Dynamics of Potentials in Bianchi Type Scalar-Tensor Cosmology
aa r X i v : . [ g r- q c ] O c t Dynamics of Potentials in BianchiType Scalar-Tensor Cosmology
M. Sharif ∗ and Saira Waheed † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
The present study investigates the nature of the field potential vianew technique known as reconstruction method for the scalar field po-tentials. The key point of this technique is the assumption that Hubbleparameter is dependent on the scalar field. We consider Bianchi typeI universe in the gravitational framework of scalar-tensor gravity andexplore the general form of the scalar field potential. In particular, thisfield potential is investigated for the matter contents like barotropicfluid, the cosmological constant and Chaplygin gas. It is concludedthat for a given value of Hubble parameter, one can reconstruct thescalar potentials which can generate the cosmology motivated by thesematter contents.
Keywords:
Scalar-tensor theory; Scalar field; Field potentials.
PACS:
The reality of cryptic dominant component of the universe distribution la-beled as dark energy (DE) and its resulting phenomena of cosmic acceleration ∗ [email protected] † [email protected] f ( R ) gravity [15], Gauss-Bonnet gravity [16], f ( T ) theory [17], f ( R, T ) gravity [18] and scalar-tensor theories [19]. The study of scalar-tensor theories in the subject of cosmology has a great worth due to its vastapplications and success [20].The complete history of the universe from the early inflationary epochto the final era of cosmic expansion can successfully be discussed by usingscalar field as DE candidate [21]. Basically, the alternating gravitational the-ories are proposed by the inclusion of some functions or terms as a possiblemodification of Einstein gravity that cannot be derived from the fundamen-tal theory. This raises a question about the appropriate choice of thesefunctions by checking their cosmological viability. However, the process ofreconstruction provides a way for having a cosmologically viable choice ofthese functions. Such a procedure has been adopted by many researchers[22]-[31]. The reconstruction procedure is not a new technique as it has along history for the reconstruction of DE models. In order to have a betterunderstanding of this technique, we may refer the readers to study some in-teresting earlier papers [32]. Basically, this technique enables one to find theform of the scalar field potential as well as scalar field for a particular valueof the Hubble parameter in terms of scale factor or cosmic time.It is worth investigating the nature of scalar field potential in the contextof scalar-tensor theories. Using reconstruction approach, the nature of thefield potential for a minimally coupled scalar-tensor theory has been discussed222]. The scalar potentials for tachyon field [23] as well as for solutions in-volving two scalar fields [24] have been reconstructed through this technique.This is also extended to the modified gravitational frameworks includingnon-minimal coupled scalar-tensor theories [25], Gauss-Bonnet gravity [26], F ( T ) theory [27] and the non-local gravity model [28]. Kamenshchik et al.[29] used this technique to reconstruct the scalar field potential for FRWuniverse in the induced gravity and discussed it for some types of matter dis-tribution which can reproduce cosmic evolution. The same authors [30] usedsuperpotential approach to reconstruct the field potential for FRW modelin a non-minimally coupled scalar-tensor gravity and explored its nature fordifferent cases like de Sitter and barotropic solutions describing the cosmicevolution.In this paper, we discuss the nature of the field potential using the recon-struction procedure for locally