Extended $f(R,L_m)$ gravity with generalized scalar field and kinetic term dependences

0.13 MB·English

by  Tiberiu Harko

Extended f (R,Lm) gravity with generalized scalar field and kinetic term dependences
Tiberiu Harko1,∗ Francisco S.N. Lobo2,† and Olivier Minazzoli3‡
1Department of Physics and Center for Theoretical and Computational Physics,
The University of Hong Kong, Pok Fu Lam Road, Hong Kong
2Centro de Astronomia e Astrof´ısica da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal and
3Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA
(Dated: October26, 2012)
We generalize previous work by considering a novel gravitational model with an action given by
anarbitrary functionof theRicciscalar, thematterLagrangian density,ascalar fieldand akinetic
termconstructedfromthegradientsofthescalarfield,respectively. Thegravitationalfieldequations
in the metric formalism are obtained, as well as the equations of motion for test particles, which
2
followfromthecovariantdivergenceofthestress-energytensor. Specificmodelswithanonminimal
1
coupling between the scalar field and the matter Lagrangian are further explored. We emphasize
0
that these models are extremely useful for describing an interaction between dark energy and dark
2
matter, and for explaining thelate-time cosmic acceleration.
t
c
O PACSnumbers: 04.50.Kd,04.20.Cv,04.20.Fy

5
2 I. INTRODUCTION energy corresponds to a new scalar particle Φ.

After the Brans-Dicke proposal,other forms of scalar-
]
c Scalar fields play a fundamental role in the cosmologi- tensor theoretical models were investigated. In particu-
q caldescriptionofourUniverse[1]. Oneofthefirstmajor lar, one can extend the coupling of the scalar field and
- extensions of general relativity, proposed by Dicke and curvaturetoanonminimalcoupling,providedbyF(φ)R,
r
g Brans [2–4], conjectured that “Mach’s principle” might where, for example, F(φ)=1 ξφ2, and ξ is a coupling
[ leadtoadependenceofthelocalNewtoniangravitational constant[9]. Inthesetypesofm−odelsnon-minimallycou-

constant, G, since in most cosmologicalmodels the total pledHiggsfieldwithalargecouplingξ mightgiveriseto
2
v mass M and radius of curvature of the Universe a(t) are asuccessfulinflation[9],whichis otherwise verydifficult
8 related by an equation of the form G−1 M/a(t)c2. tobeachieved[10]. OneofthepositivefeaturesofBrans-
∼
1 Consequently, in the variational principle of general rel- Dicke-like theories is that they seem to be generically
2 ativity [2–4], it was proposed to substitute G−1 with a driven toward general relativity during the cosmological
4
scalar field φ, and to also add to the action the kinetic evolution of the Universe [11].
.
0 energy corresponding to φ. Therefore, the variational
Couplings between the scalar and the matter fields
1 principle ofthe Brans-Dicketheorycanbe formulatedas
havebeeninvestigatedaswell[12–20]. Indeed, suchcou-
2 δ φR+L ω φ λφ/φ √ gd4x = 0, where R is
m λ plings generically appear in Kaluza-Klein theories with
1 − ∇ ∇ −
: thRe(cid:0)scalarcurvature,Lm isthe(cid:1)matterLagrangian,andω compactified dimensions [21] or in the low energy effec-
v is a coupling parameter. The scalar-tensor gravitational
tive limit of string theories [12, 13, 15]. In the latter
i models have been intensively investigated, and can be
X context, some authors proposed that the dilaton could
considered as a valid approach in explaining the recent
be a good candidate for the quintessence [12] or the in-
r
a accelerated expansion of the Universe, inferred from the flaton [13]. But even from a phenomelogical point of
Type Ia supernova observations [5]. According to stan-
view,ithasbeenarguedthatspecific restrictionssuchas
dardgeneralrelativity,the observedlate-timecosmicac-
gauge and diffeomorphism invariances essentially single
celeration can be successfully explained by introducing
out a particular set of effective theories which turns out
either a fundamental cosmological constant [6] – which
to be Brans-Dicke-like theories with scalar/matter cou-
wouldrepresentanintrinsic curvature of space-time – or
pling [14]. A good feature of such theories is that under
a dark energy – which would mimic a cosmological con-
various assumptions – and similarly to Brans-Dicke-like
stant (at least during the late stage of the cosmological
theories without scalar/matter coupling – they seem to
evolution), as the concordance of observations are still
be driven toward a weak coupling through cosmological
in favor of the λ-CDM standard model [7], where λ is a
evolution[15]. Therefore,theyseemtobeabletoexplain
constant. One of the currently main dark energy scenar-
thecurrenttightconstraintsontheequivalenceprinciple
ios is based on the so-calledquintessence [8], where dark
without fine-tuning parameters [16].
