The generalized Brans-Dicke theory and its cosmology
aa r X i v : . [ g r- q c ] J u l The generalized Brans-Dicke theory and its cosmology
Jianbo Lu, ∗ Yabo Wu, Weiqiang Yang, Molin Liu, and Xin Zhao Department of Physics, Liaoning Normal University, Dalian 116029, P. R. China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang, 464000, P. R. China
A generalized Brans-Dicke (GBD) theory is studied in this paper. The GBD theory is obtained bygeneralizing the Ricci scalar R to an arbitrary function f ( R ) in the original Brans-Dicke (BD) action.The interesting property has been found in the GBD theory, for example it can naturally solve theproblem of γ value emerging in f ( R ) modified gravity (i.e. the inconsistent problem between theobservational γ value and the theoretical γ value), without introducing the so-called chameleonmechanism. In this paper, we derive the cosmological equations and study the cosmology in theGBD theory. The cosmological solutions show that the GBD model can pass through the test ofthe observations, such as the observational Hubble data. Comparing with other theories, it can befound that the GBD theory have some other interesting properties or solve some problems existingin other theories. (1) It is well known that the f ( R ) theory are equivalent to the BD theory with apotential (abbreviated as BDV) for taking a specific value of the BD parameter ω = 0, where thespecific choice: ω = 0 for the BD parameter is quite exceptional, and it is hard to understand thecorresponding absence of the kinetic term for the field. However, for the GBD theory, it is similarto the double scalar-fields model, and both fields in the GBD own the non-disappeared dynamicaleffect. (2) One knows that in the double scalar-fields quintom model, it is required to include boththe canonical quintessence field and the non-canonical phantom field in order to make the stateparameter to cross over w = −
1, while several fundamental problems are associated with phantomfield, such as the problem of negative kinetic term and the fine-tuning problem, etc. While, in theGBD model, the state parameter of geometrical dark energy can cross over the phantom boundary w = − V ( φ ) of theBD scalar field, and an effective form of V ( φ ) could be given by studying on the GBD theory. And,it seems that a viable condition for the BD theory could be found, i.e. the BD parameter shouldbe ω > f >
0, if we assume that the effective form of the BD potential can be approximatelywritten as a popular square function of φ . PACS numbers: 98.80.-kKeywords: Modified gravity; Brans-Dicke theory; Cosmological solution; Effective potential of field. ∗ Electronic address: [email protected]
I. Introduction
There are several observational and theoretical motivations to investigate the modified or alternative theories ofgeneral relativity. Studies on the modified gravity theories of GR have been always the hot area. Several modifiedgravity theories have been widely studied [1–6], especially two simple modifications to GR: the f ( R ) theory [7, 8] andthe Brans-Dicke (BD) theory [9].Recent observations in Refs. [10–14] indicate that the Newton gravitational constant G maybe depends on time.Brans-Dicke (BD) theory is a popular one to describe the time-variable G ( t ) gravity. As a simple theory in thescalar-tensor theories [15], BD theory is apparently compatible with Mach’s principle [16], and in which a scalar field φ can be introduced naturally by considering φ ( t ) ∝ /G ( t ). But in the original BD theory [9], it is hard to interpretthe cosmic acceleration indicated by the observations [17–19]. In order to obtain an accelerating universe, one usuallymodified this theory at three aspects: (1) introducing the invisible component—-dark energy in universe [20], (2)assuming the coupling constant ω to be variable with respect to time [21, 22], (3) adding a potential term to theoriginal BD theory (abbreviate as BDV) [23]. The applications of these extended BD theories have been investigatedwidely, such as at the aspects of cosmology [24–26], weak-field approximation [27], observational constraints [28, 29],and so on [30–32].In this paper, we investigate other way to explain the cosmic acceleration in the framework of the BD theory, i.e. wegeneralize the Ricci scalar R to be an arbitrary function f ( R ) in the original BD action (abbreviate as GBD), whichis different from the studies on equivalence between the BD theory and the modified f ( R ) theory [6]. The interestingproperty has been found in the GBD model. For example, by using the method of the weak-field approximation Ref.