Reconstruction of new holographic scalar field models of dark energy in Brans-Dicke Universe
Weiqiang Yang, Yabo Wu, Limin Song, Yangyang Su, Jian Li, Dandan Zhang, Xiaogang Wang
aa r X i v : . [ g r- q c ] N ov Reconstruction of new holographic scalar field models ofdark energy in Brans-Dicke universe
Wei-Qiang Yang , Ya-Bo Wu ∗ , Li-Min Song , Yang-Yang Su , Jian Li , Dan-Dan Zhang , Xiao-Gang Wang Department of Physics, Liaoning Normal University, Dalian 116029, P.R.China College of Information Science and Technology, Dalian Maritime University, Dalian 116026, [email protected] ; [email protected] by the work [K. Karami, J. Fehri,
Phys. Lett. B , 61 (2010)] and [A. Sheykhi,
Phys. Lett. B ,205 (2009)], we generalize their work to the new holographic dark energy model with ρ D = φ ω ( µH + ν ˙ H ) inthe framework of Brans-Dicke cosmology. Concretely, we study the correspondence between the quintessence,tachyon, K-essence, dilaton scalar field and Chaplygin gas model with the new holographic dark energy model inthe non-flat Brans-Dicke universe. Furthermore, we reconstruct the potentials and dynamics for these models.By analysis we can show that for new holographic quintessence and Chaplygin gas models, if the relatedparameters to the potentials satisfy some constraints, the accelerated expansion can be achieved in Brans-Dickecosmology. Especially the counterparts of fields and potentials in general relativity can describe acceleratedexpansion of the universe. It is worth stressing that not only can we give some new results in the frameworkof Brans-Dicke cosmology, but also the previous results of the new holographic dark energy in Einstein gravitycan be included as special cases given by us. Keywords : holographic dark energy ; Brans-Dicke theory ; scalar field
PACS : 95.36.+x, 98.80.-k
According to the recent observations of type Ia supernovae , we learn that the universe is undergoingan accelerated expansion, also along with the observations of CMBR anisotropy spectrum and large scalestructure (LSS) . The fact implies that there must be some unknown component in universe which can drive theaccelerated expansion of the universe. This component is named as dark energy (DE), and holds a large negativepressure. After this, lots of candidates to dark energy have been suggested: (i) the cosmological constant ; (ii)scalar fields such as quintessence , phantom , K-essence , tachyon field , dilatonic ghost condensate , and soforth; (iii) The interacting DE models including Chaplygin gas , generalized Chaplygin gas , braneworldmodels and agegraphic DE models , etc.. Besides, holographic dark energy (HDE) model is proposedon the base of the holographic ideas. According to the holographic principle, the number of degrees of freedom ∗ Corresponding author: [email protected] f a physical system scales with the area of its boundary. In this context, Cohen et al. suggested that one usesthe event horizon as the cosmological horizon, and the total energy in a region of size L should not exceed themass of a black hole of the same size. The holographic DE not only gives the observational value of DE in theuniverse, but also can drive the universe to an accelerated expansion phase. In that case, however, an obviousdrawback concerning causality appears in this propose. This motivated Granda and Oliveros to propose anew infrared cut-off