Morerecently,ithasbeenarguedthatascalar/matter
couplingcouldbe responsibleforadependency ofthe ef-
fectivemassofthescalar-fielduponthelocalmatterden-
∗Electronicaddress: [e-mail]
†Electronicaddress: fl[e-mail] sity [22]. Such a scalar-fieldhas been dubbed chameleon
‡Electronicaddress: [e-mail] as“inregionsofhighdensity,the[scalar-field]chameleon
2
blends with its environment and becomes essentially in- Byassumingthatthe matter LagrangiandensityL de-
m
visible to searches for EP violation and fifth force” [23]. pendsonlyonthemetrictensorcomponentsg ,andnot
µν
The possibility of a nonminimal coupling between on its derivatives, we obtain the stress-energy tensor as
matter and geometry, somehow similar to the coupling
betweenthescalarfieldandcurvatureintheBrans-Dicke δLm
τ =g L 2 . (3)
theory, with the matter Lagrangian playing the role of µν µν m− δgµν
φ, was considered in [24, 25], in the framework of the
so-called f(R) models of gravity [26–28], and further Furthermore, we assume that the scalar field φ is inde-
extended in [29]. An extension of the f(R) gravity pendentofthemetric,i.e., δφ/δgµν 0. Inthefollowing
≡
model, called f(R,φ) gravity was also proposed in [30], we will denote, for simplicity, ( φ)2 =gµν µφ νφ.
∇ ∇ ∇
andfurtherdiscussedandgeneralizedin[31]. Theaction By varying the action S of the gravitationalfield with
consideredfor the f(R,φ) gravitymodel is givenby S = respect to the metric tensor components gµν, we obtain
Rwh[fe(rRe ,βφ)i/s2a−cωo(nφs)t∇anµtφ.∇µInφ−thVis(φco)n+teβxLt,mt]h√e−mgadx4xim/βal, the field equations of the fhR,Lm,φ,(∇φ)2i gravita-
tional model as
extension of the standard Hilbert-Einstein action of
general relativity, S = (R/16πG+Lm)√−gd4x, was f R + g λ f 1(f f L )g
considered in [32], wherRe the f(R) type gravity models R µν µν∇λ∇ −∇µ∇ν R− 2 − Lm m µν
were maximally generalized by assuming that the gravi- (cid:0) 1(cid:1)
tational Lagrangian is given by an arbitrary function of = 2fLmτµν −f(∇φ)2∇µφ∇νφ,(4)
the Ricci scalar R and of the matter Lagrangian L , so
m
that S = f(R,L )√ gd4x. wherethesubscriptoff denotesapartialderivativewith
m
−
It is thRe purpose of the present paper to consider a respect to the arguments, i.e., fR = ∂f/∂R, fLm =
generalized f(R,Lm) type gravity model, in which, be- ∂f/∂Lm, f(∇φ)2 =∂f/∂(∇φ)2.
