Early universe in view of a modified theory of gravity
Ranajit Mandal, Dalia Saha, Mohosin Alam, Abhik Kumar Sanyal
aa r X i v : . [ h e p - t h ] J a n Early universe in view of a modified theory of gravity
Ranajit Mandal ∗ , Dalia Saha † , Mohosin Alam ‡ , Abhik Kumar Sanyal § January 11, 2021 ∗ Dept. of Physics, Rammohan College, Kolkata, West Bengal, India - 700009. † , § Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213 ‡ Dept. of Physics, Saidpur U. N. H. S., Murshidabad, West Bengal, India - 742225.
Abstract
We study the quantum evolution of the early universe, its semi-classical analogue together with inflationaryregime, in view of a generalized modified theory of gravity. The action is built by supplementing the non-minimally coupled scalar-tensor theory of gravity with scalar curvature squared term and a Gauss-Bonnet-dilatonic coupled term. It is generalized, since all the parameters are treated as arbitrary functions of thescalar field. It is interesting to explore the fact that instead of considering additional flow parameters, aneffective potential serves the purpose of finding inflationary parameters. The dilaton stabilization issue appearshere as a problem with reheating. Addition of a cosmological constant term alleviates the problem, and inflationis effectively driven by the vacuum energy density. Thus Gauss-Bonnet term might play a significant role indescribing late-time cosmic evolution.
It is well known fact that the ‘standard model of cosmology’ based on General theory of relativity (GTR) explainsa long evolution history of the universe, right from the structure formation, and the formation of CMBR (at aredshift z ≈ z ≈ a ∝ t prior to the recentaccelerated expansion. These are also true for other extended varieties of GR models, viz. the quintessence andeven more exotic models, with single scalar field. On the contrary, Higher-order curvature induced actions canaddress these issues quite comfortably [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Thus, GTR is required to be modified at bothends, the early and the late stages of cosmic evolution. In the present manuscript, we focus our attention to thevery early stage of cosmic evolution. In this context, it has been realized long back that GTR should be replacedby a viable quantum theory of gravity. However, despite serious attempts over several decades, a renormalizablequantum theory of gravity is still not at hand. Nevertheless, all attempts in this direction lead to the weak fieldeffective actions in 4 -dimension carrying different higher-order curvature invariant terms. In the absence of acomplete and viable quantum theory of gravity, it is necessarily required to test the viability of different modifiedactions in the context of early stage of cosmic evolution, and to get certain insights regarding the behaviour ofour universe near Planck’s epoch.Principle candidates of a reasonable renormalized theory of gravitational action, apart from the linearsector (Einstein-Hilbert term), are curvature squared terms, viz., R and R µν R µν [11]. Note that thereis no need to include R αβµν R αβµν , since it combines with the other two, to form the Gauss-Bonnet term, ∗ E-mail:[email protected] † E-mail:[email protected] ‡ E-mail:[email protected] § E-mail:sanyal [email protected] = R − R µν R µν + R αβµν R αβµν , which is topologically invariant in four dimensions. However, as the theoryis expanded in the perturbative series about linearized gravity, ghost degrees of freedom appear, which destroysthe unitarity. In fact, in the absence of R µν R µν term, ghosts disappear, but ultraviolet divergence reappears,rendering the theory non-renormalizable. It was therefore attempted to construct a second derivative theory withhigher order curvature invariant terms for the purpose, which ended up in Lanczos-Lovelock gravity [12, 13].Nevertheless, Lanczos-Lovelock gravity leads to second order field equations with different combinations of higher-order terms including the Gauss-Bonnet combination, which contributes only in dimensions greater than 4 . Thisis because, as already mentioned, Gauss-Bonnet term is topologically invariant and thus does not contribute to thefield equations in four dimensions. Likewise, the other different Lanczos-Lovelock combinations behave similarlyat different dimensions. However, if we just concentrate upon Gauss-Bonnet term, it is important to mention that,this combination also arises naturally as the leading order of the α ′ expansion of heterotic superstring theory,where, α ′ is the inverse string tension [14, 15, 16, 17]. Further, the low energy limit of the string theory gives riseto the dilatonic scalar field which is found to be coupled with various curvature invariant terms [18, 19]. Thereforethe leading quadratic correction gives rise to Gauss-Bonnet term with a dilatonic coupling [20]. Another veryimportant aspect of considering such a term is: it provides nothing beyond second order field equations, andtherefore appears to be free from ghost degrees of freedom.Despite the fact that Lanczos-Lovelock gravity shows unitary time evolution of quantum states, when expandedperturbatively about the flat Minkowski background, nonetheless, nonperturbatively the situation is miserable. Infact, canonical analysis of 5-dimensional Lovelock action under 4 + 1 decomposition, makes the theory intrinsicallynonlinear [21]. Even its linearized version is cubic rather than quadratic. As a result, diffeomorphic invariance isnot manifest and standard canonical formulation of the theory is not possible [22]. Such a situation arises due tothe appearance of terms quartic in velocities in the Lagrangian. Hence, the expression for velocities is multi-valuedfunctions of momenta, resulting in the so called multiply branched Hamiltonian with cusps. This makes classicalsolution unpredictable, as at any time one can jump from one branch of the Hamiltonian to the other. Further, themomentum does not provide a complete set of commuting observable, which result in non-unitary time evolutionof quantum states. It has also been shown that the Einstein-Gauss-Bonnet-dilatonic action suffers from the samedisease of the so-called ‘Branched Hamiltonian’, and presently, there is no standard technique to handle this issue[22, 23].Under such critical situation, it has been shown that the presence of scalar curvature squared term ( R ) inthe action, bypasses the pathology of Branched Hamiltonian appearing in Lanczos-Lovelock action [22, 23]. Thus,in view of the above discussions, we find it worth studying the cosmic evolution of the early universe in view ofthe following action, S c = Z d x √− g (cid:20) α ( φ ) R − M P Λ + β ( φ ) R + γ ( φ ) G − φ ,µ φ ,µ − V ( φ ) (cid:21) . (1)In the above, Gauss-Bonnet combination ( G = R − R µν R µν + R µνρσ R µνρσ ) is coupled with a dilatonic field,in view of the string theory or an ordinary scalar field in view of the Lanczos-Lovelock combination, through anarbitrary coupling parameter γ ( φ ), while the other two coupling parameters α ( φ ) and β ( φ ) are also arbitraryfunction of φ , and V ( φ ) is an arbitrary potential function. The action (1) generalizes our earlier one appearingin [23], in respect of considering non-minimal coupling in view of α ( φ ) term, and taking independent couplingfunctions β ( φ ) and γ ( φ ) into account. An additional cosmological constant term Λ appearing with appropriatedimension ( M P = (8 πG ) − ) generalizes the action even further. The advantage of considering such an apparentlycomplicated action with non-minimal coupling parameters α ( φ ), and β ( φ ) is: it encompasses different casesstarting from α ( φ ) = constant = M P and β ( φ ) = constant, and also the cases in which either of the two is aconstant. In fact, we have shown that such different choices lead to the de-Sitter solution of the classical fieldequations with identical form of potential in all the cases studied. Further, it has also been shown that all thedifferent cases lead to the same inflationary parameters. The above action could have been further general-ized by