rotationally symmetric (LRS) Bianchi typeI (BI) universe model. The paper is organized as follows. In the next sec-tion, we provide a general discussion of this technique and explore the formof scalar field potential. Section is devoted to study the field potentialsusing the barotropic fluid, the cosmological constant and the Chaplygin gasas matter contents. In the last section, we discuss and conclude the results. The scalar-tensor gravity is generally determined by the action [31] S = Z √− g [ U ( φ ) R − ω ( φ )2 g µν φ ,µ φ ,ν + V ( φ )] d x ; µ, ν = 0 , , , , (1)where U is the coupling of geometry and the scalar field, V is the self-interacting potential, R is the Ricci scalar and ω is the interaction function.We can discuss different cases of scalar-tensor theories by taking differentvalues of U ( φ ). When both U, ω are constants, the above action yieldsthe Einstein-Hilbert action with quintessence scalar field, for U = φ with ω = ω , ω ( φ ), it corresponds to simple Brans-Dicke (BD) and the general-ized BD gravity with scalar potential, respectively. For U ( φ ) = γφ , where γ is any non-zero constant and constant ω , it leads to the action of the in-duced gravity. Anisotropic and spatially homogeneous extension of flat FRWmodel, BI universe with the expansion factors A and B is given by the metric333] ds = dt − A ( t ) dx − B ( t )( dy + dz ) (2)and the respective Ricci scalar is R = −
2[ ¨ AA + 2 ¨ BB + ( ˙ BB ) + 2 ˙ AA ˙ BB ] . The average scale factor a ( t ), the universe volume V , the directional Hubbleparameters ( H along x direction while H along y and z directions) and themean Hubble parameter are given by a ( t ) = ( AB ) / , V = a ( t ) = AB , H = ˙ AA ,H = H = ˙ BB , H ( t ) = 13 ( ˙ AA + 2 ˙ BB ) . In order to deal with highly non-linear equations, we take a physicalassumption for the scale factors, i.e., A = B m ; m = 0 , σ and the expansion scalar θ , in other words, the ratio of thesequantities σθ is constant. This condition has been used by many researchersfor the discussion of exact solutions [35]. The above condition further yieldsthe relations ˙ AA = m ˙ BB and ¨ AA = m ¨ BB + m ( m − ˙ B B , consequently the Ricciscalar takes the form R = − m + 2) ¨ BB + ( m + m + 1) ˙ B B ] (3)For BI universe model, we have √− g = B ( m +2) and the respective point-like Lagrangian density constructed by partial integration [36] of the aboveaction (when ω = ω , where ω is an arbitrary constant) is given by L ( B, φ, ˙ B, ˙ φ ) = 2( m + 2) B ( m +1) dUdφ ˙ B ˙ φ + 2 B m ˙ B (1 + 2 m ) U ( φ ) − ω B m +2 ˙ φ + V ( φ ) B m +2 , (4)4here we have neglected the boundary terms. In order to formulate thecorresponding field equations, we use the Euler-Lagrange equations ∂ L ∂B − ddt ( ∂ L ∂ ˙ B ) = 0 , ∂ L ∂φ − ddt ( ∂ L ∂ ˙ φ ) = 0 , which describe the dependent field equation for the BI model and the evolu-tion equation of scalar field. Thus we have2( m + 2) d Udφ ˙ φ − m + 2) dUdφ ¨ φ − m ) dUdφ ˙ BB ˙ φ − m ) U ( φ ) ¨ BB = 0 , (5) ω ¨ φ + ω ( m + 2) ˙ φ ˙ BB + 2(1 + 2 m ) dUdφ ˙ B B − m + 2)( m + 1) dUdφ dVdφ ˙ B B − m + 2) ¨ BB dUdφ = 0 . (6)The energy relation (conserved quantity) [37] for the Lagrangian density (4)can be written as E L = ˙ B ∂ L ∂ ˙ B + ˙ φ ∂ L ∂ ˙ φ − L that yields the independent fieldequation for BI universe (when substituted equal to zero)2(1 + 2 m ) U ( φ ) ˙ B B + 2( m + 2) dUdφ ˙ BB ˙ φ − ω φ − V ( φ ) = 0 . (7)When m = 1, these equations reduce to the case of FRW universe [29].For the special choice of U , we evaluate the scalar potential in terms ofscale factor, directional Hubble parameter and scalar field. We consider thedirectional Hubble parameter as a function of scale factor or cosmic timeby taking different cases of matter contents. The scalar field is found as afunction of scale factor or cosmic time and then the scale factor as a functionof scalar field by inverting the obtained expression. Finally, we evaluatethe Hubble parameter in terms of scalar field and hence the form of scalarpotential. We shall explore the nature of the potential that can generate thecosmic evolution described by these matter contents. Equation (7) yields V ( φ ) = 2(1 + 2 m ) U ( φ ) ˙ B B + 2( m + 2) ˙ φ dUdφ ˙ BB − ω φ