yond the ordinary matter, described by its thermody- Nowvaryingtheactionwithrespecttoφ,providesthe
namic energy density ρ and pressure p, the Universe is following evolution equation for the scalar field
filled with a scalar field φ. Consequently, in the gravita-
tioµnφalaµcφtiaornetahlesoscparleasrenfite,lda,nadstwheellLaasgirtasnkgiinaentitcaeknesertghye, (cid:3)(∇φ)2φ= 21fφ, (5)
f∇orm∇L = f(R,L ,φ, φ µφ), where f is an ar-
grav m µ
∇ ∇ where f =∂f/∂φ and
bitrary function. Therefore, in these models an explicit φ
nonminimalcouplingbetweenmatterandthescalarfield
1 ∂ ∂
igsraavlsiotaatliloonwaeldfi.elAdseaqufiartsitosntsepinitnhoeumresttruidcyfowrmeaolbistamin,atnhde (cid:3)(∇φ)2 = √ g∂xµ (cid:20)f(∇φ)2√−ggµν∂xν(cid:21), (6)
−
the equations of motion for test particles, which follow
is the generalized D’Alembert operator of
fromthecovariantdivergenceofthestress-energytensor.
f(R,L ,φ, φ µφ) gravity.
Somespecificmodelswithnonminimalscalarfield-matter m µ
∇ ∇
coupling are further explored. The contractionof Eq. (4) providesthe following rela-
tion between the Ricci scalar R, the matter Lagrangian
density L , the derivatives of the scalar field, and the
m
II. GRAVITATIONAL FIELD EQUATIONS trace τ =τµµ of the “reduced” stress-energy tensor,
f R+3 µf 2(f f L )
We assumethatthe actionforthe modifiedtheoriesof R ∇µ∇ R− − Lm m
1
gravity, considered in this work, is given by = 2fLmτ −f(∇φ)2∇µφ∇µφ. (7)
S = f(R,L ,φ,gµν φ φ)√ g d4x, (1)
Z m ∇µ ∇ν − By taking the covariant divergence of Eq. (4), we ob-
tainfor the covariantdivergenceofthe “reduced”stress-
where √ g is the determinant of the metric tensor gµν, energy tensor the following expression
and f(R−,L ,φ,gµν φ φ) is an arbitraryfunction of
m µ ν
the Ricci scalar R, ∇the m∇atter Lagrangian density, Lm, 1 σ(f τ ) = 1(L f f Φ) (8)
a scalar field φ, and the gradients constructed from the 2∇ Lm µσ 2 m∇µ Lm − Φ∇µ
scalarfield,respectively. Theonlyrestrictiononthefunc- + f(∇φ)2∇µΦ∇σ∇σΦ+∇µΦ∇σΦ∇σf(∇φ)2.
tion f is to be an analytical function of R, L , φ, and
m
of the scalar field kinetic energy, respectively, that is, f This relationship was deduced by taking into account
must possess a Taylor series expansion about any point. the following mathematical identities
We define the “reduced” stress-energy tensor as [33]
1
µR = R, (9)
2 δ(√ gLm) ∇ µν 2∇ν
τµν =−√ g δ−gµν . (2) ( ν(cid:3) (cid:3) ν)fR = ( µfR)Rµν, (10)
− ∇ − ∇ − ∇
3
and used the fact that we consider torsion-free space- where uµ is the four-velocity of the matter fluid. The
times such that: scalar field satisfies the evolution equation
1 dF(φ)
[ σ ǫ ǫ σ]ψ =0, (11) (cid:3)φ= ρ, (19)
∇ ∇ −∇ ∇
−2λ dφ
where ψ is any scalar-field. Now, using Eq. (6) we get
where (cid:3) is the usual d’Alembert operator defined in a
σ(f τ )=L f . (12) curvedspace. Thetotalstress-energytensorofthescalar
∇ Lm µσ m∇µ Lm field-matter system is given by
Forφ 0,Eqs. (4)reducetothefieldequationsofthe
f(R,L ≡)modelconsideredin[32]. ForΦ=0,onerecov- T =F(φ)ρu u +2λ φ φ 1g φ αφ .