incorporating Riemann squared term ( R αβγδ R αβγδ ) with an additional coupling parameter. However,it becomes extremely difficult to handle the field equations. We shall try to pose such a generalized action in future.It is now generally believed that the early universe must have passed through an inflationary era, since such aregime not only solves the horizon and flatness problems but also gives birth to the seeds of perturbations requiredfor structure formation. Thus, inflation should be considered as a scenario rather than a model. To study how the2resent model (1) leads to a viable inflationary regime, canonical formulation of the action under considerationshould be performed as a precursor, which is the most strong mathematical tool within gravitational field, bothfrom theoretical as well as observational aspects. Canonical analysis is presented here as a mere prologue tocanonical quantization scheme.In the following section, we write the general field equations and then translate them in the isotropic and homo-geneous Robertson-Walker minisuperspace model. Classical de-Sitter solutions are thereby presented. In section3, we perform canonical formulation, which is a non-trivial task for the higher order theory under consideration.Thereafter, standard canonical quantization scheme is followed to establish hermiticity of the effective Hamiltonianoperator. The viability of the quantum equation is tested under an appropriate semiclassical approximation. Insection 4, slow-roll approximation is performed and the results are compared with recently released data sets.Finally concluding remarks are presented in section 5. It has recently been established [24] that Modified Horowitz’ formalism (MHF) and the Dirac constraint analysistowards canonical formulation of higher-order theory of gravity lead to the same Hamiltonian, provided Diracalgorithm is initiated only after taking care of the divergent terms appearing in the action. In fact, it has beenproved that otherwise, although the two Hamiltonians are canonically related, they produce completely differentquantum descriptions [24]. We shall here follow the MHF which bypasses Dirac’s algorithm even in the presenceof the Lapse function N ( t ) [22, 23, 24, 25, 26, 27, 28, 29], that essentially acts as a Lagrange multiplier. Dirac’stechnique is explored in the appendix, to exhibit equivalence. For the purpose of carrying out MHF, the action(1) is required to be supplemented by appropriate boundary terms, viz., S c = Z d x √− g (cid:20) α ( φ ) R − Λ M P + β ( φ ) R + γ ( φ ) G − φ ,µ φ ,µ − V ( φ ) (cid:21) + α ( φ )Σ R + β ( φ )Σ R + γ ( φ )Σ G . (2)In the above, α ( φ )Σ R = 2 α ( φ ) H ∂ν K √ hd x ; β ( φ )Σ R = 4 β ( φ ) H ∂ν RK √ hd x = β ( φ )(Σ R +Σ R ) = 4 β ( φ ) H ∂ν [ R +( R − R )] RK √ hd x ; γ ( φ )Σ G = 4 γ ( φ ) H ∂ν (cid:0) G ij K ij + K (cid:1) √ hd x are the supplementary boundary terms whichare modified versions of Gibbons-Hawkings-York term [30, 31, 32], while K = K − KK ij K ij + 2 K ij K ik K kj , K ij being the extrinsic curvature tensor, h is the determinant of the induced metric h ij , and K = K ij h ij . Now, thefield equation is found under the standard metric variation as [33],2 (cid:16) α ( φ ) G µν + (cid:3) α ( φ ) g µν − ∇ µ ∇ ν α ( φ ) (cid:17) + 4 h β ( φ ) RR µν − g µν (cid:3) ( β ( φ ) R ) − ∇ µ ∇ ν ( β ( φ ) R ) − g µν β ( φ ) R i + 2 γ ( φ ) H µν + 8( γ ′′ ∇ ρ φ ∇ σ φ + γ ′ ∇ ρ ∇ σ φ ) P µρνσ − Λ M P = T ( φ ) µν , (3)where, G µν = R µν − g µν R and T ( φ ) µν = ∇ µ φ ∇ ν φ − g µν ∇ λ φ ∇ λ φ − g µν V ( φ ) are the Einstein tensor and theenergy-momentum tensor respectively. Further, H µν = 2 (cid:0) RR µν − R µρ R ρν − R µρνσ R ρσ + R µρσλ R σρλ − g µν G (cid:1) and P µνρσ = R µνρσ +2 g µ [ σ R ρ ] ν +2 g ν [ ρ R σ ] µ + Rg µ [ ρ g σ ] ν . Explicit form of equation (3) together with the φ variationequation are expressed as, αG µν + (cid:3) αg µν − α ; µ ; ν + h β (cid:0) RR µν − g µν R (cid:1) − (cid:0) (cid:3) ( βR ) g µν + ( βR ) ; µ ; ν (cid:1)i + 2 γ h RR µν − R µρ R ρν − R µρνσ R ρσ + R µρσλ R σρλν − g µν (cid:0) R − R µν R µν + R µνδγ R µνδγ (cid:1) i + 4 ( γ ′′ φ ; ρ φ ; σ + γ ′ φ ; ρ ; σ ) (cid:2) R µρνσ + 2 g µ [ σ R ρ ] ν + 2 g ν [ ρ R σ ] µ + Rg µ [ ρ g σ ] ν (cid:3) − Λ M P T µν (cid:3) φ − α ′ R − β ′ R − γ ′ G − V ′ = 0 , (4)where, prime denotes derivative with respect to φ . In the homogeneous and isotropic Robertson-Walker metric,viz., ds = − N ( t ) dt + a ( t ) h dr − kr + r ( dθ + sin θ dφ ) i , (5)3he Ricci scalar as well as the Gauss-Bonnet terms read as, R = 6 N (cid:16) ¨ aa + ˙ a a + N ka − ˙ a ˙ NaN (cid:17) ; G = 24 N a ( N ¨ a − ˙ N ˙ a ) (cid:18) ˙ a N + k (cid:19) . (6)The a variation i.e. ( ii ) equation, the N variation i.e. the ( ) equation and the φ variation equation are asfollows,2 α (cid:20) aa + ˙ a a + ka (cid:21) + 2 α ′ (cid:20) ¨ φ + 2 ˙ a ˙ φa (cid:21) + 2 α ′′ ˙ φ + 12 β (cid:20) aa + 4 ˙ a ... aa + 3¨ a a −
12 ˙ a ¨ aa + 3 ˙ a a − k ¨ aa + 2 k ˙ a a − k a (cid:21) + 48 β ′ ˙ φ (cid:20) ... aa + 2 ˙ a ¨ aa − ˙ a a − k ˙ aa (cid:21) + (24 β ′′ ˙ φ + 24 β ′ ¨ φ ) (cid:20) ¨ aa + ˙ a a + ka (cid:21) + 16 γ ′ ¨ a ˙ a ˙ φa + 8 γ ′ ¨ φ (cid:20) ˙ a a + ka (cid:21) + 8 γ ′′ ˙ φ (cid:20) ˙ a a + ka (cid:21) + ˙ φ − V − Λ M P = 0 . (7)6 α (cid:18) ˙ a a + ka (cid:19) + 6 α ′ ˙ φ (cid:16) ˙ aa (cid:17) + 36 β (cid:18) a ... aa − ¨ a a + 2 ˙ a ¨ aa − a a − k ˙ a a + k a (cid:19) + 72 β ′ ˙ φ (cid:18) ˙ a ¨ aa + ˙ a a + k ˙ aa (cid:19) + 24 γ ′ ˙ φ (cid:18) ˙ a a + k ˙ aa (cid:19) − Λ M P = (cid:18) ˙ φ V (cid:19) , (8)¨ φ + 3 ˙ aa ˙ φ + V ′ − α ′ (cid:16) ¨ aa + ˙ a a + ka (cid:17) − β ′ (cid:16) ¨ a a + 2 ˙ a ¨ aa + ˙ a a + 2 k ¨ aa + 2 k ˙ a a + k a (cid:17) − γ ′ (cid:16) ˙ a ¨ aa + k ¨ aa (cid:17) = 0 . (9)Not all the above components of Einstein’s equations are independent, e.g. the ( ) equation is the energy constraintequation. Thus it is suffice to consider only the two independent components of Einstein’s equations viz. (8) and(9), for all practical purpose. A viable gravity theory must admit de-Sitter solution in vacuum. It is not difficultto see that the above field equations also admit the following de-Sitter solutions, a = a e λt ; φ = φ e − λt , (10)in the spatially flat space ( k = 0 ), under the following different conditions, Case 1: α ( φ ) = α φ ; V ( φ ) = 12 λ φ − Λ M P ; and 6 β + γ = − λ (cid:18) α φ + φ (cid:19) , (11)where, a , φ , α and λ are arbitrary constants while β ( φ ) and γ ( φ ) retain arbitrary forms of φ , being relatedas above after setting the constant of integration to zero, without any loss of generality. Since we shall encounteroperator ordering ambiguities during canonical quantization, therefore we remove the arbitrariness on β ( φ ) and γ ( φ ), following a simple assumption viz, Subcase 1a: β = − α λ φ = β φ ; γ = − φ λ = γ φ where , β = − α λ ; γ = − λ . (12)The above forms of α ( φ ) (11), β ( φ ) and γ ( φ ) (12) will be used during canonical quantization (for operatorordering, in particular), semiclassical approximation, as well as during slow roll approximation. Subcase 1b: β = − φ λ = β φ ; γ = − α λ φ = γ φ where , β = − λ ; γ = − α λ . (13)4 ase 2: If β = β = constant, then the above de-Sitter solution (10) is admissible under the same choice of the potential,under the condition, α ( φ ) = α φ ; V ( φ ) = 12 λ φ − Λ M P ; and γ = − λ (cid:18) α φ + φ (cid:19) . (14) Case 3: If α = α = constant, even then the same de-Sitter solution is admissible under the condition, V ( φ ) = 12 λ φ − Λ M P + 6 λ α ; and 6 β + γ = − (cid:18) φ λ (cid:19) . (15)It is to be noted that the additional term appearing in the potential being a constant may be absorbed in thepotential, keeping its form unaltered, as well. Case 4: If α = α = constant, and β = β = constant, then, V ( φ ) = 12 λ φ − Λ M P + 6 λ α ; and γ = − φ λ . (16)Here again the form of the potential only differs from the earlier ones only by a constant term as in case 3. In this section, our aim is to establish the canonical structure of action (2) in the Robertson-Walker minisuperspacemodel (5). This is possible once we can write it in terms of the basic variables h ij = a δ ij = zδ ij , and k ij = − ˙ h ij N = − a ˙ aN δ ij = ˙ z N δ ij , where, a = z . For this purpose, we first express the action (2) in terms of h ij = a δ ij = zδ ij using the form of the Ricci scalar and the Gauss-Bonnet term (6) as, S c = Z (cid:20) α ( φ ) √ z (cid:18) ¨ zN − ˙ z ˙ NN + 2 kN (cid:19) − N z Λ M P + 9 β ( φ ) √ z (cid:18) ¨ z N − z ¨ z ˙ NN + ˙ z ˙ N N − k ˙ z ˙ NN + 4 k ¨ zN + 4 k N (cid:19) + 3 γ ( φ ) N √ z (cid:18) ˙ z ¨ zN z − ˙ z N z − ˙ z ˙ NN z + 4 k ¨ z − k ˙ z z − k ˙ z ˙ NN (cid:19) + z (cid:18) N ˙ φ − V ( φ ) N (cid:19)(cid:21) dt + α ( φ )Σ R + β ( φ )Σ R + γ ( φ )Σ G . (17)In the above, α ( φ )Σ R = − α ( φ ) √ z ˙ zN , β ( φ )Σ R = − β ( φ ) k ˙ zN √ z , β ( φ )Σ R = − β ( φ )