for HDE, which is about the square of the Hubble parameter H and the time derivativeof the Hubble parameter ˙ H , this model is called as new HDE (NHDE) model in this paper. The NHDE modelcan avoid the problem of causality which appears using the event horizon area as the cut-off. By use of the newinfrared cut-off for HDE, Granda and Oliveros study the correspondence between the quintessence, tachyon,K-essence and dilaton energy density with NHDE model in the flat FRW universe. Subsequently, Karami andFehri generalize their work to the non-flat case.On the other hand, it is quite possible that gravity is not given by the Einstein action, at least at sufficientlyhigh energies. In string theory, gravity becomes scalar-tensor in nature. The low energy limit of string theoryleads to the Einstein gravity, coupled non-minimally to a scalar field . Although the pioneering study onscalar-tensor theories was done by Brans and Dicke several decades ago who applied Mach’s principle intogravity , it has got a new impetus as it arises naturally as the low energy limit of many theories of quantumgravity (for example, superstring theory, et al.), and scalar-tensor theories of gravity have been widely appliedin cosmology . For HDE as a dynamical model, we need a dynamical frame to accommodate it instead ofgeneral relativity. Recently, the studies on the HDE model in the framework of Brans-Dicke theory have beencarried out , the purpose of which is to construct a cosmological model of late acceleration based on theBrans-Dicke theory of gravity.In this paper, we will try to generalize the previous work in Refs.[21] and [27] to the NHDE model with ρ D = φ ω ( µH + ν ˙ H ) in the framework of Brans-Dicke cosmology. Hence, the evolution of equation of state(EoS) for the NHDE model will be discussed in non-flat Brans-Dicke universe. Moreover, we will study thecorrespondence between the quintessence, tachyon, K-essence, dilaton and Chaplygin gas model with NHDEmodel in the non-flat Brans-Dicke universe. Furthermore, we will reconstruct the potentials and dynamics forthese models. By analysis we can show that for new holographic quintessence and Chaplygin gas models, ifthe related parameters to the potentials satisfy some constraints, the accelerated expansion of universe can beachieved in Brans-Dicke cosmology.We start from the action of Brans-Dicke theory, in the canonical form it can be written S = Z d x √ g ( − ω φ R + 12 g µν ∂ µ ∂ ν φ + L M ) , (1)where R is the scalar curvature and φ is the Brans-Dicke scalar field. The non-minimal coupling term φ R replaces with the Einstein-Hilbert term R/G in such a way that G − = 2 πφ /ω , where G is the gravitationalconstant. The signs of the non-minimal coupling term and the kinetic energy term are properly adopted to(+ − −− ) metric signature. Here, we consider the non-flat Friedmann-Robertson-Walker (FRW) universe whichis described by the line element ds = dt − a ( t )( dr − kr + r d Ω ) , (2) here a ( t ) is the scale factor, and k is the curvature parameter with k = − , , ω φ ( H + ka ) −
12 ˙ φ + 32 ω H ˙ φφ = ρ D , (3) − ω φ (2 ¨ aa + H + ka ) − ω H ˙ φφ − ω ¨ φφ −