m µν µ ν µ ν µν α
6 (cid:20)∇ ∇ − 2 ∇ ∇ (cid:21)
ersthegoodconservationequationsforeithergeneralrel-
(20)
ativity and Brans-Dicke-like scalar-tensor theories (with
Through the Bianchi identities, the covariant diver-
and without scalar/matter coupling [19]). For instance,
gence of Tµν must be zero, that is, Tµν = 0. In the
the total Lagrangian of the simplest matter-scalar field- ∇ν
following, we assume that the mass density current is
gravitational field theory, with scalar field kinetic term
conserved,i.e., (ρuν)=0. Using the latter condition,
and a self-interacting potential V(φ) corresponds to the ∇ν
and the mathematical identity (11), we obtain first
choice
dF(φ)
f = R +Lm+ λgµν µφ νφ+V(φ), (13) F(φ)ρ(uν∇νuµ)+ρuµuν dφ ∇νφ+2λ(∇µφ)(cid:3)φ=0.
2 2 ∇ ∇
(21)
whereλis aconstant. The correspondingfieldequations By eliminating the term (cid:3)φ with the help of Eq. (19),
can be immediately obtained from Eqs. (4) as we obtain
d
1 uν uµ+ lnF(φ) (uµuν φ µφ)=0. (22)
R Rg = τ λ φ φ ν ν
µν µν µν µ ν ∇ (cid:20)dφ (cid:21) ∇ −∇
− 2 − ∇ ∇
+(cid:20)λ2gαβ∇αφ∇βφ+V(φ)(cid:21)gµν. (14) ΓµUsianrgetthheeidCehnrtiisttyoffueνl∇sνyumµb≡olsdd2scx2oµr+resΓpµαoβnudαinugβ,twohtehree
αβ
metric, the equation of motion of the test particles non-
The scalar field satisfies the evolution equation
minimally coupled to an arbitrary scalar field takes the
1 ∂ ∂φ 1dV(φ) form
√ ggµν = , (15)
√ g∂xµ (cid:20) − ∂xν(cid:21) λ dφ d2xµ d
− +Γµ uαuβ+ lnF(φ) (uµuν φ µφ)=0.
ds2 αβ (cid:20)dφ (cid:21) ∇ν −∇
while the stress-energy tensor satisfies the conservation
(23)
equation
A particular model can be obtained by assuming that
στ =0. (16) F(φ) is given by a linear function,
µσ
∇
Λ+1 1
F(φ)= 1+ (Λ 1)φ , (24)
2 (cid:20) 2 − (cid:21)
III. MODELS WITH NONMINIMAL
MATTER-SCALAR FIELD COUPLING
where Λ is a constant. Then the equation of motion
becomes
As an example of the application of the formalism de-
d2xµ Λ 1
veloped in the previous section, we consider a simple +Γµ uαuβ+(uµuν gµν) ln 1+ − φ =0.
ds2 αβ − ∇ν (cid:20) 2 (cid:21)
phenomenological model, in which a scalar field is non-
(25)
minimallycoupledtopressure-lessmatterwithrest mass
In order to simplify the field equations we adopt for λ
density ρ. For the action of the system, consider
thevalueλ= Λ2 1 /8. ThenEq.(19),determining
R the scalar field−, t(cid:0)akes−th(cid:1)e simple form (cid:3)φ=ρ.