18 ˙ zN √ z ( ¨ z − ˙ z ˙ NN ) and γ ( φ )Σ G = − γ ( φ ) ˙ zN √ z ( ˙ z N z + 12 k ) are the supplementary surface terms as already mentioned, but now in the isotropic andhomogeneous Robertson-Walker metric (5). It is important to mention that unlike GTR, here the lapse functionappears in the action with its time derivative, behaving like a true variable. Despite such uncanny situation, onecan still bypass Dirac’s algorithm as we demonstrate below.First, under integrating the above action (17) by parts, the counter terms α ( φ )Σ R , β ( φ )Σ R and γ ( φ )Σ G getcancelled with the divergent (total derivative) terms and we arrive at, S c = Z (cid:20)(cid:18) − α ′ ˙ φ ˙ z √ zN − α ˙ z N √ z + 6 kN α √ z (cid:19) − N z Λ M P + 9 β √ z (cid:18) ¨ z N − z ¨ z ˙ NN + ˙ z ˙ N N + 2 k ˙ z N z + 4 k N (cid:19) − β ′ k ˙ z ˙ φN √ z − γ ′ ˙ z ˙ φN √ z (cid:18) ˙ z N z + 12 k (cid:19) + z (cid:18) N ˙ φ − V N (cid:19)(cid:21) dt + β ( φ )Σ R . (18)5t this stage we construct an auxiliary variable Q under the following prescription, Q = ∂S c ∂ ¨ z = 18 βN √ z (cid:18) ¨ z − ˙ N ˙ zN (cid:19) , (19)and substitute it judiciously straight into the action (18), as a result of which it now reads as, S c = Z (cid:20)(cid:18) − α ′ ˙ φ ˙ z √ zN − α ˙ z N √ z + 6 kN α √ z (cid:19) − N z Λ M P + Q ¨ z − N √ zQ β − ˙ N ˙ zQN + 18 βk ˙ z N z + 36 βN k √ z − β ′ k ˙ φ ˙ zN √ z − γ ′ ˙ z ˙ φN √ z (cid:18) ˙ z N z + 12 k (cid:19) + z (cid:18) N ˙ φ − V N (cid:19)(cid:21) dt + β ( φ )Σ R . (20)Now, further integrating (20) by parts, the rest of the surface terms viz. β ( φ )Σ R gets cancelled with the totalderivative term yet again, and thus the action (17) being free of all the divergent terms appearing under variationis finally expressed as, S c = Z (cid:20)(cid:18) − α ′ ˙ φ ˙ z √ zN − α ˙ z N √ z + 6 kN α √ z (cid:19) − N z Λ M P − ˙ Q ˙ z − N √ zQ β − ˙ N ˙ zQN + 18 βk ˙ z N z + 36 βN k √ z − β ′ k ˙ φ ˙ zN √ z − γ ′ ˙ z ˙ φN √ z (cid:18) ˙ z N z + 12 k (cid:19) + z (cid:18) N ˙ φ − V N (cid:19)(cid:21) dt. (21)The canonical momenta now read as, p Q = − ˙ z, (22) p z = − α ′ ˙ φ √ zN − α ˙ zN √ z − ˙ Q − ˙ N QN + 36 βk ˙ zN z − β ′ k ˙ φN √ z − γ ′ ˙ φN √ z (cid:18) ˙ z N z + 4 k (cid:19) , (23) p φ = − α ′ ˙ z √ zN − β ′ k ˙ zN √ z − γ ′ ˙ zN √ z (cid:18) ˙ z N z + 12 k (cid:19) + z ˙ φN , (24) p N = − Q ˙ zN , (25)and thus one can now, in principle, construct the phase-space structure of the Hamiltonian. However, the aboverelations signal the present of a momentum constraint in the form Qp Q + N p N = 0 . Nevertheless, one can stillavoid Dirac’s algorithm by constructing the following relation in view of the above momenta as, p Q p z = 3 α ′ ˙ z ˙ φ √ zN + 3 α ˙ z N √ z + ˙ z ˙ Q + ˙ N ˙ zQN − kβ ˙ z N z + 36 β ′ k ˙ z ˙ φN √ z + 3 γ ′ ˙ z ˙ φN √ z (cid:18) ˙ z N z + 4 k (cid:19) . (26)Using the above relation (26) and the definitions of momenta p Q , p z , p φ , the phase-space structure of the Hamil-tonian constraint equation may be expressed as , H c = 3 α (cid:18) p Q N √ z − kN √ z (cid:19) − α ′ p Q p φ z + 9 α ′ p Q N √ z − p Q p z + N Q √ z β − kβ √ z (cid:18) p Q N z + 2 kN (cid:19) + γ ′ p Q N z + 648 k β ′ p Q N z + 72 k γ ′ p Q N z − kβ ′ p Q p φ z + 12 kγ ′ p Q N z − γ ′ p Q p φ N z − kγ ′ p Q p φ z + 36 kβ ′ γ ′ p Q N z + 432 k β ′ γ ′ p Q N z + 108 kα ′ β ′ p Q N z + 3 α ′ γ ′ p Q N z + 36 kα ′ γ ′ p Q N z + N p φ z + N V z + N z Λ M P = 0 . (27) Instead of Q if the auxiliary variable would have been chosen as q = NQ , then ˙ N would not have appeared in the action,and essentially it would have been free from the constraint mentioned above. This is mentioned just to appreciate the fact that thecomplication appears here is an artefact of a bad choice of co-ordinate. p Q upto sixth order, which does not allow to obtain a viable quantum counterpart understandard canonical quantization scheme, since, this is not at all convenient in order to form the operators. Evenif one does, a large number of initial (boundary) conditions are required to solve the quantum equation, whichare not available. This envisages the fundamental importance of basic variable required for quantization, whichare ( h ij , K ij ). Therefore, we need to make a canonical transformation from { Q, p Q } to { x, p x } , where, N x = ˙ z ,which is performed following the prescription Q = p x N and p Q = − N x . Therefore, the phase-space structure ofthe Hamiltonian can be expressed in terms of basic variables as, H c = N (cid:20) xp z + 3 α (cid:18) x √ z − k √ z (cid:19) + 3 α ′ xp φ z + 9 α ′ x √ z + √ zp x β − kβ √ z (cid:18) x z + 2 k (cid:19) + γ ′ x z + 648 k β ′ x z + 72 k γ ′ x z + 36 kβ ′ xp φ z + 12 kγ ′ x z + γ ′ x p φ z + 12 kγ ′ xp φ z + 36 kβ ′ γ ′ x z + 432 k β ′ γ ′ x z + p φ z + 108 kα ′ β ′ x z + 3 α ′ γ ′ x z + 36 kα ′ γ ′ x z + V z + z Λ M P (cid:21) = N H = 0 , (28)and diffeomorphic invariance is established. The action (20) can now also be expressed in the canonical form withrespect to the basic variables as, S c = Z (cid:18) ˙ zp z + ˙ xp x + ˙ φp φ − N H (cid:19) dtd x = Z (cid:18) ˙ h ij π ij + ˙ K ij Π ij + ˙ φp φ − N H (cid:19) dtd x, (29)where π ij and Π ij are momenta canonically conjugate to h ij and K ij respectively. Thus the very importance ofthe use of basic variables have also been established. In appendix B, we construct Hamilton’s equations and hencecompute the field equations to demonstrate that the above the Hamiltonian (28) gives the correct description ofthe theory (1) under consideration in the minisuperspace (5). The quantum counterpart of the Hamiltonian (28) under standard canonical quantization reads as, i ~ √ z ∂ Ψ ∂z = − ~ βx (cid:18) ∂ ∂x + nx ∂∂x (cid:19) Ψ − ~ xz ∂ Ψ ∂φ + 3 z b α ′ c p φ Ψ + 36 kz b β ′ c p φ Ψ + (cid:18) x z + 12 kz (cid:19) b γ ′ c p φ Ψ + 9 x z c α ′ Ψ+ 648 k xz c β ′ Ψ + (cid:18) x z + 12 kx z + 72 k xz (cid:19) c γ ′ Ψ + 108 kxz b α ′ b β ′ Ψ + (cid:18) x z + 36 kxz (cid:19) b α ′ b γ ′ Ψ+ (cid:18) kx z + 432 k xz (cid:19) b β ′ b γ ′ Ψ + (cid:18) αx z − kαx − kxβz − k βxz + zx (cid:0) V + Λ M P (cid:1) (cid:19) Ψ , (30)where, n is the operator ordering index which removes some but not all of the operator ordering ambiguitiesappearing between ˆ x and ˆ p x . One can take note of the fact that the above equation is still having some operatorordering ambiguities, which may only be removed knowing explicit functional forms of α ( φ ), β ( φ ) and γ ( φ ). Ascommitted, we use the form of α ( φ ), β ( φ ) and γ ( φ ) appearing in the classical de-Sitter solution (10), (11) and(12), to perform Weyl symmetric ordering carefully between ˆ α ′ and ˆ p φ , ˆ β ′ and ˆ p φ and ˆ γ ′ and ˆ p φ . As a result,the equation (30) now takes the form, i ~ √ z ∂ Ψ ∂z = " − ~ φ β x (cid:18) ∂ ∂x + nx ∂∂x (cid:19) − ~ xz ∂ ∂φ + 3 i ~ α z (cid:18) φ ∂∂φ − φ (cid:19) − i ~ γ x z (cid:18) φ ∂∂φ + 1 (cid:19) + (cid:18) α x zφ − α γ x z φ + 2 γ x φ z + 3 α x zφ + λ zφ x + Λ M P zx (cid:19) Ψ . (31)7e have expressed the quantum equation (31) in spatially flat space k = 0 , to avoid unnecessary complication.Now, under a change of variable, the above modified Wheeler-de-Witt equation, takes the look of Schr¨odingerequation, viz., i ~ ∂ Ψ ∂σ = (cid:20) − ~ φ β x (cid:18) ∂ ∂x + nx ∂∂x (cid:19) − ~ xσ ∂ ∂φ + 2 i ~ α σ (cid:18) φ ∂∂φ − φ (cid:19) − i ~ γ x σ (cid:18) φ ∂∂φ + 1 (cid:19) + V e (cid:21) Ψ= b H e Ψ , (32)where, the proper volume, σ = z = a plays the role