12 (1 + 1 ω ) ˙ φ = p D , (4)¨ φ + 3 H ˙ φ − ω ( ¨ aa + H + ka ) φ = 0 , (5)where the dot is the derivative with respect to time and H = ˙ a/a is the Hubble parameter. Here ρ D and p D are, respectively, the density and pressure of DE. And we neglect the contributions from matter and radiationin the universe, that is, ρ total = ρ D . In addition, we shall assume that Brans-Dicke field can be described as apower law of the scale factor, φ ∝ a α . Taking the derivative with respect to time, one can get˙ φ = αHφ, (6)¨ φ = α H φ + αφ ˙ H. (7)According to Ref.[16], the HDE density with the new infrared cut-off is given by ρ D = 38 πG ( µH + ν ˙ H ) , (8)this is just the NHDE model, where H = ˙ a/a is the Hubble parameter, µ and ν are constants which must satisfythe restrictions imposed by the current observational data.In the framework of Brans-Dicke theory, using φ = ω/ πG eff , one can obtain ρ D = 3 φ ω ( µH + ν ˙ H ) . (9)Thus, Eq.(3) becomes 34 ω φ ( H + ka ) −
12 ˙ φ + 32 ω H ˙ φφ = 3 φ ω ( µH + ν ˙ H ) . (10)Furthermore, we obtain dH dx + 2 s ν H = 2 kν e − x , (11)where x = lna and s = 2 ωα − α + 3 µ −
3. Integrating the above equation with respect to x yields H = 3 ks − ν e − x + Ce − sx/ ν , (12)where C is an integration constant and s = 3 ν . For the special case of s = 3 ν , from Eq.(11), we have H = kν xe − x + Ce − x . But, here we only make the discussion of s = 3 ν . For the flat case k = 0 and using˙ x = H , from Eq.(11), one can reduce to H = − νs ˙ H, (13)where the dot denotes the time derivative with respect to the cosmic time t . Furthermore, one has H = 3 νs t . (14) t is easy to see from Eq.(14) that when α = 0, Hubble parameter H = νµ − t , which exactly reduces to thecase in Ref.[20].In addition, according to the conservation equation˙ ρ D + 3 H (1 + w D ) ρ D = 0 , (15)and w D = p D /ρ D , EoS can be expressed as w D = − − αµH + 2( αν + µ ) H ˙ H + ν ¨ H H ( µH + ν ˙ H ) , (16)which is just the evolution of EoS for the NHDE model in non-flat Brans-Dicke universe. When α = 0, it is easyto see that the Brans-Dicke scalar field becomes trivial, and Eq.(16) can reduce to the case of NHDE model ingeneral relativity , i.e., w D = − − µH ˙ H + ν ¨ H H ( µH + ν ˙ H ) . (17)It follows that the results in Refs.[20,21] are included as the special cases of α = 0 and k = 0 given by us inthis paper.Furthermore, from Eqs.(12) and (16) we obtain w D = 27 e sx/ ν kν (1 + 2 α )( µ − ν ) + X (2 s − ν − αν )9 ν [ X + 9 e sx/ ν k ( ν − µ )] , (18)where X = Ce x ( s − µ )( s − ν ) (Here, s = 3 ν and µ = ν ). Eq.(18) shows that EoS parameter is time-dependent, and can cross the phantom divide w D = − . Of course, the current observations from theseven-year WMAP data and the analysis of SN1a and CMB data favor the present values w of EoS biggerthan − k = 1, ω = − α = − / w = − w in Tab.1. The values of w are basically in the rangesof w = − . ± .
14, which are supported by the seven-year WMAP data and the analysis of SN1a and CMBdata . -1 0 1 2 3-1.4-1.2-1.0-0.8-0.6-0.4-0.2 w D z C=-3(a) -1 0 1 2 3-1.4-1.2-1.0-0.8-0.6-0.4 w D z (b) C=-3 -1 0 1 2 3-1.2-1.0-0.8-0.6-0.4 w D z C=-2 C=-3 C=-4 C=-5 (c)
Figure 1.
The evolutionary trajectories of EoS in NHDE model. (a) for fixed constants C and µ but differentcoupling constants ν ; (b) for fixed constants C and ν but different constants µ ; (c) for fixed constants µ and ν but different constants C . ( C = − , µ = 0 . w ( C = − , ν = 0 . w ( µ = 0 . , ν = 0 . − . ν = 0 . − . µ = 0 . − . C = − − . ν = 0 . − . µ = 0 . − . C = − − . ν = 0 . − . µ = 0 . − . C = − − . ν = 0 . − . µ = 0 . − . C = − Table 1.