S = F(φ)ρ+λgµν φ φ √ gd4x, (17)
µ ν The gravitational field equations take the form
Z (cid:20)2 − ∇ ∇ (cid:21) −
1 Λ+1
where F(φ) is an arbitrary function of the scalar field Rµν gµνR= Tµν, (26)
− 2 2
thatcouplesnon-minimallytoordinarymatter. Thefield
equations for this model are given by with the total stress-energy tensor given by
1 Λ 1
Rµν − 2Rgµν = F(φ)ρuµuν Tµν = (cid:20)1+ −2 φ(cid:21)ρuµuν −
1 Λ 1 1
+2λ φ φ g φ αφ , (18) − φ φ g φ αφ . (27)
µ ν µν α µ ν µν α
(cid:20)∇ ∇ − 2 ∇ ∇ (cid:21) 2 (cid:20)∇ ∇ − 2 ∇ ∇ (cid:21)
4
For Λ = 1 we reobtain the general relativistic model for been computed. Specific models with a nonminimal in-
dust. Other possible choices of the function F(φ), such teraction between the scalar field and ordinary matter
as F(φ)=exp(φ), can be discussed in a similar way. were explored. These models can be extremely useful in
Amoregeneralmodelcanbeobtainedbyadoptingfor describing the interaction between dark energy, modeled
the matter Lagrangianthe general expression [34–36] as a scalar field, and dark matter, with or without pres-
sure, respectively. Moreover,they can provide a realistic
dp(ρ) description of the late expansion of the Universe, where
L = ρ+ρ p(ρ) , (28)
m
−(cid:20) Z ρ − (cid:21) a possible interaction between ordinarymatter and dark
energy cannot be excluded a priori 1. Work along these
where ρ is the rest-mass energy density, p is the ther- lines is presently underway and the cosmological conse-
modynamic pressure, which, by assumption, satisfies a quences of the presenttheory will alsobe investigatedin
barotropicequationofstate,p=p(ρ). Byassumingthat detail in a future publication.
the matter Lagrangian does not depend on the deriva-
tives of the metric, and that the particle matter fluid
currentisconserved[ ν(ρuν)=0],theLagrangiangiven Acknowledgments
∇
by Eq.(28) is the unique matter Lagrangianthat canbe
constructed from the thermodynamic parameters of the
FSNL acknowledgesfinancialsupportofthe Funda¸ca˜o
fluid [36].
para a Ciˆencia e Tecnologia through the grants
The gravitationalfield equations and the equation de-
CERN/FP/123615/2011 and CERN/FP/123618/2011.
scribing the matter-scalar field coupling are given by
OM was supported by an appointment to the NASA
PostdoctoralProgramat the JetPropulsionLaboratory,
1
Rµν gµνR=F(φ) ǫ uµuν pgµν +λQµν, (29) California Institute of Technology, administered by Oak
− 2 −
Ridge Associated Universities through a contract with
NASA.
1 dF(φ)
(cid:3)φ= ǫ, (30)
2λ dφ
where Q = φ φ 1 φ λφg , and where the
µν ∇µ ∇ν − 2∇λ ∇ µν
totalenergydensityisǫ=ρ+ρ dp/ρ p[34,36]. With
the use of the conservation equRation −ν(ρuν) = 0, one
∇
obtains the equation of motion of massive test particles
as
d2xµ dp
+Γµ uαuβ+(uµuν gµν) ln 1+ =0,
ds2 αβ − ∇ν (cid:20) Z ρ (cid:21)
(31)
The equation of motion (31) can also be derived from
the variational principle δ 1+ dp/ρ√gµνuµuνds =
q
0. Models with scalar fieRld-matteRr coupling were con-
sidered in the framework of the Brans-Dicke theory
[18], with the action of the model given by S =
[φR/2+(ω/φ) φ µφ+F(φ)L ]√ gd4x. Such
µ m
∇ ∇ −
Rmodels cangiverise to a late time acceleratedexpansion
of the Universe for very high values of the Brans-Dicke
parameter ω. Other models with interacting scalar field
and matter have been considered in [17]. We emphasize
the the gravitationaltheory consideredin this workgen-
eralizes all of the above models.
IV. CONCLUSION
In the present paper we have presented a novel gravi-
tational theory where the Lagrangian is given by an ar-
bitrary function of the Ricci scalar, matter Lagrangian,
a scalarfield and its kinetic term, respectively. The field
equations for this model were obtained for the general 1 Let us note that such an interaction may be a good alternative
case, and the divergence of the stress-energy tensor has todarkmatter [20].
5
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