5r equivalently, V ( φ ) = [2(1 + 2 m ) U ( φ ) + 2( m + 2) φ ,B B dUdφ − ω φ ,B B ] H , (8)which provides dVdφ = 2(1 + 2 m ) H dUdφ + 4(1 + 2 m ) U ( φ ) H ˙ H ˙ φ + 2( m + 2) H ˙ φ d Udφ + 2( m + 2) H dUdφ ¨ φ ˙ φ + 2( m + 2) ˙ H dUdφ − ω ¨ φ. Using this equation in Eq.(6), it follows that ω ( m + 2) ˙ φ − m + 2) dUdφ ˙ φH + 4(1 + 2 m ) U ˙ H + 2( m + 2) ˙ φ d Udφ +2( m + 2) dUdφ ¨ φ = 0 . (9)We investigate two cases for the coupling function U , i.e., when U = U ,where U is a non-zero constant and U ≡ U ( φ ). In the first case, Eq.(9)becomes ˙ φ + ( 4(1 + 2 m ) U ω ( m + 2) ) ˙ H = 0; ω = 0 , m = − B , we have φ ′ + [ 4(1 + 2 m ) U ω ( m + 2) ] H ′ H B = 0 , (10)where prime indicates derivative with respect to scale factor, yielding solution φ ( B ) = Z ( ± q − H ( B ) B ( m ) U ω ( m +2) ) dH dB H ( B ) B ) dB + c , where c is a constant of integration. One can solve this integral for particularvalues of the Hubble parameter. In the second case, we consider U ≡ U ( φ )(a non-minimal coupling of geometry and scalar field). Equation (9) can6e written for scalar field in terms of scale factor and directional Hubbleparameter as φ ′′ + φ ′ ( H ′ H ) + φ ′ [ ω / d Udφ dUdφ ] + 2(1 + 2 m ) U ( m + 2) B dUdφ H ′ H + m (1 − m )( m + 2) φ ′ B = 0 . (11)This equation is discussed for two particular choices of U .When U = φ , i.e., the simple BD gravity, it follows that φ ′′ + φ ′ ( H ′ H ) + ω φ ′ + 2(1 + 2 m ) φ ( m + 2) B H ′ H + m (1 − m )( m + 2) φ ′ B = 0 . (12)For the case of induced gravity described by U ( φ ) = γφ , Eq.(11) yields φ ′′ + φ ′ ( H ′ H ) + φ ′ [ ω / γγφ ] + (1 + 2 m ) γφ ( m + 2) B H ′ H + m (1 − m )( m + 2) φ ′ B = 0 . (13)These two equations are difficult to solve analytically unless the function H ( B ) is given. For the sake of simplicity, we introduce a new variable x ≡ φ ′ φ which yields φ ′′ φ = x ′ + x and hence Eq.(13) turns out to be x ′ + x ( 2 γ + ω γ ) + x H ′ H + (1 + 2 m )( m + 2) B H ′ H + m (1 − m ) x ( m + 2) B = 0 . (14)Further, we assume x ≡ γω +4 γ f ′ f , where f is an arbitrary function of the scalefactor B . Also, x = φ ′ φ thus integration leads to φ = f γ/ ( ω +4 γ ) . Using thisvalue of x in Eq.(14), we obtain f ′′ + f ′ H ′ H + ω + 4 γ γ ( 1 + 2 mm + 2 ) fB H ′ H + m (1 − m ) f ′ ( m + 2) B = 0 . (15)We see that Eq.(12) is difficult to transform in x by the above transformation.If we consider the scalar field as a constant then Eq.(8) yields the scalarpotential V = 2(1 + m ) U H , , where H , is constant directional Hubbleparameter. Multiplying the Klein-Gordon equation (6) both sides with thisvalue of V , we obtain the scalar potential V = V U m m +31+ m = V γ φ m m +3)1+ m , which is obviously a constant (as V and φ are constants).7hen ω ≡ ω ( φ ), the field equations (5) and (7) remain the same exceptthat the constant ω is replaced by ω ( φ ) while Eqs.(6) becomes ω ( φ ) ¨ φ + ω ( φ )( m + 2) ˙ φ ˙ BB + ˙ φ dωdφ + 2(1 + 2 m ) dUdφ ˙ B B − m + 2)( m + 1) dUdφ dVdφ ˙ B B − m + 2) ¨ BB dUdφ = 0 . (16)Solving the field equations (5), (7) and (16), we have the same expressionsas Eqs.(10), (12) and (15) except ω is replaced by ω ( φ ). In the following,we discuss Eqs.(10), (12) and (15) separately to construct potential. Now we discuss the scalar field potential by taking three different mattercontents.