The present values w of EoS in Fig. 1.We know that Brans-Dicke cosmology becomes standard cosmology when ω → ∞ , in this case α → w D = − k ( µ − νµ − ν − ) e − x + C ( ν − µ +2 ν ) e − ν ( µ − x k ( µ − νµ − ν − ) e − x + Ce − ν ( µ − x ) ! (19)which is just the same as Eq.(10) in Ref.[21].If taking k = 0 for the flat universe, we obtain w D = − − α s ν . (20)In addition, by use of w D < − / H , t , a , we can obtain the constraintsof the parameters (such as µ , ν , ω and α etc.). For the non-flat case ( k = 0), in Eq.(12), when tak-ing H = H , a = a = 1 (i.e, x = lna = 0), the integration constant C can be expressed as C = H − k/ ( s − ν ). Thus, by means of EoS parameter w D < − / k ( − − ν − α − αν +4 ωα ) − H (2 ωα − α − − µ − ν − α − αν +4 ωα )27 kν +9 H (3+6 α − ωα ) < − /
3. Specially, for the flat case of k = 0, from Eqs.(14) and (20), we can obtain H t = 3 ν/s , and w D = − − α + H t < − /
3, which leadsto α > H t −
1. Furthermore, according to H t = √ Ω D ln h √ Ω D −√ Ω D i , we find when Ω D → H t → ∞ .Hence, we can conclude that for the flat case of k = 0, the constraint on α is α > − α = 0, Eq.(20)reduces to the case of NHDE model in general relativity w D = − µ − ν . (21)In order to obtain accelerated expansion, the constants µ and ν must satisfy the restrictions: for thequintessence-like phase with − < w D < − /
3, we obtain µ > ν > µ −
1, or, µ < ν < µ −
1. For µ < ν >
0, or, µ > ν <
0, it describes a phantom-like phase with w D < − k = 0, we cangive the related results of scalar fields and potentials for the NHDE model in the flat Brans-Dicke universe.Furthermore, we can obtain the fields and potentials in general relativity, which describe accelerated expansionof the universe. To establish this correspondence, we compare energy density of the NHDE model given byEq.(9) with the corresponding energy density of scalar field model, and also equate the EoS for these scalar odels with the EoS given by Eq.(18). Here, we take scalar field as ϕ in order to differ from the parameter φ in Brans-Dicke theory.1. New holographic quintessence modelThe energy density and pressure of the quintessence scalar field ϕ are as follows ρ Q = 12 ˙ ϕ + V ( ϕ ) , (22) p Q = 12 ˙ ϕ − V ( ϕ ) . (23)The EoS for the quintessence scalar field is given by w Q = p Q ρ Q = ˙ ϕ − V ( ϕ )˙ ϕ + 2 V ( ϕ ) . (24)Therefore, we have w D = 27 e sx/ ν kν (1 + 2 α )( µ − ν ) + X (2 s − ν − αν )9 ν [ X + 9 e sx/ ν k ( ν − µ )] = ˙ ϕ − V ( ϕ )˙ ϕ + 2 V ( ϕ ) , (25)and ρ D = 3 φ ω ( αH + β ˙ H ) = 12 ˙ ϕ + V ( ϕ ) . (26)Furthermore, the kinetic energy term and the quintessence potential energy are˙ ϕ = e α − s/ ν − x [27 e sx/ ν kν (1 − α )( µ − ν ) + X (3 αν − s )]18 ων ( s − ν ) , (27) V ( ϕ ) = e α − s/ ν − x [27 e sx/ ν kν (2 + α )( µ − ν ) + X ( s − ν − αν )]36 ων ( s − ν ) . (28)Using ˙ ϕ = Hϕ ′ , where prime denotes the derivative with respect to x , we obtain ϕ ′ = 1 H s e α − s/ ν − x [27 e sx/ ν kν (1 − α )( µ − ν ) + X (3 αν − s )]18 ων ( s − ν ) , (29)where H is given by Eq. (12). Consequently, after integration with respect to x we can obtain the evolutionaryform of the quintessence scalar field as ϕ ( a ) − ϕ (0) = Z lna H s e α − s/ ν − x [27 e sx/ ν kν (1 − α )( µ − ν ) + X (3 αν − s )]18 ων ( s − ν ) dx, (30)where we take a = 1 for the present time.For the flat case of k = 0, assuming ϕ (0) = 0 for the present t = 0, then Eqs.