[33] shows that the GBD theory can naturally solve the problem of γ value emerging in f ( R ) modified gravity (i.e. theinconsistent problem between the observational γ value and the theoretical γ value), without introducing the so-calledchameleon mechanism. The chameleon mechanism is introduced to solve the problem of γ value in the f ( R ) modifiedgravity. Here γ is the parametrized post-Newtonian (PPN) parameter.The GBD cosmology is studied in this paper, and the structure of our paper is as follows. In section II, we brieflyintroduce the GBD theory, and derive to gain the field equations and the cosmological equations in the GBD theory.In section III, we give the cosmological solutions of the GBD model. It is shown that the GBD model can pass throughthe test of the observation, such as the observational Hubble data. In section IV and V, we investigate the propertiesof the geometrical dark energy and the effective potential of the BD scalar field in the GBD theory. By comparingwith the preceding studies (such as the studies on the f ( R ), the BDV, and the quintom models), some new ingredientsand significant progresses of this work could be shown as follows. (1) In the GBD theory one can take an arbitraryvalue of ω and the kinetic-energy term of scalar field in the action is non-disappeared, which is obviously different fromthe f ( R ) theory. The f ( R ) gravity theory becomes equivalent to the BDV theory for a specific value of ω = 0 under atransformation, where the kinetic-energy term for the scalar field is absent. (2) The GBD theory tends to investigatethe physics from the viewpoint of geometry, while the BDV tends to solve physical problems from the viewpoint ofmatter. Several special characteristics of scalar fields could be revealed through studies of geometrical gravity in theGBD, such as we can investigate to given an effective form of potential of the BD scalar field. (3) Comparing withthe two scalar-fields quintom model, the effective state parameter of geometrical dark energy in the GBD model cancross over the phantom boundary: w = − w = −
1, wherethe puzzling problems are emergent, such as the negative kinetic term and the fine-tuning problem. Section VI is theconclusion.
II. Field equations and cosmological equations in the GBD theory
In framework of the time-variable gravitational constant, we study a generalized Brans-Dicke theory by using afunction f ( R ) to replace the Ricci scalar R in the original BD action. The action of system is written as S = S g ( g µν , φ ) + S m ( g µν , ψ ) = 12 Z d x L T , (1)with the total Lagrange quantity L T L T = L g + L φ + L m = √− g [ φf ( R ) − ω φ ∂ µ φ∂ µ φ + 16 πc L m ] . (2)Obviously, the system contains three dynamical variable: the gravitational field g µν , the matter field ψ and the BDscalar field φ . ω is the couple constant. According to Eq. (2), it is easy to see that the GBD theory can be consideredas a special case of the more general f ( R, φ ) theory [34–36]. It is well known that the so-called f ( φ ) R theory [37, 38],as a special case of the f ( R, φ ) theory, has been widely studied [39–41]. Given that f ( R, φ ) is a more complex theoryand the more simple theory is usually more favored by the researcher in physics, here we investigate the GBD modelinduced by the directly observational motivation of the accelerating universe and some other motivations exhibitedin the introduction. Concretely, we discuss some interesting cosmological contents in the GBD model, such as thecomparison with observation, the properties of effective state parameter for the geometrical dark energy, the effectivepotential of the BD scalar field, etc.Taking c = 1 and varying the action with respect to metric δSδg µν = δS g ( g µν ,φ ) δg µν + δS m ( g µν ,ψ ) δg µν = 0, one can get thegravitational field equation φ (cid:20) f R R µν − f ( R ) g µν (cid:21) − ( ∇ µ ∇ ν − g µν (cid:3) )( φf R ) + 12 ωφ g µν ∂ σ φ∂ σ φ − ωφ ∂ µ φ∂ ν φ = 8 πT µν , (3)where f R ≡ ∂f /∂R , ∇ µ is the covariant derivative associated with the Levi-Civita connection of the metric, (cid:3) ≡∇ µ ∇ µ , and T µν = − √− g δS m δg µν is the energy momentum tensor of the matter. Varying the action (1) with respect to thescalar field φ and the matter field ψ give respectively f ( R ) + 2 ω (cid:3) φφ − ωφ ∂ µ φ∂ µ φ = 0 , (4) δSδψ = δS g ( g µν , φ ) δψ + δS m ( g µν , ψ ) δψ = 0 . (5)The trace of Eq. (3) is f R R − f ( R ) + 3 (cid:3) ( φf R ) φ + ωφ ∂ µ φ∂ µ φ = 8 πTφ . (6)From Eqs. (4) and (6), one can see that the curvature of the spacetime could be caused by the motion of φ . Andfrom Eq. (3), it is shown that the BD scalar field does not exert any direct influence on matter, while it coupleswith another scalar field f R . Furthermore, the standard f ( R ) modified gravity is recovered for φ =constant, whileabove equations reduce to the Einstein’s general relativity (GR) for both BD scalar field φ =constant and f ( R ) = R .Combining Eqs. (4) and (6), we get (cid:3) φ − ∂ µ φ∂ µ φ φ = 14 ω [8 πT − φRf R − (cid:3) ( φf R )] . (7)One can read from Eq. (7) that, for ω → ∞ the constant- G theory can be recovered, which is same to the result inthe standard BD theory.In the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric ds = − dt + a ( t ) d~x , (8)using Eqs. (3) and (4), we can derive the evolutional equations of the background universe in the GBD theory,3 f R H = 8 πρ m φ + f R R − f ( R )2 − H ˙ f R + 12 ω ˙ φφ ! − Hf R ˙ φφ , (9) − f R ˙ H = 8 πφ ( ρ m + p m ) + ¨ f R − H ˙ f R + ω ˙ φφ ! − Hf R ˙ φφ + f R ¨ φφ + 2 ˙ φφ ˙ f R , (10) f ( R ) − ω ˙ φφ ! + 2 ω ·· φφ + 6 ωH ˙ φφ = 0 . (11)Here a is the cosmic scale factor, H is the Hubble parameter, R = 6 (cid:16) H + · H (cid:17) , and ”dot” denotes the derivativewith respect to cosmic time t . For case of φ =constant ( ˙ φ = 0 and ¨ φ = 0) in Eqs.(9-11), they are reduced to the f ( R )theory, while for case of f ( R ) = R they are reduced to the original Brans-Dicke theory. III. Cosmological solutions in the GBD theory
For solving the cosmological equations (9-11), we define the dimensionless variables: y H = H /m − a − , (12) y R = R/m − a − , (13) y φ = φ/φ , (14) y ′ φ = φ ′ /φ , (15)Thus using Eqs. (9) and (11), we get the differential equations for { y H , y R , y φ , y ′ φ } as follows y ′ H = 13 y R − y H , (16) y ′ R = − [( y H + a − ) f R − f R ( y R + 3 a − ) + f m − ω ( y ′ φ y φ ) ( y H + a − ) + f R y ′ φ y φ ( y H + a − ) − a − φ ]( y H + a − ) m f RR + 9 a − , (17) y ′ φ = φ ′ /φ , (18) y ′′ φ = φ ω ( y H + a − ) [ − fm + ω ( y ′ φ y φ ) ( y H + a − ) − ω y ′ φ y φ ( 13 y R − y H − a − ) − ω y ′ φ y φ ( y H + a − )] . (19)Here the subscript ”0” denotes the current value of parameters, the superscript ′ denotes the derivative with respectto ln a , the parameter m is defined as m = (8315 M pc ) − ( Ω m h . ) and Ω m is the current dimensionless energy densityof the matter. To solve above differential equations, the initial conditions ( a = 1) are expressed respectively as y H | a =1 = H /m − , (20) y R | a =1 = 6 H (1 − q ) /m − , (21) y φ | a =1 = 1 , (22) y ′ φ | a =1 = 0 . . (23)Here q = − ¨ aaH is the deceleration parameter, and its current value q can be given by the cosmic observations.The value of the initial condition y ′ φ | a =1 can be indicated by the following observations. For example, the limits onthe variation of G can be exhibited by: | ˙ GG | = | ˙ φφ | ≤ . × − y − from Pulsating white dwarf G117-B15A [10], − × − y − ≤ ˙ φφ ≤ . × − y − from Nonradial pulsations of white dwarfs [11], | ˙ φφ | ≤ . × − y − fromMillisecond pulsar PSR J0437-4715 [12], | ˙ φφ | ≤ − y − from Type-Ia supernovae [13], ˙ φφ = (0 . ± . × − y − from Neutron star masses [14], | ˙ φφ | ≤ . × − y − from Helioseismology [42], and ˙ φφ = (4 ± × − y − fromLunar laser ranging experiment [43], etc. Taking a stringent bound | ˙ φφ | ≤ − y − and considering the currentvalue of the dimensionless Hubble constant h = 0 . ± .