First we consider the barotropic fluid (a particular case of the perfect fluid)with equation of state (EoS), p = kρ, < k < , where p and ρ are pressureand density, while k is the EoS parameter. In order to find the evolutionof Hubble parameter due to barotropic fluid, we consider the Einstein fieldequations for BI universe model as(1 + 2 m ) H = ρ, ( m + 32 ) ˙ H + ( m + m + 42 ) H = − p, (17)where we have used the condition A = B m and also combined the two de-pendent field equations. The integration of the energy conservation equationyields ρ = ρ B − (1+ k )( m +2) , where ρ is an integration constant. Consequently,the directional Hubble parameters are found to be H ( B ) = H ( B ) m = [( m + m + 41 + 2 m + 2 k ) 2 ρ (1 + k )( m + 2)( m + 3) ] / × B − (1+ k )( m +2)2 , (18)where the integration constant is taken to be zero. The evolution of Hubbleparameter is H ′ ( B ) H ( B ) = − (1+ k )( m +2)2 B . The corresponding deceleration parameter8urns out to be positive, i.e., q = − k +1)2 which is consistent with thebarotropic fluid. Using these values in Eq.(10), we obtain φ ( B ) = ln( φ B ± r k )(1+2 m ) U ω ) , where φ is a non-zero integration constant. This shows that the constantcoupling of geometry and scalar field, i.e., U = U for the barotropic fluidleads to the logarithmic form of scalar field which further corresponds toexpanding or contracting scalar field versus scale factor B on the basis ofsign. Consequently, the scale factors turn out to be A ( φ ) = ( exp( φ ) φ ) ∓ m r k )(1+2 m ) U ω , B ( φ ) = ( exp( φ ) φ ) ∓ r k )(1+2 m ) U ω . We see that the scale factors are of exponential form which indicate rapidcosmic expansion for the expanding scalar field. The corresponding fieldpotential is V ( B ) = [2(1 + 2 m ) U − (1 + k )(1 + 2 m ) U ]( m + m + 41 + 2 m + 2 k ) × ρ (1 + k )( m + 2)( m + 3) B − (1+ k )( m +2) . (19)This is of power law nature and indicates inverse power law behavior for m > < k < ω ( φ ), we consider the ansatz ω ( φ ) = ω φ n ; n > φ ( B ) = [ c (ln( B ) n + 4 ln( B ) n − B ) n c − B ) nc + 4 ln( B ) − B ) c + c n + 4 nc + 4 c )] / ( n +2) (2 n +2 ) − , where c is an integration constant and c = m )(1+ k ) U ω . For the sake ofsimplicity, we take c = 0 and hence the scalar field becomes φ ( B ) = c / ( n +2)2 (ln( B n +8 n +8 )) / ( n +2) (1 / ( n +2)) . Thus the scale factors in exponential form are A ( φ ) = exp( 4 m (2 n + 8 n + 8) c − φ n +2 ) , B ( φ ) = exp( 42 n + 8 n + 8 c − φ n +2 ) . V ( B ) = [2(1 + 2 m ) U − ω c / ( n +2)2 ln( B n +8 n + n )2 (1 / ( n +2)) ) n c / ( n +2)2 (2 / ( n +2) ) − × (2 n + 8 n + 8) ( n + 2) (ln( B n +8 n + n )) − (1+ n ) n +2 ]( m + m + 41 + 2 m + 2 k ) × ρ (1 + k )( m + 2)( m + 3) B − (1+ k )( m +2) , (20)which contains the product of inverse power law and logarithmic functionsof the scale factor.For U = φ , Eq.