(28) and (30) reduce to ϕ ( t ) = r ( s − µ )(3 αν − s )18 ων t αν/s α ( α = 0) , (31) V ( ϕ ) = ν ( s − µ )( s − ν − αν )4 ωs "s ων ( s − µ )(3 αν − s ) αϕ − s/ αν ) . (32)It is easy to see that when taking 3 αν/s = −
1, Eq.(32) can reduce to V = V ϕ ( V is a constant) whichdescribes the accelerated expansion of universe . By using the decelerated parameter q ≡ − a ¨ a/ ˙ a < a = t − /α , we have − < α < α >
0. Furthermore, w D = − − α/ s/ ν < − / α > − / o we obtain − / < α < α >
0. In addition, considering that ων ( s − µ )(3 αν − s ) > ω < − / − / < α < α >
0. Therefore we believe that the potential(32) for the new holographic quintessence model, which can reduce to V = V ϕ , can describe the acceleratedexpansion of universe if the parameters satisfy the constraints 3 αν/s = − − / < α < ω < − / k = 0 and α = 0 in Eqs.(28) and (29), the scalar field and potential can reduce tothe case of NHDE model in general relativity ϕ ( t ) = r νµ − M p lnt, (33) V ( ϕ ) = 3 ν − µ + 1( µ − M p exp − r µ − ν ϕM p ! . (34)According to Ref.[40], this potential can describe an accelerated expansion provided that ν/ ( µ − >
1, andalso has cosmological scaling solutions .2. New holographic tachyon modelThe energy density and pressure for the tachyon field are as follows ρ T = V ( ϕ ) p − ˙ ϕ , (35) p T = − V ( ϕ ) p − ˙ ϕ , (36)where V ( ϕ ) is tachyon potential. The EoS for the tachyon scalar field is obtained as w T = p T ρ T = ˙ ϕ − . (37)By means of w D = w T and ρ D = ρ T , we can obtain the kinetic energy term and the tachyon potentialenergy in the new holographic tachyon model as follows˙ ϕ = 54 e sx/ ν kν ( α − µ − ν ) + 2 X ( s − αν )9 ν [ X + 9 e sx/ ν k ( ν − µ )] , (38) V ( ϕ ) = e α − s/ ν − x [9 e sx/ ν k ( µ − ν ) − X ] Y ω ( s − ν ) , (39)where Y = q e sx/ ν kν (1+2 α )( ν − µ ) − X (2 s − ν − αν ) ν [ X +9 e sx/ ν k ( ν − µ )] .Furthermore, the evolutionary form of the tachyon scalar field can be obtained ϕ ( a ) − ϕ (0) = Z lna H s e sx/ ν kν ( α − µ − ν ) + 2 X ( s − αν )9 ν [9 e sx/ ν k ( ν − µ ) + X ] dx. (40)For the flat case k = 0, using the initial condition ϕ (0) = 0, we have ϕ ( t ) = r s − αν )9 ν t, (41) V ( ϕ ) = 3(3 µ − s ) ν ωs r ν + 6 αν − sν "s ν s − αν ) ϕ αν/s − . (42)Furthermore, when taking α = 0, Eqs.(41) and (42) can reduce to the counterparts in general relativity ϕ ( t ) = r µ − ν t, (43) ( ϕ ) = 2 M p r − µ − ν (cid:18) νµ − (cid:19) ϕ . (44)According to Refs.[20] and [40], the inverse square potential in Eq.(44) can describe the accelerated expansionof universe when the parameters satisfy the condition − / √ < ( µ − /ν < / − √ ρ ( ϕ, χ ) = f ( ϕ )( − χ + 3 χ ) , (45) p ( ϕ, χ ) = f ( ϕ )( − χ + χ ) . (46)The EoS is obtained as w KE = p ( ϕ, χ ) ρ ( ϕ, χ ) = χ − χ − . (47)Equating Eq.