010 from the Planck 2015 results [44], we can calculateto limit | y ′ φ ( a = 1) | ≤ .
015 by using the center value H = 67 . kms − M pc − = 6 . × − y − . Here we take y ′ φ ( a = 1) = 0 .
01 as an initial condition in Eq. (23). For comparison, the cases of other initial values of y ′ φ ( a = 1)(less than 0.01) are also discussed.To find a cosmological solution of the GBD theory, we need to take a concrete form of f ( R ) function at prior. Asan example, we consider an interesting model called exponential gravity f ( R ) = R − βR s (1 − e − R/R s ) , (24)which is proposed by Refs.[45–47]. Here β and R s are two constants with βR s ≃ H Ω m [47]. This model has animportant feature that it has only one more parameter than the ΛCDM model. The first and the second derivativesof Eq. (24) with respect to R are f R = 1 − βe − R/R s , (25) f RR = βR s e − R/R s . (26) −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna H ( a ) / H y ,φ (a=1)=0.01, β=1.0y ,φ (a=1)=0.01, β=1.5y ,φ (a=1)=0.01, β=2.0 - - - - - H €€€€€€€€€ H y Φ ' H a = L =- Φ ' H a = L = Φ ' H a = L = Β= Φ ' H a = L = FIG. 1: The numerical solutions of H ( a ) /H in the GBD model with the different model parameter β or the different initialcondition y ′ φ ( a ). −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna φ ( a ) / φ y ,φ (a=1)=0.01, β=1.0y ,φ (a=1)=0.01, β=1.5y ,φ (a=1)=0.01, β=2.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna φ ( a ) / φ β=1.5, y ,φ (a=1)= −0.01β=1.5, y ,φ (a=1)=0β=1.5, y ,φ (a=1)=0.01 FIG. 2: The numerical solutions of φ ( a ) /φ in the GBD model with the different model parameter β or the different initialcondition y ′ φ ( a ). Thus using the system of the ordinary differential equations (16)-(19) and the initial conditions (20)-(23), we cannumerically exhibit the solutions: H ( a ) and φ ( a ) in the GBD theory, which are illustrated in Fig.1 and Fig.2.In Fig.1 (left), we show the dependence of H ( a ) on the parameter β . Fig.1 (right) illustrates the evolution of H ( a )with respect to ln a with taking the different values of y ′ φ ( a = 1). In the following, we use a to denotes the currentvalue: a = 1. We can see that the evolutions of H ( a ) almost have the same trajectory for two cases: y ′ φ ( a ) = 0 . y ′ φ ( a ) = − .
01, while the evolutions of H ( a ) are obviously different for two cases: y ′ φ ( a ) = 0 .
01 and y ′ φ ( a ) = 0.It seems that the effect to H ( a ) from the BD field is notable. Using the observational Hubble data listed in table I,we display these observational H ( z ) value in Fig.1 (right). Here z = (1 − a ) /a is the cosmic redshift. It is shown fromFig.1 (right) that the most observational H ( z ) data are located in the region between the case of y ′ φ ( a ) = 0 and thecase of y ′ φ ( a ) = ± .
01. It seems that the GBD model could pass through the test of the observation, such as theobservational Hubble data, since the evolution of H ( a ) with y ′ φ ( a ) = 0 .