of internal time parameter. In the above equation, theeffective potential V e is given by, V e = 3 α xσ φ − α γ x σ φ + 4 γ x φ σ + α xσ φ + λ σ φ x + 2 σ Λ M P x . (33) c H e and probabilistic interpretation: We split the effective Hamiltonian operator b H e (32) into three components apart from the effective potential as, b H e = − ~ φ β x (cid:18) ∂ ∂x + nx ∂∂x (cid:19) − ~ xσ ∂ ∂φ + 2 i ~ σ (cid:18) α φ − γ φx σ (cid:19) ∂∂φ + 2 i ~ σ (cid:18) γ x σ − α φ (cid:19) + V e = b H + b H + b H + b V e , (34)where, b H = − ~ φ β x (cid:18) ∂ ∂x + nx ∂∂x (cid:19) , (35) b H = − ~ xσ ∂ ∂φ , (36) b H = 2 i ~ σ (cid:18) α φ − γ φx σ (cid:19) ∂∂φ + 2 i ~ σ (cid:18) γ x σ − α φ (cid:19) , (37) b V e = V e . (38)Now, let us consider the first term, Z (cid:0) b H Ψ (cid:1) ∗ Ψ dx = − ~ φ β Z (cid:18) x ∂ Ψ ∗ ∂x + nx ∂ Ψ ∗ ∂x (cid:19) Ψ dx = − ~ φ β Z (cid:18) Ψ x ∂ Ψ ∗ ∂x + n Ψ x ∂ Ψ ∗ ∂x (cid:19) dx. (39)Twice we integrate the above integral (39) by parts, and drop the first term due to fall-of condition, to obtain, Z (cid:0) b H Ψ (cid:1) ∗ Ψ dx = − ~ φ β Z Ψ ∗ (cid:20) x ∂ Ψ ∂x − n + 2 x ∂ Ψ ∂x + 2( n + 1) x Ψ (cid:21) dx. (40)Under the particular choice of the operator ordering index viz., n = − Z (cid:0) b H Ψ (cid:1) ∗ Ψ dx = − ~ φ β Z Ψ ∗ (cid:20) x ∂ Ψ ∂x − x ∂ Ψ ∂x (cid:21) dx = Z Ψ ∗ b H Ψ dx. (41)rendering b H hermitian, of-course for a particular choice of operator ordering parameter n = − b H istypically hermitian. We therefore take up the third term viz., b H , Z ( b H Ψ) ∗ Ψ dφ = − i ~ σ Z (cid:18) α φ − γ φx σ (cid:19) ∂ Ψ ∗ ∂φ Ψ dφ − i ~ σ Z (cid:18) γ x σ − α φ (cid:19) Ψ ∗ Ψ dφ. (42)8nder integration by parts and dropping the integrated out terms due to fall-of condition as usual, we obtain, Z ( b H Ψ) ∗ Ψ dφ = 2 i ~ σ Z Ψ ∗ (cid:18) α φ − γ φx σ (cid:19) ∂ Ψ ∂φ dφ + 2 i ~ σ Z (cid:18) γ x σ − α φ (cid:19) Ψ ∗ Ψ dφ = Z Ψ ∗ b H Ψ dφ, (43)again rendering the fact that the effective Hamiltonian b H is hermitian too. Thus, b H e finally turns out to be ahermitian operator. The hermiticity of b H e now allows one to write the continuity equation, as, ∂ρ∂σ + ∇ . J = 0 , (44)which requires to find ∂ρ∂σ , where, ρ = Ψ ∗ Ψ . A little algebra leads to the following equation: ∂ρ∂σ = − ∂∂x (cid:20) i ~ φ β x (cid:0) ΨΨ ∗ ,x − Ψ ∗ Ψ ,x (cid:1)(cid:21) − ∂∂φ (cid:20) i ~ xσ (cid:0) ΨΨ ∗ ,φ − Ψ ∗ Ψ ,φ (cid:1) − σ (cid:16) α φ − γ φx σ (cid:17) Ψ ∗ Ψ (cid:21) − i ~ φ β ( n + 1) x (cid:0) ΨΨ ∗ ,x − Ψ ∗ Ψ ,x (cid:1) , (45)where, we have used the symbols for derivatives as, ∂ Ψ ∂x = Ψ ,x , ∂ Ψ ∂φ = Ψ ,φ etc. Clearly, the continuity equation canbe written, only under the same earlier choice n = − ∂ρ∂σ + ∂J x ∂x + ∂J φ ∂φ = 0 . (46)In the above, ρ = Ψ ∗ Ψ and J = ( J x , J φ ,
0) are the probability density and current density respectively, where, J x = i ~ φ β x (cid:0) ΨΨ ∗ ,x − Ψ ∗ Ψ ,x (cid:1) , (47) J φ = i ~ xσ (cid:0) ΨΨ ∗ ,φ − Ψ ∗ Ψ ,φ (cid:1) − σ (cid:16) α φ − γ φx σ (cid:17) Ψ ∗ Ψ . (48)Here, the variable σ plays the role of internal time parameter, as already mentioned. Thus, the operator orderingindex has been fixed to n = − Semiclassical approximation is essentially a method of finding an approximate wavefunction associated with aquantum equation. If the integrand in the exponent of the semiclassical wavefunction is imaginary, then theapproximate wave function is oscillatory, and falls within the classical allowed region. Thus, a quantum theoryis justified, when the semiclassical approximation is working, i.e. admits classical limit. In such case most of theimportant physics lies in the classical action. A quantum theory therefore, may only be accepted as viable, ifit admits and also found to be well-behaved under, an appropriate semiclassical approximation. To justify thepresent quantum equation (32) in the context of the acid test mentioned above, we therefore need to study itsbehaviour under certain appropriate semiclassical limit in the standard WKB approximation. For this purpose itis much easier to handle the equation (31), and express it in the following form, (cid:20) − ~ √ zφ β x (cid:18) ∂ ∂x + nx ∂∂x (cid:19) − ~ xz ∂ ∂φ − i ~ ∂∂z + i ~ z (cid:18) α φ − γ x φz (cid:19) ∂∂φ − i ~ z (cid:18) α φ + γ x z (cid:19) + V (cid:21) Ψ = 0 , (49)where, V = (cid:20) α x √ zφ − α γ x z φ + 2 γ x φ z + 3 α x √ zφ + λ z φ x + Λ M P z x (cid:21) . (50)9quation (49) may be treated as time independent Schr¨odinger equation with three variables ( x , z , φ ), andtherefore, we seek the solution of equation (49) as usual, in the following form ,Ψ = Ψ e i ~ S ( x,z,φ ) (51)and expand S in power series of ~ as, S = S ( x, z, φ ) + ~ S ( x, z, φ ) + ~ S ( x, z, φ ) + ..... (52)Now inserting the expressions (51) and (52) together with appropriate derivatives (Ψ ,x , Ψ ,xx , Ψ ,φ , Ψ ,φφ , Ψ ,z etc.)in equation (49) and equating the coefficients of different powers of ~ to zero, one obtains the following set ofequations (upto second order), √ zφ β x S ,x + S ,φ xz + S ,z − z (cid:18) α φ − γ x φz (cid:19) S ,φ + V ( x, z, φ ) = 0 , (53) − i √ zφ β x S ,xx − in √ zφ β x S ,x − iS ,φφ xz + S ,z + √ zφS ,x S ,x β x + S ,φ S ,φ xz − iz (cid:18) α φ + γ x z (cid:19) − z (cid:18) α φ − γ x φz (cid:19) S ,φ = 0 , (54) − i √ zφS ,xx β x − in √ zφS ,x β x + √ zφ β x (cid:16) S ,x + 2 S ,x S ,x (cid:17) + 12 xz (cid:16) S ,φ + 2 S ,φ S ,φ (cid:17) − iS ,φφ xz + S ,z − z (cid:18) α φ − γ x φz (cid:19) S ,φ = 0 , (55)which are to be solved successively to find S ( x, z, φ ) , S ( x, z, φ ) and S ( x, z, φ ) and so on. Now identifying S ,x as p x , S ,z as p z and S ,φ as p φ one can recover the classical Hamiltonian constraint equation H c = 0 , presentedin equation (28) from equation (53). Thus, S ( x, z ) can now be expressed as, S = Z p z dz + Z p x dx + Z p φ dφ, (56)apart from a constant of integration which may be absorbed in Ψ . The integrals in the above expression can beevaluated using the classical solution for k = 0 presented in equation (10), (11), (12), and the definitions of p z given in (23), p φ in (24)) and also using the relation p x = Q . For the last expression for the momentum, it isrequired to recall the expression for Q given in (22), remembering the relation, x = ˙ z , where, z = a . Further,we choose n = − n . Now, using classical de-Sittersolution together with the expressions for α ( φ ) given in (11), and that of β ( φ ), γ ( φ ) presented in (12): thevariable x (= ˙ z ) along with all the expressions of momenta, viz. p x , p z and p φ may be expressed in term of x , z and φ as, α ′ = − α φ , (57a) x = 2 λz, (57b) p x = 36 β λxa φ , (57c) p z = − α λza φ − β λ za φ + 24 γ a φ λ √ z , (57d) p φ = 6 α a φ λφ − γ a φ λ φ − a φ λφ , (57e) Although, in the semiclassical approximation, the amplitude should be treated as slowly varying function with respect to thephase, however, the Hamilton-Jacobi equation remains the same and the semiclassical wavefunction, at least up to the first orderapproximation that we perform, remains unaltered. Therefore, to avoid complication, we consider the amplitude to be a constant. Z p x dx = 18 β λx a φ ; (58a) Z p z dz = − α λz a φ − β λ z a φ + 48 γ a φ λ √ z ; (58b) Z p φ dφ = − α a φ λ φ + 16 γ a φ λ φ + a φ λφ . (58c)Hence, explicit form of S in terms of z is found as, S = − α λz a φ + 16 γ a φ λ √ z. (59)For consistency, one can trivially check that the expression for S (59) so obtained, satisfies equation (53) identi-cally. In fact it should, because, equation (53) coincides with Hamiltonian constraint equation (28) for k = 0 . Thisunambiguously establishes the fact that equation (53) is the Hamilton-Jacobi equation, while S is the Hamilton-Jacobi function. Moreover, one can also compute the zeroth order on-shell action (20). Using the classical solution(10), (11) and (12), one may express all the variables in terms of t and substitute in the action (20) to obtain, A = A cl = Z (cid:20) − α a λ φ e λt + 16 γ a φ λ e λt (cid:21) dt. (60)Integrating we have, A = A cl = − α a λφ e λt + 16 γ a φ λ e λt , (61)which is the same as the Hamilton-Jacobi function obtained in (59), since z = a e λt . This proves consistency ofthe present approach towards semiclassical approximation. At this end, the wave function reads as,Ψ = Ψ e i ~ (cid:20) − α λz a φ +16 γ a φ λ √ z (cid:21) , (62)exhibiting oscillatory behaviour. First order approximation:
