Classical and quantum exact solutions for a FRW multi-scalar field cosmology with an exponential potential driven inflation

A flat Fiedmann-Robertson-Walker (FRW) multi-scalar field cosmology is studied with a particular potential of the form $ \rm V(\phi,\sigma)=V_0 e^{-\lambda_1 \phi-\lambda_2 \sigma}$, which emerges as a relation between the time derivatives of the scalars field momenta. Classically, by employing the Hamiltonian formalism of two scalar fields $\rm(\phi,\sigma)$ with standard kinetic energy, exact solutions are found for the Einstein-Klein-Gordon (EKG) system for different scenarios specified by the parameter $\rm\lambda^2=\lambda_1^2+\lambda_2^2$, as well as the e-folding function $\rm N_{e}$ which is also computed. For the quantum scheme of this model, the corresponding Wheeler-DeWitt (WDW) equation is solved by applying an appropriate change of variables.

fields, and in many cases, the employed potentials are simple polynomial powers of the scalar fields or in other cases the employed potential is a series of lineally summed exponentials, however, it has been shown that a potential of the form V(φ, σ) = V 0 e −λ1φ−λ2σ is a good candidate to model the inflation phenomenon for multi-scalar field theory, as discussed in previous work [27], and might provide a richer post inflation scenario.
Generally, in the studies of inflationary cosmology one employs the usual slow-roll approximation with the objective to extract simple expression for basics observable, such as the scalar and tensor spectral indices, the running of the scalar spectral index and the tensor-to-scalar ratio. Moreover, in the slow-roll regime the set of EKG equations reduces in such a way that one can quickly obtain the solution of the scale factor. Nevertheless, there is an alternative approach which allows for an easy derivation of many inflation results. It is called the Hamilton's formulation, widely used in analytical mechanics. Using this method we obtain the exact solutions of the complete set of EKG equations without using the aforementioned approximation.
On the other hand, we implement a basic formulation in quantum cosmology by means of the Wheeler-DeWitt (WDW) equation. The WDW equation has been analyzed with different approaches in order to solve it, and there are several papers on the subject, such is the case in [28], where they debate what a typical wave function for the universe is. In ref. [29] has a review on quantum cosmology where the problem of how the universe emerged from big bang singularity can no longer be neglected in the GUT epoch. Moreover, the best candidates for quantum solutions are those that have a damping behavior with respect to the scale factor, since only such wave functions allow for good classical solutions when using a Wentzel-Kramers-Brillouin (WKB) approximation for any scenario in the evolution of our universe [30,31]. Furthermore, in the context of a single scalar field a family of scalar potentials is obtained in the Bohmian formalism [32,33], where among others a general potential of the form V(φ) = V 0 e −λφ is examined.
Given this insight, for a two scalar field scenario we consider a potential of the form V(φ, σ) = V 0 e −λ1φ−λ2σ in order to solve the WDW equation.
This work is arranged as follows. In section II we present the model with the action and the corresponding EKG equations for our cosmological model and the associated Hamiltonian density. In section III general equations for the classical solutions of scale factor, scalar fields and their associated momenta are derived in terms of the free parameters of the model. In subsections III A, III B, III C and III D the particular solutions and their number of e-folds is computed for different cases of the λ parameter. in section IV we use the Hamiltonian density to compute the corresponding WDW equation, which is solved by using a change of variables, an ansatz for the wave function is employed in terms of a generic function and parameters which are to be determined. In subsections IV A and IV B the corresponding wave function and their constants relations are presented for different cases of the δ parameter, which in turn is related to the λ parameter of the classical solutions. Finally, in section V we present our conclusions for this work.

II. THE MODEL
We begin with the construction of two scalar fields cosmological paradigm, which requires canonical scalar fields φ, σ. The action of a universe with the constitution of such fields is where R is the Ricci scalar, V(φ, σ) is the corresponding scalar field potential, and the reduced Planck mass M 2 P = 1/8πG = 1. The corresponding variations of Eq.(1), with respect to the metric and the scalar fields give the Einstein-Klein-Gordon field equations The line element to be considered in this work is the flat FRW where N is the lapse function, which in a special gauge one can directly recover the cosmic time t phys (Ndt = dt phys ), the scale factor A(t) = e Ω(t) is in the Misner's parametrization, and the scalar function has an interval, Ω ∈ (−∞, ∞).
Consequently the field equations are φφ σσ By building the corresponding Lagrangian and Hamiltonian densities for this cosmological model, classical solutions to Einstein-Klein-Gordon Eqs. (2)(3)(4) can be found using the Hamilton's approach, and the quantum formalism can be determined and solved. In that sense, we use the metric Eq.(5) into Eq.(1) having where upper "•" represents the first time derivative, and the corresponding momenta are defined in the usual way Π q = ∂L/∂q. We obtain By performing the variation of the canonical Lagrangian with respect to N, i.e. δL canonical /δN = 0, where L canonical = Π qq − NH, it implies the constraint H = 0. Hence the Hamiltonian density is In the gauge N = 24e 3Ω and using the Hamilton equationsq = ∂H/∂Π q andΠ q = −∂H/∂q, we have the following where U = 24V(φ, σ)e 6Ω . Given a particular form of the potential V(φ, σ) one can derive a relation between the time derivative of the momenta such asΠ φ ∝Π σ , provided that ∂V /∂φ = α∂V /∂σ, where α is a constant. Such connection can be obtained considering two different configurations of the potential: where V 1 and V 2 are constants, and f(φ, σ) is an arbitrary function. We select the simplest form of the potential V(φ, σ) = A(φ)B(σ): where V 0 is a constants and λ 1 and λ 2 are distinguishing parameters. This class of potential has been obtained by other methods, see for instance [27,[34][35][36][37]. Therefore the time derivative of the momenta are simplyΠ φ = −λ 1 U andΠ σ = −λ 2 U, which solutions are where p φ and p σ are integration constants. Henceforth we will employ this scheme in order to find analytic classic and quantum solutions.