(12) takes the form φ ′′ + ( m (1 − m ) m + 2 − (1 + k )( m + 2)2 ) φ ′ B + ω φ ′ − (1 + 2 m )(1 + k ) φB = 0 . When ω = ω or ω ( φ ) = ω φ n , the solution to this differential equation isquite complicated and cannot provide much insights. However, if we take m = − / ω = ω , then this leads to φ ( B ) = 2 ω ln[ ω (4 c B / k ) + 9 c + 3 c k )3 + k ] , (21)where c and c are integration constants and ω = 0. The respective scalefactors are A ( φ ) = [ 14 c ( 6 ω exp( ω φ ) − c − c k )] m/ (3 / k )) ,B ( φ ) = [ 14 c ( 6 ω exp( ω φ ) − c − c k )] / (3 / k )) and the corresponding scalar field potential turns out to be V ( B ) = 2( m + 2)( 3 c (3 + k ) B / k ) ω (4 c B / k ) + 9 c + 3 c k ) ) − ω c (3 + k ) B / k ) ω (4 c B / k ) + 9 c + 3 c k ) ) . (22)We can conclude that the scalar field is described by logarithmic functionand the scale factors are of exponential nature which yields expansion forincreasing scalar field while the potential turns out to be of power law nature.10ow we discuss the induced gravity case and evaluate the function f byusing the Hubble parameter and its evolution in Eq.(15) which leads to f ′′ + [ m (1 − m ) m + 2 − (1 + k )( m + 2)2 ] f ′ B − ω + 4 γ γ (1 + 2 m )(1 + k ) fB = 0whose solution is f ( B ) = c B r + c B r ; r , = 1 − c ± q c + 1 − c − c , (23)where c and c are arbitrary constants while c and c are given by c = − m + (3 + k ) m + 4 + 4(1 + m ) k m + 2) , c = − (1 + 2 m )(1 + k ) ω + 4 γ γ . The corresponding scalar field is φ ( B ) = ( c B r + c B r ) γω γ which is clearlyof power law nature. Since it is difficult to invert this expression for the scalefactor B in terms of φ , so we take either c = 0 or c = 0, which leads toeither A ( φ ) = 1 c m φ m ( ω γ )4 r γ , B ( φ ) = 1 c φ ω γ r γ , or A ( φ ) = 1 c m φ m ( ω γ )4 r γ , B ( φ ) = 1 c φ ω γ r γ . We see that the scale factors are also of power law nature and show ex-panding or contracting behavior depending upon the values of the involvedparameters. The scalar field potential (8) then turns out to be V ( B ) = [(1 + 2 m ) γ + 4 γ ( m + 2) r , ω + 4 γ − ω γ r , ( ω + 4 γ ) ]( m + m + 41 + 2 m + 2 k ) × ( ρ c γω γ , (1 + k )( m + 2) ) B γr , ω γ − (1+ k )( m +2) . (24)This may be of positive or inverse power law nature depending upon thevalues of parameters.For variable ω , the analytical solution of Eq.(15) is not possible. How-ever, the corresponding numerical solution can be found by using the initial11igure 1: Plots show the field potential versus scale factor B . Plots (a), (b),(c) and (d) correspond to the field potentials given by Eqs.(19), (20), (22)and (24), respectively. Here m = 2 , ρ = 1 , U = 3 , k = 0 . ω = 0 . m = − . f (1) = 0 .
67 and f ′ (1) = 1 .
