aa r X i v : . [ a s t r o - ph ] A ug Comments on Scalar-TensorRepresentation of f ( R ) Theories
Yousef Bisabr ∗ Department of Physics, Shahid Rajaee Teacher Training University, Lavizan, Tehran 16788, Iran.
Abstract
We propose a scalar-tensor representation of f ( R ) theories with use of conformaltransformations. In this representation, the model takes the form of the Brans-Dickemodel with a potential function and a non-zero kinetic term for the scalar field. In thiscase, the scalar field may interact with matter systems and the corresponding matterstress tensor may be non-conserved. PACS Numbers: 98.80.-k
Recent observations on expansion history of the universe indicate that the universe is expe-riencing a phase of accelerated expansion [1]. This can be interpreted as evidence either forexistence of some exotic matter components or for modification of the gravitational theory. Inthe first route of interpretation one can take a perfect fluid with a sufficiently negative pres-sure, dubbed dark energy [2], to produce the observed acceleration. There is also a large classof scalar field models in the literature including, quintessence [3], phantom [4] and quintomfields [5] and so forth. In the second route, however, one attributes the accelerating expansionto a modification of general relativity. A particular class of models that has recently drawna significant amount of attention is the so-called f ( R ) gravity models [6][7]. These modelspropose a modification of Einstein-Hilbert action so that the scalar curvature is replaced bysome arbitrary function f ( R ). It is well known that f ( R ) theories of gravity can be writtenas a scalar-tensor theory by applying a Legendre transformation [8][9]. This scalar-tensor rep-resentation corresponds to a class of Brans-Dicke theory with a potential function and ω = 0 ∗ e-mail: [email protected].
1n the metric formalism. There is also such a correspondence for ω = − in the Palatiniformalism in which metric and connections are taken as independent variables, see [10] andreferences therein. Here we do not consider Palatini formalism.Although f ( R ) gravity theories exhibit a natural mechanism for accelerated expansion withoutrecourse to some exotic matter components, because of vanishing of the kinetic term of thescalar field in the scalar-tensor representation there are criticisms that emphasized inability ofthese models to pass solar system tests [11]. In the present note we will focus on the dynamicalequivalence of f ( R ) theories and the Brans-Dicke theory with use of conformal transforma-tions. We will show that this equivalence holds for an arbitrary Brans-Dicke parameter. Inthis case, however, the gravitational coupling of matter systems may be anomalous in the sensethat the scalar field inter the matter field action. To begin with, we offer a short review on the equivalence of f ( R ) theories with a particularclass of Brans-Dicke theory with a potential. We consider the following action † S = 12 Z d x √− gf ( R ) + S m ( g µν , ψ ) (1)where f ( R ) is an arbitrary function of the scalar curvature R . The matter action S m ( g µν , ψ )is S m ( g µν , ψ ) = Z d x √− g L m ( g µν , ψ ) (2)in which the Lagrangian density L m corresponds to matter fields which are collectively denotedby ψ . One usually introduces a new field χ = R by which the action (1) can then be writtenas S = 12 Z d x √− g { f ( χ ) + f ′ ( χ )( R − χ ) } + S m ( g µν , ψ ) (3)where prime denotes differentiation with respect to R . Now variation with respect to χ leadsto the equation f ′′ ( R )( χ − R ) = 0 (4)If f ′′ ( R ) = 0, we have the result χ = R . Inserting this result into (3) reproduces the action(1). Then redefining the field χ by Φ = f ′ ( R ) and setting V (Φ) = χ (Φ)Φ − f ( χ (Φ)) (5)the action (3) takes the form S = 12 Z d x √− g { Φ R − V (Φ) } + S m ( g µν , ψ ) (6)This is the Brans-Dicke action with a potential V (Φ) and a Brans-Dicke parameter ω = 0.Therefore there is a dynamical equivalence between f ( R ) theories and a class of Brans-Dicke † We work in the unit system in which ¯ h = c = 8 πG = 1. f ( R ) theoriesthe matter action S m ( g µν , ψ ) is independent of the scalar field Φ. Thus in both actions (1) and(6) the weak equivalence principle holds and test particles follow geodesics lines of g µν .In original form of the Brans-Dicke theory [12], where the potential term of the scalar field isnot present, it is found that in order to get agreement between predictions and solar systemexperiments ω should be large and positive. The current observations set a lower bound on ω which is ω > f ( R ) theories is ill-defined in the scale of solar system. The underlying logic isbased on the fact that there is no observed deviation from general