003 is well consistent with those observationaldata. From Fig.2 (right), one can see that evolutional tendency of BD scalar field depends on the initial value of y ′ φ ( a ). z 0.0708 0.09 0.12 0.17 0.179 0.199 0.20 0.24 0.27H(z) 69 . ± .
68 69 . ± . . ± . . ± . . ± . . ± . . ± . . ± .
65 77 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
68 82 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . IV. Effective state parameter of geometrical dark energy in the GBD
Probing properties of the dark energy is important, and it has been studied in the standard cosmology or the severalmodified gravity theories [59–75]. Next we investigate the properties of geometrical dark energy in this GBD theory,and analyze the effects of the BD scalar field. Rewriting the Eq.(3) as follows G µν = R µν − Rg µν = 8 πT effµν φ , (27)with T effµν = φ φ T µν f R + φ πf R [( ∇ µ ∇ ν − g µν (cid:3) ) f R + 12 f ( R ) g µν − f R Rg µν + f R φ ( ∇ µ ∇ ν − g µν (cid:3) ) φ − ωφ g µν ∂ σ φ∂ σ φ + ωφ ∂ µ φ∂ ν φ ] , (28)then the effective energy density and the effective pressure are derived as ρ eff = φ φ ρ m f R + φ πf R " − H ˙ f R − f ( R ) + 12 f R R − Hf R ˙ φφ + 12 ω ( ˙ φφ ) (29) p eff = φ φ p m f R + φ πf R " ¨ f R + 2 H ˙ f R + 12 f ( R ) − f R R + f R ¨ φφ + 2 Hf R ˙ φφ + 12 ω ( ˙ φφ ) + 2 ˙ φφ ˙ f R . (30)Here ρ m and p m are the energy density and the pressure of matter, respectively. According to Eqs. (27) and (28), wecan define the effective Newton gravitational constant G eff = φf R . To keep the attractive property of gravity, we getan constraint: φf R >
0. If we assume φ >
0, then f R >
0. The effective state parameter for geometrical dark energyhas a form w effg = p eff − p m ρ effg − ρ m = 8 πφ p m − πφp m f R + φ φ h ¨ f R + 2 H ˙ f R + f ( R ) − f R R + f R ¨ φφ + 2 Hf R ˙ φφ + ω ( ˙ φφ ) + 2 ˙ φφ ˙ f R i πφ ρ m − πφρ m f R + φ φ h − H ˙ f R − f ( R ) + f R R − Hf R ˙ φφ + ω ( ˙ φφ ) i . (31) −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna −1.0−0.50.0 w e ff g y ,φ (a=1)=0.01, β=1.0y ,φ (a=1)=0.01, β=1.5y ,φ (a=1)=0.01, β=2.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna −3−2−10123 w e ff g β=1.5, y ,φ (a=1)= −0.01 β=1.5, y ,φ (a=1)=0 β=1.5, y ,φ (a=1)=0.01 β=1.5, y ,φ (a=1)=0.005 FIG. 3: The evolutions of w effg ( a ) in the GBD model with the different model parameter β or the different initial condition y ′ φ ( a ). Taking the function f ( R ) = R + βR s (1 − e − R/R s ) as an example, we plot the evolution of w effg ( a ) in Fig.3 byusing the different values of model parameter β and the initial values y ′ φ ( a ). In this GBD model, the dependence of w effg ( a ) on the model parameter β are illustrated in Fig.3 (left). In the Fig.3 (right), one can see that w effg ( a ) almosthave the same evolutions for the two cases: y ′ φ ( a ) = ± .
01, i.e. the trajectories of w effg ( a ) are not sensitive to thesymbol of initial condition y ′ φ ( a ), while the effect on w effg ( a ) from the BD scalar field is notable since the evolution of w effg ( a ) with y ′ φ ( a ) = 0 is obviously different from other three cases: y ′ φ ( a ) = ± .