95 and is given by the polynomialinterpolation f ( B ) = 0 . B − . B + 6 . B − . B + 206 . B − . B + 772 . B − . B + 180 . , (25)where we have taken m = 2 , γ = 0 . , k = 0 . ω = 0 . φ . Thecorresponding scalar field is φ ( B ) = ( f ( B )) γω γ , yielding the form of thefield potential in polynomial form which represents positive power law nature.Here the scalar field is in polynomial form which cannot be inverted for scalefactor B .We have plotted the potentials given by Eqs.(19), (20), (22) and (24)versus scale factor B as shown in Figure . It is found that in all cases, thescalar field potentials are positive decreasing functions except for the plot(c) which has a signature flip from positive to negative with the increasein scale factor (this graph corresponds to the negative value of m ). We canconclude that for a positive behavior of the field potential (which is physicallyacceptable), we should take positive range of m . In this case, we take p = − ρ and hence the energy density becomes a con-stant, i.e., ρ = ρ . The corresponding directional Hubble parameters and itsevolution are given by H ( B ) m = H ( B ) = r ρ m + 3 (1 − m + m + 41 + 2 m ) B, H ′ ( B ) H ( B ) = 12 B ln( B ) . (26)The deceleration parameter turns out to be a dynamical quantity q = − (1 + m +2) ln( B ) ). It is interesting to mention here that in our case, the directionalHubble parameters are dependent on the scale factor B (due to anisotropy)whereas in the case of FRW universe, the Hubble parameter is independentof the scale factor, i.e., it turns out to be constant. We use these valuesin the previously discussed three cases, i.e., U = U , φ and U = γφ .Equation (10) provides ( φ ′ ) = (1+2 m ) U ω ( m +2) 1 B ln( B ) whose integration leads to φ ( B ) = ± p − B ) c + c , where c is an integration constant while c = m ) U ω ( m +2) . This leads to the scale factor as an exponential function of the13calar field B ( φ ) = exp( − / c ( φ − c )). Likewise, for ω = ω φ n , the scalarfield is found to be φ ( B ) = (2 − / ( n − ) [ ± p − B ) c + c ( n − B ) c + c ) ] , (27)where c is an integration constant while c is the same as above. Usingthese values in Eq.(8), the field potential can be determined which wouldinclude the product terms of scale factor and logarithmic function.In the case of simple BD gravity, Eq.(12) is not easy to solve for bothcases ω = ω and ω = ω φ n . However, the corresponding numerical solutionscan be constructed in a similar way as we have discussed in the previous case.The scalar field as well as the potentials constructed, in this way, would beof polynomial nature. For m = − /
2, it leads to φ ′′ + ω φ ′ = 0 and hence φ ( B ) = 2 ln( c Bω + c ω ) ω , B ( φ ) = 2 c ω (exp( ω φ/ − c ω , where c and c are integration constants. The field potential correspondingto these values can be obtained from Eq.(8) which would be of power lawnature. For the case of induced gravity, Eq.(15) provides f ′′ + f ′ B ln B + m (1 − m ) m + 2 f ′ B + ( ω + 4 γ )4 γ (1 + 2 m )( m + 2) fB ( 12 B ln( B ) ) = 0 . (28)Solving this equation, we have the solution in terms of Kummer functions f ( B ) = c KummerM ( 14 ( − m (1 − m ) γ + (2 ω + 9 γ ) m + 6 γ + ω ( m + 2)+ 3( m + 2) γ ( m + 2 − m (1 − m )))(( m + 2) γ ( m + 2 − m (1 − m ))) − , / , ( − m − m (1 − m ))( m + 2) ln( B ) p ln( B ) B / (2( m +2)( m +2 − m (1 − m )))( m +2)2 )+ c KummerU ( 14 ( − m (1 − m ) γ + (2 ω + 9 γ ) m + 6 γ + ω ( m + 2)+ 3( m + 2) γ ( m + 2 − m (1 − m )))(( m + 2) γ ( m + 2 − m (1 − m ))) − , / , ( − m − m (1 − m ))( m + 2) ln( B ) p ln( B ) B / (2( m +2)( m +2 − m (1 − m )))( m +2)2 ) , (29)where c and c are integration constants. Since φ = f γ/ ( ω +4 γ ) , conse-quently the scalar field potential can be determined (it would be a lengthy14xpression in Kummer function). For ω = − γ , the solution is f ( B ) = c + ( Z B − m (1 − m ) /m +2 p ln( B ) dB ) c , (30)where c and c are integration constants. The corresponding potential canbe determined by using the value of the scalar field φ = f γω γ in Eq.(8). Itwould include the integral term and hence cannot be categorized as powerlaw, exponential or logarithmic form. Finally, we consider the Chaplygin gas EoS as DE candidate which is definedby p = − Cρ , where C is some positive constant. In order to discuss the po-tential, we use the above EoS parameter in the energy conservation equationand then integration leads to ρ ( B ) = ( C + c B − m +2) ) / , where c is anintegration constant. Using this value in Eq.