ)+i (Aχψ A∗χ¯ψ¯ )
√2e(cid:0)ψ¯ σ¯bσaχ¯∂ A+χσaσ¯bψ(cid:1)∂ A∗ 2f a − a √2f a− a
a b a b
− 2 3
+ 1eεab(cid:0)cd(A∗∂ A A∂ A∗)ψ σ ψ¯ (cid:1) +4f(A2−A2∗)(ψaσaψ¯b+ψ¯aσ¯aψb)
a a b c d
4 −
3 3
1 1 +i ω m i ∂ [em(A2 A2∗)]
eAχσaσ¯bψab eA∗χ¯σ¯aσbψ¯ab. (16) 2f ma − 2f m a −
− √2 − √2
3 ∂
M
L ,
−ef ∂ba
The 4 fermions interactions read
4
− L 6∂
M
M =C 3χ¯χ¯+ L ,
1 e√2 1 − e∂M∗
= + eχσcσ¯bψ ψ¯ χ¯ i ψ¯ σ¯bηac (cid:18) (cid:19)
4 c b a
L 4 − 8 6∂
+ σ¯aσcσ¯b ψcψ¯bχ¯A + h.c., (cid:0) (17) F∗ =C2 −3χ¯χ¯+ e∂MLM∗ , (20)
(cid:18) (cid:19)
(cid:1)
5
1 1
where ie m ba+ ψ¯ψ¯M ψ σaψ¯ bc
1 − a Dm 2 − 2 a c
C (A,A∗) ,
1 ≡ 6|A|2−A2−A2∗ + 1εabcd ψ¯aσ¯bψcd+ψaσbψ¯cd . (A.2)
1 8 !#)
C (A,A∗) C (A,A∗) A∗ A .
2 1
≡ 3 −
(cid:18) (cid:19)
In eqs. (20) we omitted the dependence from A and Finally, the action of the chiral projector
A∗ ofthe functions f,C1 andC2. WhenLM is specified, D¯α˙D¯α˙ −8R on the field Φ† is given by [8]
from eqs. (20) one can explicitly compute the functions
h and h introduced in section IV. (cid:0) (cid:1)
1 2 ¯ ¯α˙ 8R Φ† =
α˙
D D −
(cid:0) 4 (cid:1) 2
VI. CONCLUSIONS 4F∗+ MA∗+θ 4i√2σcDˆ χ¯ √2σab χ¯
c a
− 3 "− − 3
In this paper we have studied the minimal supersym- 4
metric extension of the BD theory (MSBD) defined by + A∗ 2σabψab iσaψ¯aM +iψaba
3 − !#
eq. (8). The underlying motivation was the research of
possible connections between a Scalar-Tensor interpre- 8
+θθ 4em DˆaA∗ ib DˆaA∗
tation of Dark Energy and a supersymmetric descrip- (− a Dm − 3 a
tion of Dark Matter. Eq. (8) is obtained replacing the
2 8
Planck mass with a chiral superfield in the supergrav- √2ψ¯ σ¯abχ¯+2√2ψ¯ Dˆaχ¯ M∗F∗
ab a
ity Lagrangian (6). We called this extra superfield the −3 − 3
Planck superfield. Although this approach looks very 2i√2ψ¯ χ¯ba+ 1i√2ψ¯ σ¯aσcχ¯b
a a c
natural, the resulting phenomenology is radically differ- −3 3
entfromtheoneoftheoriginalBDtheory. IntheMSBD
4 1 2 1
theory the extra degrees of freedom of the Planck super- + A∗ +iψ¯aσ¯bψab+ MM∗+ baba
field intrinsically couple to matter and a Jordan frame 3 "− 2R 3 3
formulation can not be achieved through a suitable viel- ie m ba+ 1ψ¯ψ¯M 1ψ σaψ¯ bc
beinredefinition. As aconsequence,thistheorydoesnot − a Dm 2 − 2 a c
satisfy the weak equivalence principle by construction.