(47) with the EoS of NHDE, w KE = w D , one has χ = 27 e sx/ ν kν ( α + 2)( µ − ν ) + X ( s − ν − αν )3[27 e sx/ ν kν ( α + 1)( µ − ν ) + X ( s − ν − αν )] . (48)By use of Eq.(48), ˙ ϕ = 2 χ , and ˙ ϕ = ϕ ′ H , we obtain the evolutionary form of the K-essence scalar field as ϕ ( a ) − ϕ (0) = Z lna H s e sx/ ν kν ( α + 2)( µ − ν ) + X ( s − ν − αν )]3[27 e sx/ ν kν ( α + 1)( µ − ν ) + X ( s − ν − αν )] dx. (49)Furthermore, using ρ D = ρ ( ϕ, χ ), Eqs. (3) and (12), the expression of f ( ϕ ) can be given as follows f ( ϕ ) = e α − s/ ν − x [9 e sx/ ν k ( µ − ν ) − X ]4 ω ( s − ν ) χ (3 χ − . (50)For the flat case of k = 0, assuming ϕ (0) = 0, then Eqs. (48), (49) and (50) become into χ = s − ν − αν s − ν − αν ) , (51) ϕ ( t ) = s s − ν − αν )3( s − ν − αν ) t, (52) f ( ϕ ) = 9 ν ( s − µ )( s − ν − αν ) ωs ( s − ν − αν ) "s s − ν − αν )2( s − ν − αν ) ϕ αν/s − , (53)Thus, when taking α = 0, Eqs.(52) and (53) can reduce to ϕ ( t ) = r (cid:16) ν − µ +12 ν − µ +1 (cid:17) t and f ( ϕ ) = 6 M p ν h ν − µ +1( µ − i ϕ .Hence, the potential of new holographic K-essence corresponds to the power-law expansion like new holographictachyon model.4. New holographic dilaton modelThe energy density and pressure of the dilaton DE model are given by ρ DI = χ − c ′ e λϕ χ , (54) p DI = χ − c ′ e λϕ χ , (55) here c ′ and λ are positive constants and χ = ˙ ϕ /
2. The EoS for the dilaton scalar field is given by w DI = p DI ρ DI = χ − c ′ e λϕ χ χ − c ′ e λϕ χ . (56)By means of ω DI = ω D , we find the following solution c ′ e λϕ χ = 27 e sx/ ν kν ( α + 2)( µ − ν ) + X ( s − ν − αν )3[27 e sx/ ν kν ( α + 1)( µ − ν ) + X ( s − ν − αν )] . (57)Moreover, using χ = ˙ ϕ / ϕ = ϕ ′ H , we have e λϕ ( a )2 = e λϕ (0)2 + λ √ c ′ Z lna H s e sx/ ν kν ( α + 2)( µ − ν ) + X ( s − ν − αν )27 e sx/ ν kν ( α + 1)( µ − ν ) + X ( s − ν − αν ) dx, (58)where H is given by Eq. (12). Therefore, the evolutionary form of the dilaton scalar filed can be written as ϕ ( a ) = 2 λ ln [ e λϕ (0)2 + λ √ c ′ Z lna H s e sx/ ν kν ( α + 2)( µ − ν ) + X ( s − ν − αν )27 e sx/ ν kν ( α + 1)( µ − ν ) + X ( s − ν − αν ) dx ] . (59)For the flat case k = 0, using ˙ x = H , assuming ϕ (0) → −∞ for the present time t = 0, then one can give ϕ ( t ) = 2 λ ln [ s s − ν − αν c ′ ( s − ν − αν ) λt ] . (60)By using α = 0, Eq.(60) becomes into ϕ ( t ) = λ ln [ λ √ c ′ q ν − µ +12 ν − µ +1 t ] . According to Ref.[40], we know thatfor the dynamics system of this model, the condition of accelerated expansion of universe is − < ( µ − /ν < proposed a model of DE involving a fluid known as a Chaplygin gas(CG). This fluidalso leads to the acceleration of the universe at late times, and its simplest form has the following specificequation of state p CG = − Aρ CG , (61)where A is a positive constant. With the continuity equation, Eq. (61) can be integrated to give ρ CG = p A + Be − x , (62)where B is an integration constant and x = lna . We will establish the correspondence between NHDE modeland CG model, that is, the new holographic CG model. To do this, we treat CG