relativity at this scale and f ( R ) theories must be reduced to general relativity in an appropriate limit. The problem isthat general relativity corresponds to f ( R ) = R for which f ′′ ( R ) = 0, while the dynamicalequivalence requires f ′′ ( R ) = 0.We intend here to use a different scalar-tensor representation of f ( R ) theories suggested byconformal transformations. To this aim, we apply the following conformal transformation˜ g µν = Ω g µν , Ω = f ′ ( R ) (7)to the action (1). This together with a redefinition of the conformal factor in terms of a scalarfield φ = q ln Ω , yields [9] S = 12 Z d x q − ˜ g { ˜ R − ˜ g µν ∇ µ φ ∇ ν φ − V ( φ ) } + S m (˜ g µν , ψ, φ ) (8)where S m (˜ g µν , ψ, φ ) = Z d x q − ˜ g e − √ φ L m (˜ g µν , ψ ) (9)In the action (8), φ is minimally coupled to ˜ g µν and appear as a massive self-interacting scalarfield with a potential V ( φ ) = 12 e − √ φ r [Ω ( φ )] − e − √ φ f ( r [Ω ( φ )]) (10)where the function r (Ω ) is the solution of the equation f ′ [ r (Ω )] − Ω = 0. Thus the variables(˜ g µν , φ ) in the action (8) provide the Einstein frame variables for f ( R ) theories.Variation of (8) with respect to ˜ g µν and φ give, respectively,˜ G µν = t µν + e − √ φ T µν (˜ g µν , ψ ) (11)˜ ✷ φ − dV ( φ ) dφ = s e − √ φ T (12)where T µν = − √− g δS m δg µν (13)3 µν = ∂ µ φ∂ ν φ −
12 ˜ g µν ∂ γ φ∂ γ φ − V ( φ )˜ g µν (14)and T ≡ g µν T µν . Now the Bianchi identity implies˜ ∇ µ T µν = s ∂ µ φ T µν − s e − √ φ ∂ ν φ T (15)There is also a scalar-tensor representation by applying the conformal transformation˜ g µν = Ω ¯ g µν , Ω = e φα (16)to the action (8) with α being a constant parameter. This together with φ = α ln ϕ transformthe action (8) to S = 12 Z d x √− ¯ g { ϕ ¯ R − ωϕ ¯ g µν ∇ µ ϕ ∇ ν ϕ − U ( ϕ ) } + S ′ m (¯ g µν , ψ, ϕ ) (17)where S ′ m (¯ g µν , ψ, ϕ ) = Z d x √− ¯ g ϕ n L m (¯ g µν , ψ, ϕ ) (18)and ω = α −
92 (19) n = 2 − α s
23 (20) U ( ϕ ) = 2 ϕ V ( ϕ ) (21)This is the scalar-tensor representation of the action (1) obtaining by conformal transforma-tions (7) and (16). In contrast with this representation, the gravitational part of the action (1)consists only of the metric tensor g µν which obeys fourth-order field equations. We may callthe conformal frames corresponding to the actions (1) and (17) the Jordan frame representa-tion of (8) ‡ . Here a question which arises is that which of the conformal frames correspondingto the actions (1), (8) and (17) should be considered as the physical frame. It should bepointed out that reformulation of a theory in a new conformal frame leads, in general, to adifferent physically inequivalent theory. The ambiguity of the choice of a particular frame asthe physical one is a longstanding problem in the context of conformal transformations. Theterm “physical” theory denotes one that is theoretically consistent and predicts values of someobservables that can, at least in principle, be measured in experiments [15]. In this respectdifferent authors may consider different conformal frames as physical according to their at-titude towards the issue of the conformal frames § . For instance, while in f ( R )-theories one ‡ Note that we define here Jordan frame in terms of how the geometry is described in the vacuum sectorrather than in terms of how it couples with matter systems [9]. The action (1) is in Jordan frame since theresulting field equations are fourth-order in terms of metric tensor. On the other hand, the action (17) is alsoin Jordan frame since it describes the geometry by a metric tensor and a scalar field (nonminimal coupling ofthe scalar field). § For a good review on this issue, see [15] and references therein. f ( R ) theories, namely the actions(6) and (17). There are two important differences: Firstly, in the action (17) the Brans-Dickescalar field ϕ has a non-zero kinetic term and the Brans-Dicke parameter ω is only constrainedby observations. Secondly, the scalar field ϕ enter the matter part of the action (17). Thelatter means that the scalar field interacts with matter systems and tests particles do not followthe geodesic lines of the metric ¯ g µν . We will return to this issue later.Variation with respect to ¯ g µν and ϕ leads to the field equations ϕ ¯ G µν − ωϕ ( ∇ µ ϕ ∇ ν ϕ −