Under the choice, n = − − √ zφ β x (cid:18) iS ,xx − S ,x S ,x − ix S ,x (cid:19) − xz (cid:18) iS ,φφ − S ,φ S ,φ (cid:19) − z (cid:18) α φ − γ x φz (cid:19) S ,φ − iz (cid:18) α φ + γ x z (cid:19) + S ,z = 0 . (63)One can now compute appropriate derivatives of the expression of S given in (59), to obtain the expression of S ,z from the above equation (63) as, S ,z = i C √ z + C z + C z D z + D z + D , (64)11here, C = − α a φ , C = 12 γ λ , C = − γ a φ β , D = − α a φ , D = γ a φ β , and D = (cid:16) − α β λ (cid:17) are allconstants. The above equation (64) may be integrated in principle, and S may be expressed in the form, S = iF ( z ) . (65)Therefore the wave function up to first-order approximation reads as,Ψ = Ψ e i ~ (cid:20) − α λz a φ +16 γ a φ λ √ z (cid:21) , (66)where,Ψ = Ψ e − F ( z ) , (67)which only tells upon the pre-factor keeping the exponent part unaltered. We have therefore exhibited a techniqueto find the semiclassical wavefunction, on-shell. One can proceed further to find higher order approximations.Nevertheless, it is clear that higher order approximations too, in no way would affect the form (exponent) ofthe semiclassical wavefunction, which has been found to be oscillatory around the classical inflationary solution.Since, the semiclassical wavefunction exhibits oscillatory behaviour around classical de-Sitter solution (10), it istherefore strongly peaked around classical inflationary solutions. Thus we prove that the quantum counterpart ofthe action (1) produces a reasonably viable theory. It is important to mention that the semiclassical approximationis validated only if it occurs at sub-Planckian energy scale. Indeed it is so, as we shall find in the next section thatthe energy scale is H ∗ ≈ − M P , where H ∗ is the Hubble parameter which determines the energy scale duringinflationary regime. Having proved the viability of the action (1) in the quantum domain, we now proceed to test inflation with currentlyreleased data sets in this regard [34, 35]. Inflation is a quantum phenomena, which was initiated sometime between(10 − and 10 − ) sec., after gravitational sector transits to the classical domain. To be more specific, inflationis a quantum theory of perturbations on top of a classical background, which means the energy scale of thebackground must be much below Planck scales. There are also recent hints from the string theory swamplandthat the energy scale must be rather low for inflation. In the previous sub-section we have mentioned that if aquantum theory admits a viable semiclassical approximation, then most of the important physics may be extractedfrom the classical action itself. Clearly, the above semiclassical approximation is validated if the energy scale ofinflation is below Planck’s scale. For a complicated theory such as the present one, the computation of inflationaryparameters is of-course a very difficult job. However, we follow a unique technique to make things look rathersimple. Let us first rearrange the ( ) and the φ variation equations of Einstein, viz., (8) and (9) respectively as,6 α H + 6 α ′ ˙ φ H + 36 β H (cid:20) (cid:18) (cid:19) + 4 ˙HH (cid:18) (cid:19) + 2 (cid:18) ¨HH − H (cid:19) − (cid:18) (cid:19) − (cid:21) + 72 β ′ ˙ φ H (cid:20)(cid:18) (cid:19) + 1 (cid:21) + 24 γ ′ ˙ φ H = V + Λ M P + ˙ φ , (68)¨ φ + 3H ˙ φ + V ′ = 6 α ′ H (cid:20)(cid:18) (cid:19) + 1 (cid:21) + 36 β ′ H (cid:20)(cid:18) (cid:19) + 2 (cid:18) (cid:19) + 1 (cid:21) + 24 γ ′ H (cid:18) (cid:19) , (69)where, H = ˙ aa denotes the expansion rate, which is assumed to be slowly varying. Note that we have alreadypresented inflationary solutions in (10) of the classical field equations (8) and (9) in standard de-Sitter form. Now,instead of standard slow roll parameters, we introduce a hierarchy of Hubble flow parameters [36, 37, 38, 39, 40,41, 22, 23] in the following manner, which appears to be much suitable and elegant to handle higher order theories.Firstly, the background evolution of the theory under consideration is described by a set of horizon flow functions(the behaviour of Hubble distance during inflation) starting from, ǫ = d H d H i , (70)12here, d H = H − is the Hubble distance, also called the horizon in our chosen units. We use suffix i to denotethe era at which inflation was initiated. Now hierarchy of functions is defined in a systematic way as, ǫ l +1 = d ln | ǫ l | d N , l ≥ . (71)In view of the definition N = ln aa i , implying ˙N = H, one can compute ǫ = d ln d H d N , which is the logarithmicchange of Hubble distance per e-fold expansion N, and is the first slow-roll parameter: ǫ = ˙ d H = − ˙HH . Thisensures that the Hubble parameter almost remains constant during inflation ǫ ≪ ǫ = d ln ǫ d N =
1H ˙ ǫ ǫ , which implies ǫ ǫ = d H ¨ d H = − (cid:16) ¨HH − ˙H H (cid:17) . In the same manner higherslow-roll parameters may be computed. Equation (71) essentially defines a flow in space with cosmic time beingthe evolution parameter, which is described by the equation of motion ǫ ˙ ǫ l − d H i ǫ l ǫ l +1 = 0 , l ≥ . (72)In view of the slow-roll parameters, equations (68) and (69) may therefore be expressed as, − α H − α ′ ˙ φ H − β H h (cid:0) − ǫ (cid:1) − (cid:0) ǫ ǫ (cid:1) − i − β ′ ˙ φ H (cid:2)(cid:0) − ǫ (cid:1) + 1 (cid:3) − γ ′ ˙ φ H + (cid:16) ˙ φ V + Λ M P (cid:17) = 0 , (73)and ¨ φ + 3H ˙ φ = − V ′ + 6 α ′ H (cid:2) − (cid:0) ǫ (cid:1)(cid:3) + 36 β ′ H h(cid:0) − ǫ (cid:1) + 2 (cid:0) − ǫ (cid:1) + 1 i + 24 γ ′ H (cid:0) − ǫ (cid:1) , (74)respectively, which may therefore be approximated using the slow roll hierarchy to,6 α H = ˙ φ h V + Λ M P − (cid:0) α ′ ˙ φ H + 144 β ′ ˙ φ H + 24 γ ′ ˙ φ H (cid:1)i , (75)and ¨ φ + 3H ˙ φ + (cid:2) V ′ − (cid:0) α ′ H + 144 β ′ H + 24 γ ′ H (cid:1)(cid:3) = 0 . (76)Before imposing the standard slow roll conditions, viz. | ¨ φ | ≪ | ˙ φ | and ˙ φ ≪ V ( φ ), we try to reduce equations(75) and (76) in a much simpler form. For example redefining the potential as, U = V − ( α + 12H β + 2H γ ) , (77)equation (76) takes the standard form of Klien-Gordon Equation,¨ φ + 3H ˙ φ + U ′ = 0; . (78)In view of the reduced equation (78), it is now quite apparent that the evolution of the scalar field is drivenby the re-defined potential gradient U ′ = dUdφ , subject to the damping by the Hubble expansion 3H ˙ φ , as in thecase of single field equation, while the potential U ( φ ) carries all the information in connection with the couplingparameters of generalised higher order action under consideration. Further, assuming U = V + M P Λ −
6H ˙ φ (cid:0) α ′ + 24H β ′ + 4H γ ′ (cid:1) , (79)13quation (75) may be reduced to the following simplified form, viz,6 α H = ˙ φ U ( φ ) . (80)It is important to mention that, the two choices on the redefined potential U ( φ ) made in (77) and (79), do notconfront in any case and may be proved to be consistent as demonstrated underneath. During slow roll, theHubble parameter H almost remains unaltered. Thus replacing H by λ , and using the forms of the parameters α ( φ ) presented in (11), along with β ( φ ) and γ ( φ ) assumed in (12), the two relations (77) and (79) lead to thefollowing first order differential equation on φ , (cid:18) φ − α λ φ − M P φ (cid:19) dφ = 12 λ dt, (81)which can immediately be solved to yield, φ ( t ) = InverseFunction log (cid:0) − λ (cid:1) λ + 3 α λ tanh − (cid:16) λ √ √ Λ (cid:17) √ / − α (cid:20) c + t λ (cid:21) , (82)where, c is a constant of integration. Obviously, the solution is complicated to explore the behaviour of φ against t . However, we shall show later, separately both in the cases without and with the cosmological constant (undercertain reasonable assumption based on the data set), that indeed φ falls-of with time, as required.Having proven consistency of our assumptions, we enforce the standard slow-roll conditions ˙ φ ≪ U and | ¨ φ | ≪ | ˙ φ | , on equations (80) and (78), which thus finally reduce to,6 α H ≃ U, (83)and 