01 and y ′ φ ( a ) = 0 . w effg with y ′ φ ( a ) = 0 has the more small value than other cases of y ′ φ ( a ) = 0, while w effg ( a ) with y ′ φ ( a ) = 0 has the more large value than the cases of y ′ φ ( a ) = 0 at the high redshift ln a = −
1. Andthe values of w effg are located in the range [-2.63,-0.75] for using the different initial conditions: from y ′ φ ( a ) = 0 to y ′ φ ( a ) = ± .
01. The evolutions of w effg ( a ) with y ′ φ ( a ) = ± .
01 in Fig.3 show that they vary from w ∼ (radiation)to w < w = −
1, whereseveral fundamental problems are associated with phantom field, such as the problem of negative kinetic term andthe fine-tuning problem, etc. It is shown that the GBD model can paly a role of the quintom without bearing theproblems existing in the two-fields quintom model, which is also a motivation to appeal us to study the GBD model.
V. A effective potential of the BD scalar field in the GBD theory
One knows that the potential of a scalar field usually paly an important role in the early inflation universe andthe late accelerating universe. Determining the forms of the potential function for a scalar field is significative, sincethe potential display some properties for a scalar field. By using Eqs. (4) and (6), the equations of motion for theBrans-Dicke scalar field φ and the other scalar field ϕ = f R are expressed as follows (cid:3) φ = ∂ µ φ∂ µ φ φ − φf ω = − V φ ( φ ) , (32) (cid:3) f R = 13 [ 8 πTφ − f R R + 2 f − f R (cid:3) φφ − f R ˙ φφ − ω∂ µ φ∂ µ φφ ] = −V ϕ ( ϕ ) + 8 πT φ . (33)Obviously, this GBD theory is similar to the two scalar-fields theory. Here V ( φ ) and V ( ϕ ) denotes the effectivepotential of fields, and the subscript φ (or ϕ ) denotes the derivative with respect to scalar field. The effective formsof V φ and V ϕ can be gained by comparing with the standard form of equation of motion for the canonical scalar field,i.e. the standard Klein-Gordon equation in the vacuum: (cid:3) Φ + V Φ (Φ) = 0.One knows that the geometrical representation may be more appealing to relativists due to its more apparentgeometrical nature, whereas the scalar-field representation seems more appealing to particle physicists. Obviously,the GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV tends to solvephysical problems from the viewpoint of matter. Given that the equivalence between the BDV theory and the f ( R )theory, some properties of the BD scalar field could be found. So, it is possible that several special characteristics ofscalar fields could be revealed through studies of geometrical gravity in the GBD. Next we investigate the effectiveform of potential V ( φ ) of the BD scalar field φ . Assuming that the variable φ is independent of its derivative ∂ µ φ and the geometrical quantity f ( R ), we can gain an effective form of V ( φ ) by integrating Eq.(32) with respect to φV ( φ ) = H φ ′ φ + f ω φ + C ( ˙ φ, f ) . (34)Here superscript ′ denotes the derivative with respect to ln a , C is a parameter that is independent of φ . UsingEq.(34) and taking C = 0, we can plot the shapes of BD effective potential in Fig.4. We can see from Fig.4 that thetrajectories of BD effective potential V ( φ ) are not sensitive to the variation of β values, while the shapes of V ( φ )much depend on the initial condition y ′ φ ( a ) for the smaller ln a (ln a < − . a > − . V ( φ ) ∼ φ, f ) in Eq. (34) is a undetermined freedom, whose uncertainty can be used to modify the trajectoriesof the BD effective potential.Furthermore, an interesting property can be found in the GBD theory by comparing with f ( R ) theory. It is wellknown that, the f ( R ) theory are equivalent to the BDV theory with taking a specific value of the BD parameter ω = 0[77, 78]. However, the specific choice: ω = 0 for the BD parameter is quite exceptional, and it is hard to understandthe corresponding absence of the kinetic-energy term for the scalar field. But in the GBD theory, one can read fromEq.(2) that the value of ω is arbitrary and the kinetic-energy term of the scalar field is non-disappeared. In addition,if we assume that the effective form of the BD potential can be approximately written as a popular square functionof φ , i.e. we assume V ( φ ) ≃ V + m φ [79, 80], then we have V φ ≃ m φ with m owning the mass dimension. Thus,we need to require ˙ φ ≃ m = f ω . Obviously, if f >
0, weget an constraint on the BD parameter ω >