III. CLASSICAL SOLUTIONS
We start from the Hamilton equations Eq.(13) in order to find relations between the scale factor and the scalar fields, such asφ which solutions are where φ 1 and σ 1 are integration constants, and they be determined by suitable conditions. These expression are indeed general relations, since they satisfy the Einstein-Klein-Gordon equations Eqs. (6)(7)(8)(9). Then by taking into account the constraint H = 0, we obtain the temporal dependence for Π Ω (t) which allows us to construct a master equation: where the parameters m i , i = 1, 2, 3, are Subsequently by analyzing the parameter λ 2 = λ 2 1 + λ 2 2 we will obtain three different solutions.

D. Number of e-folds
Inflation is characterised by the number of e-folds it expands during such period, that corresponds to A ′′ phys > 0, where the primes represent the derivatives with respect to the cosmic time t phys . The e-folding function N e = dt phys H(t phys ) is described by t phys : computing the integral from t phys * to t phys end ; where t phys * represents the time when the relevant cosmic microwave background (CMB) modes become superhorizon at 50-60 e-folds before inflation ends at t phys end ; and H(t phys ) = H phys = A ′ phys /A phys is the Hubble parameter. Although, in our prescription we use a proper time t, we can evaluate the Hubble function in the corresponding gauge as H phys =Ω/N.
At the end of inflation the expansion rate of the scale factor must be null which translates to −H ′ phys = H 2 phys orΩ = 2Ω 2 . From here we can compute the time when inflation ends (t end ) given each particular case. In Table I appears the computation of the e-folding function N e and t end for each case given by the λ parameter. (1)

IV. QUANTUM SOLUTIONS
The Wheeler-DeWitt equation for this model is acquired by replacing Π q µ = −i ∂ q µ in (12). The factor e −3Ω may be factor ordered withΠ Ω in several forms. Hartle and Hawking [30] have suggested what might be called a semi-general factor ordering, which in this case would order e −3ΩΠ2 Ω as where p is any real constant that measures the ambiguity in the factor ordering for the variable Ω, in the following we will assume such factor ordering for the Wheeler-DeWitt equation, which becomes ∂σ 2 is the d'Alambertian in the coordinates q µ = (Ω, φ, σ) and the potential is U = +24V 0 e 6Ω−λ1φ−λ2σ . Then we transform the coordinates to obtain a potential that only depends on a single variable, employing the following transformation Now we find the partial derivatives of ψ with respect to the old coordinates (a, φ, σ) but in terms of the new variables from here we use these new relations in the quantum Hamiltonian density, obtaining At this point, we propose the following ansatz, Ψ = e 1 (c2κ+c3η) G(ζ), where the parameters c i are constants and G(ζ) is a function to be determined. By introducing the aforementioned into Eq.(51) we obtain the following differential equation of the function G, where the constants are The solution of Eq.(52) is dependant to the value of constant δ 0 , which turns in three different cases, I) δ = 0 implying that λ 2 1 + λ 2 2 = 3, II) δ < 0 implying that λ 2 1 + λ 2 2 < 3 and III) δ > 0 implying that λ 2 1 + λ 2 2 > 3, which can be analyzed in two different cases.
For this case, the Eq.(52) becomes which solution is therefore, the corresponding wave function for this case becomes Note that wave function has a damping behavior with respect to the scale factor, which is a required feature.
For this case, the Eq.(52) becomes, which is similar to that in [38], where Z ν is the Bessel function and ν = √ a 2 −4c κ the corresponding order, and its relations are which according to the constant b, the solution to the function G becomes where the constants are , .
Whilst c 5 < 0 and c 6 > 0 the wave functions Eqs.(61,62) will remain suppressed by the growth of the scale factor.
Yielding an expected damped wave function.

V. CONCLUSIONS
We studied a flat Friedmann-Robertson-Walker (FRW) multi-scalar field cosmological model. We introduce the corresponding Einstein-Klein-Gordon (EKG) system of equations and the associated Hamiltonian density. Exact solutions to the EKG system are derived by means of Hamilton's approach where a particular scalar potential of the form V = V 0 e −λ1φ−λ2σ was utilized, which gave rise to different cases dependant of the free parameter λ, for which the scalar fields, the scale factor and the e-folding function were found. The Hamiltonian density was employed in order to compute the Wheeler-DeWitt (WDW) equation, which was solved by means of a change of variables. An ansatz for the wave function was proposed which in turn allowed us to find the exact form of the generic function and its constants which was composed by, the aforementioned in terms of the free parameter λ. We found the model to be rather simple and its solutions to be quite interesting for a model building inflation.