as an ordinary scalar field ϕ .So we have ρ ϕ = ˙ ϕ / V ( ϕ ) = p A + Be − x , (63) p ϕ = ˙ ϕ / − V ( ϕ ) = − A √ A + Be − x . (64)Hence, it is easy to obtain the kinetic energy terms and the scalar potential for the CG as˙ ϕ = Be − x a √ A + Be − x , (65) V ( ϕ ) = 2 A + Be − x a √ A + Be − x . (66) y use of ρ CG = ρ D , one can obtain B = e − x/ ν [ M − Ae (6+4 s/ ν ) x ω ( s − ν ) ]16 ω ( s − ν ) , (67)where M = C e (6+4 α ) x ( s − µ ) ( s − ν ) +81 e (2+4 α +4 s/ ν ) x k ( µ − ν ) +18 Ce x (6+6 α + s/ν ) / k ( s − µ )( s − ν )( ν − µ ).And using p CG = ρ CG w D , we have B = Ae x (cid:20) − − νX + 81 νe sx/ ν k ( ν − µ )27 e sx/ ν kν (1 + 2 α )( µ − ν ) + X (2 s − ν − αν ) (cid:21) . (68)So it is easy to give the expression of A and BA = − e − (6+4 s/ ν ) x M [27 e sx/ ν kν (1 + 2 α )( µ − ν ) + X (2 s − ν − αν )]144 ω ν ( s − ν ) [ X + 9 e sx/ ν k ( ν − µ )] , (69) B = e − sx/ ν M [27 e sx/ ν kν ( α − µ − ν ) + X ( s − αν )]72 ω ν ( s − ν ) [ X + 9 e sx/ ν k ( ν − µ )] . (70)Thus we can obtain the kinetic energy terms and the scalar potential for the Chaplygin gas as˙ ϕ = [27 e sx/ ν kν ( α − µ − ν ) + X ( s − αν )] N ν [ X + 9 e sx/ ν k ( ν − µ )] , (71) V ( ϕ ) = [27 e sx/ ν kν ( α + 2)( ν − µ ) − X ( s − ν − αν )] N ν [ X + 9 e sx/ ν k ( ν − µ )] , (72)where N = q e − (6+4 s/ ν ) x Mω ( s − ν ) . Using ˙ ϕ = Hϕ ′ and integrating Eq.(72), one can obtain the evolutionary form as ϕ ( a ) − ϕ (0) = Z lna H s [27 e sx/ ν kν ( α − µ − ν ) + X ( s − αν )] N ν [ X + 9 e sx/ ν k ( ν − µ )] dx. (73)For the flat case k = 0, assuming the initial condition ϕ (0) = 0, one can obtain ϕ ( t ) = r ( s − µ )(3 αν − s )18 ων t αν/s α ( α = 0) , (74) V ( ϕ ) = ν ( s − µ )( s − ν − αν )4 ωs "s ων ( s − µ )(3 αν − s ) αϕ − s/ αν ) , (75)The discussion about the physical statement of the potential is the same as one in the new holographicquintessence model because the both has the same potential.In summary, by generalizing the previous work to the NHDE model with ρ D = φ ω ( µH + ν ˙ H ) inthe framework of Brans-Dicke cosmology, we have obtained the evolution of EoS and given the present valuesof EoS w (see Tab.1), which are basically consistent with the present observation data w = − . ± . αν/s = − − / < α < ω < − /
2, the accelerated expansion can beachieved in Brans-Dicke cosmology. It is worth stressing that not only have we given some new results of theNHDE model in the framework of Brans-Dicke theory, but also the previous results of the new holographic darkenergy in Einstein gravity can be included as special cases of α = 0 or k = 0 given by us, which can describethe accelerated expansion. cknowledgments Authors thank the anonymous reviewers for constructive comments. This work has been Supported by theNational Natural Science Foundation of China (Grant No.10875056), the Natural Science Foundation of LiaoningProvince, China (Grant No.20102124), and the Scientific Research Foundation of the Higher Education Instituteof Liaoning Province, China (Grant No.2009R35).
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