3H ˙ φ ≃ − U ′ . (84)Now, combining equations (83) and (84), it is possible to show that the potential slow roll parameter ǫ equals theHubble slow roll ( ǫ ) parameter under the condition, ǫ = − ˙HH = α (cid:18) U ′ U (cid:19) − α ′ (cid:18) U ′ U (cid:19) ; η = 2 α (cid:18) U ′′ U (cid:19) , (85)while η remains unaltered. Further, since H˙ φ = − U αU ′ , therefore, the number of e-folds at which the presentHubble scale equals the Hubble scale during inflation, may be computed as usual in view of the following relation:N( φ ) ≃ Z t f t i H dt = Z φ f φ i H˙ φ dφ ≃ Z φ i φ f (cid:16) U αU ′ (cid:17) dφ, (86)where, φ i and φ f denote the values of the scalar field at the beginning ( t i ) and the end ( t f ) of inflation. Thus,slow roll parameters reflect all the interactions, as exhibited earlier [42, 43, 44], but here only via the redefinedpotential U ( φ ). Let us first consider cosmological constant Λ = 0 , so that the potential V ( φ ) = λ φ , while the forms of α ( φ )(11), β ( φ ) and γ ( φ ) (12) satisfy classical de-Sitter solutions as well. Now, in order to compute inflationary14 in M P φ f in M P n s r N0.0084 0.3694 0.9706 0.1008 790.0082 0.3664 0.9713 0.0984 810.0080 0.3634 0.9720 0.0960 830.0078 0.3604 0.9727 0.0936 850.0076 0.3573 0.9734 0.0912 870.0074 0.3541 0.9741 0.0888 900.0072 0.3509 0.9748 0.0864 920.0070 0.3476 0.9755 0.0840 95Table 1: Data set for the inflationary parameters tak-ing φ i = 2 . M P and varying α . r n s Varying α Figure 1: This plot depicts the variation of n s with r , varying α .parameters numerically, it is necessary to find the form of the re-defined potential U ( φ ). As mentioned, duringinflation the Hubble parameter remains almost constant, and therefore while computing U ( φ ), one can replace itby the constant λ , without any loss of generality. Thus,12H (cid:0) α + 12H β + 2H γ (cid:1) ≈ − H φ , such that , U = 12 m φ , where , m = λ + H ≈ λ . (87)In view of the above form of U , we find the following relation from (80),6 α φ H = 12 m φ . (88)Now choosing, α = 0 . M P , m = 2 × − M P , φ i = 2 . M P , it is possible to calculate from the aboveequation, H ≈ . × − M P , and hence the energy scale of inflation (H ∗ ≈ − M P ) is sub-Planckian. Notethat the inflationary parameters do not depend on m . Further, in view of the above quadratic form of there-defined potential, the slow roll parameters ( ǫ, η ) (85) and the number of e-folding N (86) take the followingforms, ǫ = 6 α φ , η = 4 α φ , N = 14 α Z φ i φ f φ dφ = 112 α ( φ i − φ f ) . (89)Now, comparing expression for the primordial curvature perturbation on super-Hubble scales produced by single-field inflation P ζ ( k ) with the primordial gravitational wave power spectrum P t ( k ), one obtains the tensor-to-scalarratio for single-field slow-roll inflation r = P t ( k ) P ζ ( k ) = 16 ǫ , while, the scalar tilt, conventionally defined as n s − n s = 1 − ǫ + 2 η . In view of all these expressions we compute the inflationary parametersand present them for different values of the parameter α in table 1. We also present respective n s versus r plotsin figure 1.Table 1 depicts that under the variation of α within the range 0 . M P ≤ α ≤ . M P , the spectral index ofscalar perturbation and the scalar to tensor ratio lie within the range 0 . ≤ n s ≤ .
976 and 0 . ≤ r ≤ . (cid:18) φ − α λ φ (cid:19) dφ = 12 λ dt, (90)15 ϕ Variation of ϕ with t Figure 2: This plot depicts fall of φ with t .which can immediately be solved to yieldln φ + 2 α φ = λ t − t ) . (91)Clearly, the scalar decays as φ ∼ t − (91), as depicted in figure 2, if φ ( ≤ M P ), is not too large, and quickly fallsbelow Planck’s mass, φ < M p .To exhibit the fact that as φ ≪ M p the field approaches an oscillatory solution, let us express equation (80) as,3H = 12 α (cid:18)
12 ˙ φ + 12 m φ (cid:19) , (92)where, U = m φ , and m ≈ λ . In view of the expression of α ( φ ) = α φ , the above equation reads as,3H m = 3H λ = φ α ˙ φ λ + φ ! . (93)Note that for single scalar field, the above equation leads to: 3H = M p ( ˙ φ + m φ ). At the end of inflation, φ α ∼ M p , according to the present data set. Once the Hubble rate falls below √ λ , this equation (93) may beapproximated to,˙ φ ≈ − λ φ , (94)which exhibits oscillatory behaviour of φ ∼ e i √ λ t . The field therefore starts oscillating many times over a Hubbletime, driving a matter-dominated era at the end of inflation. Despite all these excellent features, the model runsinto problem, since the number of e-folds varies within the range 79 ≤ N ≤
95 , being much larger than usualN ≈
60 . In fact, to fit the currently released datasets, r ≤ .
07 , the number of e-folds shoots up to N ≈
100 , whichis too large. As a result, the universe cools down much more than the usual slow roll inflation, ending up with avery cold universe at the end of inflation, which could make filling the universe with matter rather difficult. Thisproblem is somewhat related to dilaton stabilization issue. Note that in the present model, the same dilatonicfield is responsible for inflation. It is known that generic string models suffer from dilaton runaway problem, sincedepending on the initial conditions, the field can acquire a large amount of kinetic energy and can easily overshootany local minima in the potential. Such runaway behavior of the dilaton and other moduli fields prevents a viableinflation [45]. Although the moduli stabilization issue has been studied by several authors earlier, so that it cansuccessfully play the role of inflaton field without creating problems like reheating [46, 47, 48], nevertheless, sincewe have encountered problem, therefore let us consider next the role of cosmological constant in this respect.16 in M P φ f in M P n s r N0.00019 4.7077 0.9671 0.08883 460.00018 4.7079 0.9688 0.08415 480.00017 4.7080 0.9705 0.07948 510.00016 4.7082 0.9723 0.07480 550.00015 4.7084 0.9740 0.07013 580.00014 4.7086 0.9757 0.06545 620.00013 4.7088 0.9775 0.06078 670.00012 4.7090 0.9792 0.05610 73Table 2: Data set for the inflationary parameters tak-ing φ i = 4 . M P ; m = 0 . M P ; Λ = 1 . M P andvarying α . r n s Varying α Figure 3: This plot depicts the variation of n s with r , varying α . It has been shown that an additive cosmological constant( M P Λ) does not affect the solutions to the classicalfield equations other than the fact that it only adds to the potential function. As mentioned, during inflationthe Hubble parameter remains almost constant, and therefore while computing U ( φ ), one can replace it by theconstant λ , without any loss of generality. Thus, taking into account the case 1a, we readily obtain,12H (cid:0) α + 12H β + 2H γ (cid:1) ≈ − H φ , such that , U = 12 m φ − Λ M P , where , m = λ + H ≈ λ . (95)However, inflationary parameters ǫ, η and the number of e-foldings N depend on the potential function itself andare likely to modify the situation. For example, in view of the above form of U , we obtain the following expressionfrom equation (80),6 α φ H = 12 m φ − Λ M P . (96)Now, for the above form of re-defined potential, the inflationary parameters read as, ǫ = 4 m α φ ( m φ − M P ) + 2 m α ( m φ − φ Λ M P ) , η = 4 m α m φ − φ Λ M P . (97)N = 14 α Z φ i φ f ( m φ − M P ) m dφ = 112 α ( φ i − φ f ) − Λ M P m α ( φ i − φ f ) . (98)We present data set in Table 2, which depicts that under the variation of α within the range 0 . M P ≤ α ≤ . M P , the spectral index of scalar perturbation and the scalar to tensor ratio lie within the range0 . ≤ n s ≤ .
979 and 0 . ≤ r ≤ .
089 respectively, which show excellent agreement with the recently releaseddata [34, 35]. The number of e-foldings now varies within the acceptable range 46 < N <
73 , which is sufficient tosolve the horizon and flatness problems. For the sake of visualization we present the the spectral index of scalarperturbation versus the scalar to tensor ratio plot in figure 3. We also present another pair of data sets in Table3 and Table 4, for lower values of φ i . In both the tables, the spectral index of scalar perturbation and the scalarto tensor ratio lie within the range 0 . ≤ n s ≤ .
979 and 0 . ≤ r ≤ .