1
Thisconclusioncouldmaketheminimalsupersymmetric + εabcd ψ¯aσ¯bψcd+ψaσbψ¯cd , (A.3)
8
extension of the BD idea phenomenologically inconsis- !#)
tent. 1
In spite of this result, we find that if a consistent su- where
persymmetric Scalar-Tensor theory were constructed, it
could provide a natural framework to achieve a Dark 1
Matter-Dark Energy unification. For instance, in such DˆaA∗ = eam∂mA∗− 2√2ψ¯aα˙χ¯α˙
a scenario Dark Matter and Dark Energy could be iden-
i
tifiedwithdifferentcomponentsofthePlancksuperfield. Dˆaχ¯α˙ = eamDmχ¯α˙ − 2√2σ¯bα˙δψaδDˆbA∗
This issue is at present under analysis.
1
√2ψ¯α˙F∗. (A.4)
− 2 a
APPENDIX
We listhere someusefulθ expansionsthat weusedfor
ACKNOWLEDGMENTS
deriving the Lagrangians of section V. Let us start with
the chiral density . Its component fields expansion is
E
given by [8] I sincerely thank Massimo Pietroni for many useful
2 =e 1+iθσaψ¯ θθ(M∗+ψ¯ σ¯abψ¯ ) . (A.1) suggestionsanddiscussionsonScalar-Tensortheoriesand
a a b
E − their possible supersymmetric extensions. I would also
Thecurvaturesuperfieldhasthefollowingpowerseries
(cid:2) (cid:3) liketothankWilfriedBuchmuellerforinterestingdiscus-
expansion [8]
sions on the topic and Massimo Pietroni and Gonzalo
1 Palma for having read and commented on a draft of the
R = M +θ σaσ¯bψ iσaψ¯ M +iψ ba
−6( ab− a a ! paper. I finally acknowledges a Research Grant funded
by the VIPAC Institute.
1 2 1
+ θθ +iψ¯aσ¯bψ + MM∗+ bab
ab a
"− 2R 3 3
6
1 Hereby“inconsistent”wemeanthattheweakequivalenceprin-
instance[4]andreferencestherein.
ciple is not satisfied by construction. For any other possible
inconsistency or constraint that apply to ST theories, see for
[1] D. N.Spergel et al.,[arXiv:astro-ph/0603449]. [arXiv:astro-ph/0604492].
[2] W. de Boer, “Is dark matter supersymmetric?,” Prepared [6] N. Bartolo and M. Pietroni, Phys. Rev. D 61
forCargeseSchoolofParticlePhysicsandCosmology: the (2000) 023518 [arXiv:hep-ph/9908521]; G. Esposito-
Interface, Cargese, Cosica, France, 4-16 Aug 2003. Farese and D. Polarski, Phys. Rev. D 63, 063504
[3] E. J. Copeland, M. Sami and S. Tsujikawa, (2001) [arXiv:gr-qc/0009034]; R. Catena, N. Fornengo,
[arXiv:hep-th/0603057]. A. Masiero, M. Pietroni and F. Rosati, Phys. Rev. D 70,
[4] T. Damour, “Gravitation, experiment and cosmology,” 063519 (2004) [arXiv:astro-ph/0403614].
[arXiv:gr-qc/9606079]. [7] C. Brans and R. H.Dicke, Phys.Rev. 124 (1961) 925.
[5] P.Jordan,SchwerkaftundWeltall(Vieweg,Braunschweig, [8] JuliusWessandJonathanBagger,SupersymmetryandSu-
1955); M. Fierz, Helv. Phys. Acta 29, 128 (1956); pergravity, Princeton UniversityPress.
T. Damour and K. Nordtvedt, Phys. Rev. D48, 3436 [9] B. Bertotti, L. Iess and P. Tortora, Nature 425 (2003)
(1993);T.DamourandA.M.Polyakov,Nucl.Phys.B423, 374.
532 (1994); R. Catena, M. Pietroni and L. Scarabello,