(17), it follows that H ( B ) = 4 C / m + 3 (1 − m + m + 41 + 2 m ) ln( B ) + c ( m + 2)( m + 3) C / × (1 + m + m + 42(1 + 2 m ) ) B − m +2) , (31)whose evolution yields H ′ H = p − m + 2) p B − m +2) B ( p ln( B ) + p B − m +2) ) , (32)where p = C / m +3 (1 − m + m +41+2 m ) and p = c ( m +2)( m +3) √ C (1 + m + m +41+2 m ). For theconstant coupling of scalar field and geometry ( U = U ) with ω = ω , wehave φ ( B ) = Z ±√ ω ( m + 2)( B − m ) p + B p ln( B )) U (1 + 2 m )( − p + 2 mp B − m +2) + 4 p B − m +2) )(( B − m +2) p + B p ln( B )) × ω ( m + 2)) − ) / . Thus we can determine the field potential that can generate the cosmic evo-lution of Chaplygin gas matter (it would be in integral form). For ω = ω φ n ,15he scalar field is2 φ ( B ) ( n +2) / n + 2 + Z [( ω ( m + 2)( p + ln( B ′ ) B ′ m +4 p )) − ( φ ( B ) n/ B ′ m ) × ( − U ω (2 m + 5 m + 2) φ ( B ) − n ( − B ′ − mp m − p ln( B ′ ) mp B ′ − m + B ′ p ln( B ′ ) − p ln( B ′ ) p B ′ − m + B ′ − m p p − B ′− m p )) / ) B ′− ] = 0 . Clearly, it is not possible to have an explicit expression for scalar field in termsof scale factor B and hence the form of the respective field potential cannotbe determined. For simple BD gravity with ω = ω and ω = ω φ n , we couldnot find analytical solutions but numerical solutions can be constructed in asimilar pattern as we have discussed earlier. For induced gravity, analyticalsolution is only possible if we take p = 0, which further implies the samecases as we have found in the cosmological constant case (as H ′ H = B ln( B ) ). This paper investigates scalar field potentials by a new technique known asthe reconstruction technique for the field potentials. We have applied thistechnique to BI universe model in the context of general scalar-tensor theory.The general form of the field potential without assigning any values of
U, V and H has been explored. We have also discussed two particular cases of U ,i.e., when it is a constant and U = U ( φ ). In both cases, the field potentialdepends upon the scale factor B , the scalar field and the directional Hubbleparameter H . Further, we have taken two cases for ω , i.e., ω = ω and ω = ω φ n . It is found that an explicit form of the field potential cannot befound in terms of scale factor unless we choose some particular value of theHubble parameter. For this purpose, we have taken the evolution of Hubbleparameter motivated by the barotropic fluid, the cosmological constant andthe Chaplygin gas matter contents. In literature [33, 38], four types of scalarfield potentials have usually been discussed, i.e., the positive and inversepower laws, the exponential and the logarithmic potentials while other formsare multiple of these four types.For the barotropic fluid, the potential can be found but it is not possiblefor the simple BD gravity. We have also observed that for constant U , thescalar fields are logarithmic functions for both ω = ω and ω = ω φ n , whilethe scale factors are of exponential nature. Also, for simple BD gravity with16 = − . ω = ω , the scale factors are exponential functions whilefor the induced gravity, they turn out to be of power law form. In orderto examine their behavior, we have plotted the field potentials versus scalefactor B as shown in Figure . It is concluded that the field potentials arepositive and decrease to zero except for the case of simple BD gravity wherewe have taken negative value of m . We may conclude that for positive fieldpotential, we should impose the condition m >
0. We have also discussed anumerical approach (polynomial interpolation) for the cases where no ana-lytical solution exists. Likewise, for the cosmological constant candidate ofDE with constant coupling function U , we can determine the form of the fieldpotential without taking any condition for both ω , however in other cases,we have to impose some certain conditions.In the case of Chaplygin gas matter contents, the scalar field potential canbe discussed only for ω = ω with U = U . However, in other cases, eitherthe explicit analytical solution is not possible or we have the same expressionof the field potential as in the case of cosmological constant. It would beworthwhile to investigate the form of the field potential for the exponentialform of coupling function of scalar field and geometry. This procedure maylead to some interesting results when the chameleon mechanism is taken intoaccount in the framework of scalar-tensor gravity. References [1] Riess, A.G. et al.: Astrophys. J. 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