083 respectively, which again showexcellent agreement with the recently released data [34, 35]. The number of e-folds (N) also lie very much withinadmissible region (Table 3 and Table 4).Now, for a consistency check, let us chose sub-Planckian energy scale H ∗ ≈ − M P , as in the previoussubsection, together with a mid range value of α = 0 . M P , while Λ = 1 M P , φ i = 4 . M P , are chosen asdepicted in the table 2. Therefore, the term appearing in the left hand side of equation (96) may be neglected,and we simply have,12 m φ − Λ M P ≈ , resulting in , m ≈ . M P , (99)17 in M P φ f in M P n s r N0.000100 1.9929 0.9696 0.08367 510.000095 1.9931 0.9711 0.07949 530.000090 1.9933 0.9726 0.07530 560.000085 1.9935 0.9741 0.07111 600.000080 1.9937 0.9756 0.06694 630.000075 1.9939 0.9772 0.06275 680.000070 1.9941 0.9787 0.05857 72Table 3: Data set for the inflationary parameters tak-ing φ i = 2 . M P ; m = 0 . M P ; Λ = 1 . M P andvarying α . α in M P φ f in M P n s r N0.000095 1.4828 0.9673 0.09089 480.000090 1.4830 0.9690 0.08610 500.000085 1.4832 0.9708 0.08132 530.000080 1.4834 0.9729 0.07654 570.000075 1.4836 0.9742 0.07175 610.000070 1.4839 0.9759 0.06697 650.000065 1.4841 0.9776 0.06219 700.000060 1.4844 0.9794 0.05740 76Table 4: Data set for the inflationary parameters tak-ing φ i = 1 . M P ; m = 0 . M P ; Λ = 1 . M P andvarying α .which shows consistency with our data set presented in table 2. Further, since r = 0 .
07 for the above choice of α , so we can calculate the energy scale of inflation H ∗ using the formula used in a single scalar field as [49],H ∗ = 8 × r r . GeV = 4 . × GeV ≈ . × − M P , (100)which further reveals consistency in our choice of the energy scale H ∗ during inflation. From the analysis presentedin the above two subsections, it is quite clear that inflation is supported by the vacuum energy density, ratherthan the dilaton, and thus the problem of moduli field stabilization does not appear. Here, we exhibit that all the different choices of the coupling parameters α ( φ ), β ( φ ) and γ ( φ ) presented insubcase 1b, and cases 2, 3 and 4 do not tell upon the inflationary parameters. Subcase 1b:
Following equation (95), here again we make the choice,12H (cid:0) α + 12H β + 2H γ (cid:1) ≈ − H φ , U = 12 m φ − Λ M P , where , m = λ + H ≈ λ . (101)Since the re-defined potential U ( φ ) together with the mass relation remain unaltered, so all the computationsmade above remains unaltered. Case 2:
Even under the choice of β = β = constant made in case 2 (14), we obtain the re-defined potential U ( φ ) in viewof (95) as,12H (cid:0) α + 2H γ (cid:1) ≈ − H φ , U = 12 m φ − Λ M P , where , m = λ + H ≈ λ . (102)Thus, U ( φ ) together with the mass formula remain unaltered here again, and hence as mentioned all the compu-tations remain valid. Case 3:
Again the choice of α = α = constant made in case 3 in (15) does not affect the re-defined potential U ( φ ) andthe mass formula since,12H (cid:0) β + 2H γ (cid:1) ≈ − H φ , U = 12 m φ − Λ M P + 6 λ α , where , m = λ + H ≈ λ . (103)18 ϕ Variation of ϕ with t Figure 4: This plot depicts fall of φ with t . Case 4:
Finally, keeping both the coupling parameters α = α = constant and β = β = constant, made in case 4 (16)one can find the form of the re-defined potential U ( φ ) as,12H (cid:0) γ (cid:1) ≈ − H φ , U = 12 m φ − Λ M P + 6 λ α , where , m = λ + H ≈ λ , (104)which is the same as before. Note that the additional constant term (6 λ α ) appearing in the potential of cases3 and 4, is extremely small to make any difference This also validates the choice of the action (1), we begin with. Although the theory under consideration is highly complicated, we have been able to reduce the system of fieldequations considerably to study inflation. We now pose to prove that the theory does not also suffer from gracefulexit problem. For this purpose, we need to show that as φ ≪ M P , it executes oscillatory behaviour. Thus, at firstwe have to prove that φ indeed falls of with time. Since both λ as well as α are about five order of magnitudesmaller than the other parameters, we therefore express equation (81) under suitable approximation as:˙ φ ≈ − M P Λ λφ , (105)which may immediately be solved (taking λ ≈ − M P , Λ = 1 M P ) as, n φ ( t ) = −√ √ c − tM P o or n φ ( t ) = √ √ c − tM P o . (106)We present Figure 4 to depict the behaviour of φ with time, taking the integration constant, c = 3 . Havingshown that φ decays, let us now express equation (80) as,3H = 12 α (cid:18)
12 ˙ φ + 12 m φ − Λ M P (cid:19) , (107)taking, U ( φ ) = m φ − Λ M P , and m ≈ λ . In view of the expression of α ( φ ) = α φ , the above equation readsas, 3H m = 3H λ = φ α ˙ φ λ + φ − Λ M P λ ! . (108)19ote that for single scalar field, the above equation reads as 3H = M p ( ˙ φ + m φ − M P ). Since at the end ofinflation, φ α ∼ M p , according to the present data set, so once the Hubble rate falls below √ λ , this equation(108) may be approximated to,˙ φ ≈ − (2 λ φ − Λ M P ) , (109)which may immediately be integrated to yield, φ ( t ) = ± √ Λ M p tan (cid:0) √ λ { t − t } (cid:1) λ q (cid:0) √ λ { t − t } (cid:1) + 1] = ± √ Λ M p √ λ sin (cid:16) √ λ { t − t } (cid:17) . (110) t being a constant of integration. Thus the scalar field (dilaton) starts oscillating many times over a Hubbletime, driving a matter-dominated era at the end of inflation. Although, there are alternatives to inflation, such as matter bounce or ekpyrosis originating from string theory,branes and extra dimensions [50, 51], which do explain the origin of large scale structure and flatness of the uni-verse [52, 53], inflation is indeed the mainstream choice. Inflation perhaps, is the simplest scenario, which not onlysolves the horizon and the flatness problems but also can explain elegantly, the origin of the seeds of perturbation.It is therefore required to test every alternative/modified theory of gravity in the context of inflation. As men-tioned, inflation is a quantum theory of perturbation, and it occurred when gravity becomes classical, i.e., at thesub-Planckian epoch. It is therefore also required to test the viability of the alternative/modified theory modelsunder quantization. In the present manuscript, a generalized action has been constructed out of non-minimallycoupled [ α ( φ )] scalar-tensor theory of gravity, being associated with a curvature squared term having functionallydependent coupling parameter [ β ( φ )], the Gauss-Bonnet-Dilatonic [ γ ( φ )] term and the cosmological constantterm. Such an action has not been treated in the context of inflation, to the best of our knowledge. A varietyof classical de-Sitter solutions have been presented, which exhibit identical form of the potential under the choiceof different coupling parameters, and also with constant coupling parameters α ( φ ) = α , or β ( φ ) = β , or both.This validates the choice of a complicated action to begin with. The phase-space structure has been formulatedfollowing ‘MHF’. The effective Hamiltonian under quantization has been found to be hermitian, establishingunitarity, also a viable semiclassical approximation has been presented which depicts that the quantum equationis classically allowed, and the semiclassical wave function is strongly peaked around a classical de-Sitter solution.This motivates to study inflation.One important finding is that, in the presence of dilatonic coupling, a combined hierarchy of Hubble andGauss–Bonnet flow parameters are usually required to handle the situation, since additional condition apart fromthe standard slow roll condition are needed [23, 43]. Also, in the case of a non-minimally coupled scalar-tensortheory of gravity in the presence of scalar curvature squared term with constant coupling (without dilaton) againa combined hierarchy of Hubble and non-minimal flow parameters was introduced [41]. It therefore appears thatadditional conditions are essential corresponding to the number of coupling parameters present in the theory.However, although in the present case, there are three independent functional coupling parameters, we do not re-quire any further condition, like the hierarchy of flow parameters associated with non-minimal coupling parameter α ( φ ), β ( φ ) coupled with R and γ ( φ ): the dilatonic Gauss–Bonnet coupling. In fact, a combined hierarchy ofHubble flow parameter has been found to be sufficient to study inflation, while all the information regarding thecoupling parameters is carried by the re-defined