0. And ω should own a large value with the requirement of a small valueof m .0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna V ( φ ) y ,φ (a=1)=0.01, β=1.0y ,φ (a=1)=0.01, β=1.5y ,φ (a=1)=0.01, β=2.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 lna −0.0006−0.0004−0.00020.00000.00020.00040.0006 V ( φ ) β=1.5, y ,φ (a=1)= −0.01β=1.5, y ,φ (a=1)=0β=1.5, y ,φ (a=1)=0.01 FIG. 4: The evolutions of effective potential for Brans-Dicke scalar field with the different model parameter β and the differentinitial condition y ′ φ ( a ). VI. Conclusion
The GBD theory is investigated in this paper, which is obtained by generalizing the Ricci scalar R to an arbitraryfunction f ( R ) in the original Brans-Dicke action. This theory can be reduced to the original BD theory and the f ( R )modified gravity under certain conditions. We give the gravitational field equation and the BD scalar-field equationin the GBD theory. Using the FLRW metric and the field equations, we can obtain the cosmological equations inthis theory. The evolutional equations of universe and BD field are numerically solved by taking a concrete form of f ( R ) function. It is shown that the modification to H ( a ) from the dynamical BD scalar field is notable, and the GBDmodel can pass through the test of the observation, such as the observational Hubble data.The trajectories of the effective state parameter for the geometrical dark energy is studied in the GBD universe,which indicates that the evolutions of w effg ( a ) with y ′ φ ( a ) = 0 can vary from radiation ( w effg ∼ /
3) to dark energy( w effg < y ′ φ ( a ), especially they are sensitive to the given symbol of y ′ φ ( a ).Ref. [33] shows an interesting property of the GBD theory, where the GBD theory can naturally solve the problem of γ value emerging in f ( R ) modified gravity (i.e. the inconsistent problem between the observational γ (PPN parameter)value and the theoretical γ value), without introducing the so-called chameleon mechanism. In this paper, we alsocompare our results with other theories. It can be seen that the GBD theory have some other interesting propertiesor solve some problems existing in other theories. (1) We can notice that it is required to include both the canonicalquintessence field and the non-canonical phantom field in the double scalar-fields quintom model, in order to makethe state parameter to cross over the phantom boundary: w = −
1, while several fundamental problems are associatedwith the non-canonical phantom field, such as the problem of negative kinetic term and the fine-tuning problem, etc.It can be found that in this paper the effective state parameter of geometrical dark energy in the GBD model cancross over the phantom boundary without bearing the problems relating with the phantom field. (2) It is well knownthat, the f ( R ) theory are equivalent to the BDV theory with a specific value of the BD parameter ω = 0. However,the specific choice: ω = 0 for the BD parameter is quite exceptional, and it is hard to understand the corresponding1absence of the kinetic term for the scalar field in the action of the BDV theory, while in the GBD theory the valueof ω is arbitrary and the dynamical effect of the scalar field is non-disappeared. (3) One knows that the geometricalrepresentation may be more appealing to relativists due to its more apparent geometrical nature, whereas the scalar-field representation seems more appealing to particle physicists. Obviously, the GBD theory tends to investigate thephysics from the viewpoint of geometry, while the BDV or the quintom scalar-field model tends to solve physicalproblems from the viewpoint of matter. Given that the equivalence between the BDV theory and the f ( R ) theory,some properties of the BD scalar field could be found. So, it is possible that several special characteristics of scalarfields could be revealed through studies of geometrical gravity in the GBD. As shown in this paper, an effective formof the BD potential can be gained by studying the GBD theory. And, it seems that a viable condition for the BDtheory could be found, i.e. the BD parameter should be ω > f >
0, if we assume that the effective form of theBD potential can be approximately written as a popular square function of φ . Acknowledgments
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