12 ¯ g µν ∇ γ ϕ ∇ γ ϕ ) − ( ¯ ∇ µ ∇ ν ϕ − ¯ g µν ¯ ✷ ϕ ) + 12 U ( ϕ )¯ g µν = ¯ T µν (22)2 ωϕ ¯ ✷ ϕ − ωϕ ∇ γ ϕ ∇ γ ϕ + ¯ R − dU ( ϕ ) dϕ = ϕ − ¯ T (23)where ¯ T µν = − √− ¯ g δS m δ ¯ g µν (24)and ¯ T = ¯ g µν ¯ T µν is the trace of the stress tensor ¯ T µν . Now applying the Bianchi identity¯ ∇ µ ¯ G µν = 0 and using the field equation of the scalar field (23), we obtain¯ ∇ µ ¯ T µν = − a ν (25)where a ν = 12 ¯ T ∂ ν ln ϕ (26)The equation (25) implies that the matter stress tensor ¯ T µν is not conserved due to interactionof the scalar field ϕ with the matter part of (17). Except for the case that the matter fieldaction (18) is traceless [9], the scalar field ϕ influences the motion of any gravitating matter.In fact, a ν indicates an anomalous acceleration corresponding to a fifth force.It should be noted that there are different types of models in literature that concern mattersystems that are not conserved due to interaction with an arbitrary function of scalar curva-ture [16] or some scalar fields [17]. However, the important difference between (25) and thecorresponding equations in those models is that the former is the result of the well-knownproperty of conformal transformations, namely that conservation equation of a matter stresstensor with a nonvanishing trace is not conformally invariant [18].It is possible to apply this result in the scale of solar system. To do this, we first note thatcombining (7) and (16) gives the relation between the scalar field ϕ and the function f ( R ) ϕ = [ f ′ ( R )] − n (27)Then we take ¯ T µν to be the stress tensor of dust (or perfect fluid with zero pressure) withenergy density ¯ ρ . In this case and for a static spacetime we obtain for the spatial part of a ν , a i = 12 − n ¯ ρ ∂ i R f ′′ ( R ) f ′ ( R ) (28)5s one expects, in the case that f ( R ) is a linear function of R like the Eistein-Hilbert action,the anomalous acceleration is zero. For CDTT model [6] in which f ( R ) = R − µ R we obtain a i = 2 n − ρ ∂ i ln R (1 + R µ ) − (29)where µ is an arbitrary mass scale. While the scalar-tensor representation (6) of f ( R ) theories is useful in cosmological scales itsuffers problems in weak field limit and solar system scales. Firstly, the kinetic term of thescalar field vanishes which is in conflict with current bounds on the value of ω . Secondly, itis recently reported that since f ′′ ( R ) = 0 in this scales, the dynamical equivalence of f ( R )theories and scalar-tensor theories represented by (6) breaks down. The main feature of ouranalysis is to show that there is also a dynamical equivalence between f ( R ) theories andscalar-tensor theories with use of conformal transformations. In this representation the scalarfield has a non-vanishing kinetic term and a non-zero Brans-Dicke parameter. Therefore, theobservational constraints can be applied on ω in this representation. However, it should benoted that the action (17) differs from the Brans-Dicke action in two ways. Firstly, it containsa potential function U ( ϕ ). Secondly, the matter system interacts with the scalar field ϕ . Thatthe scalar field possess a potential function clearly alters the usual bound on the Brans-Dickeparameter ω so that the new bound depends on the functional form of the potential [19]. On theother hand, the coupling of the scalar field with matter sector should be strongly suppressedso as not to lead to observable effects. In fact, one can use chameleon mechanism [20] toimplement constraints on the potential function U ( ϕ ). Then using a relation between ϕ andthe curvature scalar (see the relation (27)), this can provide some viable forms of f ( R ) theorieswhich are in accord with local gravity tests. Indeed, this is the method that is recently used bysome authors to deal with f ( R ) theories which are consistent with Solar System experiments[21].It is important to note that non-conservation of the stress tensor ¯ T µν should not be consideredas an intrinsic behavior of the model presented here. It is simply related to the fact that beforeapplying the conformal transformations, the matter action is introduced in the nonlinear action(1). The matter action might be added after the conformal transformations in the Jordanframe. In that case there would be no anomalous acceleration. This ambiguity of introducingmatter systems to equivalent conformal frames is closely related to the well-known problemthat which of these frames should be taken as the physical one [9][15]. Without dealingwith this long-standing problem, we would like to point out that the advantage of the non-minimal coupling of matter in (17) is that it can potentially explain the possible deviations of f ( R ) theories (or its scalar-tensor representation) from newtonian gravity in local experiments.Moreover, as earlier stated, the scalar field may also be interpreted as a chameleon field whichcan suppress detectable effects of anomalous gravitational coupling of matter in solar systemscales. This chameleon behavior of the scalar field is under progress by the author and isdeserved to be investigated elsewhere. 6 eferences [1] A. G. Riess et al., Astron. J. , 1009 (1998)S. Perlmutter et al., Bull. Am. Astron. Soc., , 1351, (1997)S. 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