effective potential U ( φ ). The choice of the quadratic potential,and the functional forms of coupling parameters are not arbitrary, rather they satisfy classical de-Sitter solution.In the absence of the cosmological constant term, although the inflationary parameters (scalar to tensor ratio, r and the spectral index, n s ) obtained show reasonably good agreement with the latest released Planck’s data[34, 35], nevertheless, the number of e-fold is too large, which might cause problem in reheating the universe. Thisproblem is due to the fact that the same dilaton has been called for inflation, and it is well known that dilatonsuffers from stabilization issue. In the presence of the cosmological constant term, the problem disappears andthe fit is excellent including the number of e-folds. Since the numerical value of the cosmological constant has20een chosen to be Λ = 1 . M P , which corresponds to the energy density ρ Λ ≈ GeV , which is the computedvalue of the vacuum energy density, ρ vac , it is therefore apparent that inflation is essentially driven by the vacuumenergy density, and the dialton stabilization issue is bypassed. Thus the scalar field may be treated as the modulifield (dilaton), rather than an ordinary scalar field.Last but not the least, it has been expatiated that all the different choices of the coupling parameters whichlead to the same classical de-Sitter solution with identical potential, lead to the same re-defined effective potential[ U ( φ )], and as a result inflationary parameters remain unaltered. This not only validates the choice of thecomplicated action we begin with, but also the technique of using effective potential instead of taking non-minimalflow parameters into account. References [1] A.K. Sanyal, If Gauss–Bonnet interaction plays the role of dark energy, Phys. Lett. B 645, 1 (2007),arXiv:astro-ph/0608104.[2] A.K. 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A Canonical formulation following Dirac’s algorithm:
In Dirac’s constraint analysis, it is customary to fix h ij and K ij at the boundary, and therefore there is noneed to supplement the action by surface terms. Hence, one can consider the action (17) without the supplemen-tary boundary terms and integrate the divergent terms appearing in the action by parts, may express the pointLagrangian in the form, L = (cid:18) − α ′ ˙ φ ˙ z √ zN − α ˙ z N √ z + 6 kN α √ z (cid:19) − N z Λ M p + 9 β √ z (cid:18) ¨ z N − z ¨ z ˙ NN + ˙ z ˙ N N + 2 k ˙ z N z + 4 k N (cid:19) − β ′ k ˙ z ˙ φN √ z − γ ′ ˙ z ˙ φN √ z (cid:18) ˙ z N z + 12 k (cid:19) + z (cid:18) N ˙ φ − V N (cid:19) , (111)which is essentially the action (18) without the boundary term Σ R . Thereafter, it is required to substitute˙ z = N x , i.e.; ¨ z = N ˙ x + ˙ N x , so that the point Lagrangian (111) may be express in the following form, L = (cid:18) − α ′ ˙ φ √ zx − αN x √ z + 6 kN α √ z (cid:19) − N z Λ M p + 9 β √ z (cid:18) ˙ x N + 2 kN x z + 4 k N (cid:19) − β ′ kx ˙ φ √ z − γ ′ x ˙ φ √ z (cid:18) x z + 12 k (cid:19) + z (cid:18) N ˙ φ − V N (cid:19) + u (cid:18) ˙ zN − x (cid:19) , (112)23here the expression (cid:0) ˙ zN − x (cid:1) is being treated as a constraint and introduced through the Lagrangian multiplier u .The canonical momenta are, p x = 18 β ˙ xN √ z ; p z = uN ; p φ = − α ′ √ zx − β ′ kx √ z − γ ′ x √ z (cid:0) x z + 12 k (cid:1) + ˙ φz N ; p N = 0 = p u . (113)The primary Hamiltonian is Therefore, H p = 3 α (cid:18) N x √ z − kN √ z (cid:19) + u ˙ zN + 3 α ′ N xp φ z + 9 α ′ N x √ z + √ zN p x β − kβN √ z (cid:18) x z + 2 k (cid:19) + γ ′ N x z + 648 k N β ′ x z + 72 N k γ ′ x z + 36 kβ ′ N xp φ z + 12 kN γ ′ x z + γ ′ N x p φ z + 12 kγ ′ N xp φ z + 36 kβ ′ γ ′ N x z + 432 k β ′ γ ′ N x z + N p φ z + 108 kα ′ β ′ N x z + 3 α ′ γ ′ N x z + 36 kα ′ γ ′ N x z + V N z + N z Λ M p − u (cid:0) ˙ zN − x (cid:1) . (114)Now introducing the constraints φ = N p z − u ≈ φ = p u ≈ u and u respectively, we get (Note that, these are second class constraints , since { φ i , φ j } 6 = 0 . Further, since the lapsefunction N is non-dynamical, so the associated constraint vanishes strongly, ie. p N = 0 and therefore it maysafely be ignored), H p = N (cid:20) α (cid:18) x √ z − k √ z (cid:19) + 3 α ′ xp φ z + 9 α ′ x √ z + √ zp x β − kβ √ z (cid:18) x z + 2 k (cid:19) + γ ′ x z + 648 k β ′ x z + 72 k γ ′ x z + 36 kβ ′ xp φ z + 12 kγ ′ x z + γ ′ x p φ z + 12 kγ ′ xp φ z + 36 kβ ′ γ ′ x z + 432 k β ′ γ ′ x z + p φ z + 108 kα ′ β ′ x z + 3 α ′ γ ′ x z + 36 kα ′ γ ′ x z + V z + z Λ M p (cid:21) + ux + u (cid:0) N p z − u (cid:1) + u p u . (115)Note that the Poisson brackets { x, p x } = { z, p z } = { φ, p φ } = { u, p u } = 1 , hold. Now constraint should remainpreserved in time, which are exhibited through the following Poisson brackets˙ φ = { φ , H p } = − u − N ∂H p ∂z ≈ ⇒ u = − N ∂H p ∂z ; ˙ φ = { φ , H p } ≈ u − x ⇒ u = x. (116)Therefore the primary Hamiltonian is modified to H p = N (cid:20) xp z + 3 α (cid:18) x √ z − k √ z (cid:19) + 3 α ′ xp φ z + 9 α ′ x √ z + √ zp x β − kβ √ z (cid:18) x z + 2 k (cid:19) + γ ′ x z + 648 k β ′ x z + 72 k γ ′ x z + 36 kβ ′ xp φ z + 12 kγ ′ x z + γ ′ x p φ z + 12 kγ ′ xp φ z + 36 kβ ′ γ ′ x z + 432 k β ′ γ ′ x z + p φ z + 108 kα ′ β ′ x z + 3 α ′ γ ′ x z + 36 kα ′ γ ′ x z + V z + z Λ M p (cid:21) − N p u ∂H p ∂z . (117)As the constraint should remain preserved in time in the sense of Dirac, so˙ φ = { φ , H p } = − N (cid:20) ∂H p ∂z − N p u ∂ H p ∂z (cid:21) + N ∂H p ∂z ≈ ⇒ p u = 0 . (118)Finally the phase-space structure of the Hamiltonian, being free from constraints reads as, H = N (cid:20) xp z + 3 α (cid:18) x √ z − k √ z (cid:19) + 3 α ′ xp φ z + 9 α ′ x √ z + √ zp x β − kβ √ z (cid:18) x z + 2 k (cid:19) + γ ′ x z + 648 k β ′ x z + 72 k γ ′ x z + 36 kβ ′ xp φ z + 12 kγ ′ x z + γ ′ x p φ z + 12 kγ ′ xp φ z + 36 kβ ′ γ ′ x z + 432 k β ′ γ ′ x z + p φ z + 108 kα ′ β ′ x z + 3 α ′ γ ′ x z + 36 kα ′ γ ′ x z + V z + z Λ M p (cid:21) = N H , (119) Although, second class constraints are handled with Dirac brackets, one can trivially check that the Poisson bracket is identicalto the Dirac bracket, in the present situation.
B Field equations from Hamilton’s equations:
In this appendix, we pose to find all the field equations (7), (8), and (9), obtained in the Robertson-Walker min-isuperspace (5) under the variation of the action (4) from the Hamiltonian (28). We therefore find the Hamilton’sequations and combine to form the Euler-Lagrange equations. The Hamilton’s equations are,˙ x = ∂H∂p x = √ zp x β (120)˙ φ = ∂H∂p φ = 3 α ′ xz + 36 kβ ′ xz + γ ′ xz (cid:18) x z + 12 k (cid:19) + p φ z (121)˙ p φ = − ∂H∂φ = − (cid:20) α ′ (cid:18) x √ z − k √ z (cid:19) − β ′ √ zp x β + 3 α ′′ xp φ z + 36 kβ ′′ xp φ z + γ ′′ xp φ z (cid:18) x z + 12 k (cid:19) + 9 α ′ α ′′ x √ z − kβ ′ √ z (cid:18) x z + 2 k (cid:19) + 1296 k β ′ β ′′ x z + 108 kα ′′ β ′ x z + 108 kα ′ β ′′ x z + 3 α ′′ γ ′ x z (cid:18) x z + 12 k (cid:19) + 3 α ′ γ ′′ x z (cid:18) x z + 12 k (cid:19) + 36 kβ ′′ γ ′ x z (cid:18) x z + 12 k (cid:19) + 36 kβ ′ γ ′′ x z (cid:18) x z + 12 k (cid:19) + γ ′ γ ′′ x z (cid:18) x z + 12 k (cid:19) + V ′ z (cid:21) . (122)Now inserting p x and p φ in the above equation (122) from equations (120) and (121) respectively, one canimmediately reduced equation (122) as˙ p φ = (cid:20) − α ′ (cid:18) x √ z − k √ z (cid:19) − α ′′ x √ z ˙ φ + 9 β ′ ˙ x √ z + 18 kβ ′ √ z (cid:18) x z + 2 k (cid:19) − γ ′′ x ˙ φ √ z (cid:18) x z + 12 k (cid:19) − kβ ′′ x ˙ φ √ z − V ′ z (cid:21) , (123)performing temporal derivative on equation (121), we obtained˙ p φ = (cid:20) ¨ φz + 3 √ z ˙ z ˙ φ − α ′′ x √ z ˙ φ − α ′ ˙ x √ z − α ′ x ˙ z √ z − kβ ′′ x ˙ φ √ z − kβ ′ ˙ x √ z + 18 kβ ′ x ˙ zz − γ ′′ x ˙ φ √ z (cid:18) x z + 12 k (cid:19) − γ ′ ˙ x √ z (cid:18) x z + 12 k (cid:19) + γ ′ x ˙ z z (cid:18) x z + 12 k (cid:19)(cid:21) . (124)Therefore, comparing (123) and (124) one can readily arrive at, − α ′ (cid:18) a ¨ a + a ˙ a + ka (cid:19) − β ′ (cid:18) a ¨ a + 2 ˙ a ¨ a + ˙ a a + k a + 2 k ˙ a a + 2 k ¨ a (cid:19) + 3 a ˙ a ˙ φ − γ ′ (cid:18) ˙ a ¨ a + k ¨ a (cid:19) + a (cid:18) ¨ φ + V ′ (cid:19) = 0 . (125)If we now insert all the momenta in the Hamiltonian constraint equation (28), the ( ) equation of Einstein isobtained as, − αa (cid:18) ˙ a + k (cid:19) − α ′ ˙ a ˙ φa + Λ M p − β (cid:18) a ... aa − ¨ a a + 2 ˙ a ¨ aa − a a − k ˙ a a + k a (cid:19) − β ′ ˙ φ (cid:18) ˙ a ¨ aa + ˙ a a + k ˙ aa (cid:19) − γ ′ ˙ φ (cid:18) ˙ a a + k ˙ aa (cid:19) + (cid:18) ˙ φ V (cid:19) = 0 ..