Hamilton's approach in cosmological inflation with an exponential potential and its observational constraints
HHamilton’s approach in cosmological inflation with an exponential potential and itsobservational constraints
Omar E. N´u˜nez, ∗ J. Socorro, † and Rafael Hern´andez-Jim´enez ‡ Departamento de F´ısica, DCeI, Universidad de Guanajuato-Campus Le´on, C.P. 37150, Le´on, Guanajuato, M´exico School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom
The Friedmann-Robertson-Walker (FRW) cosmology is analyzed with a general potential V( φ ) inthe scalar field inflation scenario. The Bohmian approach (a WKB-like formalism) was employedin order to constraint a generic form of potential to the most suited to drive inflation, from here afamily of potentials emerges; in particular we select an exponential potential as the first non trivialcase and remains the object of interest of this work. The solution to the Wheeler-DeWitt (WDW)equation is also obtained for the selected potential in this scheme. Using Hamilton’s approach andequations of motion for a scalar field φ with standard kinetic energy, we find the exact solutionsto the complete set of Einstein-Klein-Gordon (EKG) equations without the need of the slow-rollapproximation (SR). In order to contrast this model with observational data (Planck 2018 results),the inflationary observables: the tensor-to-scalar ratio and the scalar spectral index are derived inour proper time, and then evaluated under the proper condition such as the number of e-foldingcorresponds exactly at 50-60 before inflation ends. The employed method exhibits a remarkablesimplicity with rather interesting applications in the near future. PACS numbers: 4.20.Fy, 4.20.Jb, 98.80.Cq, 98.80.Qc
I. INTRODUCTION
The inflation paradigm is considered the most accepted mechanism to explain many of the fundamental problemsof the early stages in the evolution of our universe [1–4], such as the flatness, homogeneity and isotropy observed inthe present universe. Another important aspect of inflation is its ability to correlate cosmological scales that wouldotherwise be disconnected. Fluctuations generated during this early phase of inflation yield a primordial spectrumof density perturbation [5–8], which is nearly scale invariant, adiabatic and Gaussian, which is in agreement withcosmological observations [9]. The single-field scalar models have been broadly used to describe such expansion, themost phenomenological successful are those with a quintessence scalar field and slow-roll inflation [10–18].Research on the inflationary topic is primarily done in two forms, one of them is to modify the General Relativityin a way that allows the inflationary solutions. The other one is the introduction of new forms of matter, with thecapability of driving inflation into the General Relativity, where one introduces a canonical scalar field. Essentially,in the studies of inflationary cosmology one imposes the usual slow roll approximation (SR) with the objective toextract expressions for basic observables, such as the scalar spectral index and the tensor-to-scalar ratio. The slow rollapproximations reduce the set of Einstein-Klein-Gordon (EKG) equations in such a way that one can quickly obtainthe solution to the scale factor in this approximation. However, there is also an alternative approach which allowsfor an easier derivation of many inflation results without such approximation, and that is known as the
Hamilton’sformulation , which is widely used in analytic mechanics. Using this approach we obtain the exact solution of thecomplete set of EKG equations without using the aforementioned approximation.There are many works in the literature that have extensively treated this type of problems, such as [19–26], where ∗ Electronic address: neophy@fisica.ugto.mx † Electronic address: socorro@fisica.ugto.mx ‡ Electronic address: [email protected] a r X i v : . [ g r- q c ] O c t the authors obtained exact and SR dynamical solutions of a system such as the scalar field potential employed inthe form V ( φ ) ∝ e − √ p φ . Prominently in [19] a radiation fluid was included and an extensive study of the evolutionof perturbation in power-law inflation was performed. And a thoroughly classical and quantum analysis, consideringdynamical and perturbative aspects, was implemented in [20–26]. However, even if the above mentioned worksrigorously examined the scalar field fluctuations, the observational parameters were not contrasted with the up todaymost accurate astronomical surveillance data, therefore one of the main reasons for this analysis. Further ahead theresults are discussed.Even more, in [23] the author deals with different values of λ in a V( φ ) ∝ e − λφ like potential as a model forthe primordial inflation. Indeed, Russo’s model is similar to ours albeit treated with a different approach. Russo’ssolution’s are equivalent as well, we were even able to find a direct transformation for the critical value of λ = √ τ ) = e τ (2 τ ) and φ ( τ ) = √ (2 τ − ln 2 τ ), we can see that under aparticular transformation we can change our solutions to his proper time τ , which is τ = e √ t , for this particularcase, the solutions we obtain with our method are φ (t) = φ − t + ˜P6P e √ t , A(t) = A Exp (cid:34) ˜P12 √ e √ t + 2 √ t (cid:35) where (˜P , P ) are integration constants, substituting τ in such solutions yields similar results. However, for theparticular case of λ <
1, Russo’s analysis concludes that the obtained inflation is eternal, but we found otherwise,we were able to compute the observables at the appropriate number of e-folding that ensures the end of inflation andcompare it with the observational data; thus, imposing a more rigorous analysis of the solutions, such results arediscussed within the work.There are other works of interest that have treated with a similar model, and even under the premise of SuperSymmetric Quantum Mechanics [28–30] where similar solutions are found.In our approach we use the quantum solution of the Wheeler-DeWitt (WDW) equation, a basic equation in quantumcosmology, from where we obtain a family of scalar potentials in the Bohmian formalism as in previous works [31, 32],which are found using a constraint equation on the superpotential function S and the amplitude of probability W,which are the solution to the Hamilton-Jacobi equation at quantum level, from here we select an exponential potentialfor the scalar field as the first non trivial case, which becomes the model of interest for this work. Thus, in that sense,we analyze the case of scalar field cosmology, constructed using a scalar field, and a general potential of the formV( φ ) = V e − λφ , for different values of λ for which we find the exact solutions to the EKG equations.In order to thoroughly contrast the model we analyze the inflationary observables such as the number of e-folds, thetensor-to-scalar ratio and the primordial tilt or scalar spectral index. The analysis of the observables was performedin the same framework as the Planck Collaboration (2018) [9]. The acceleration parameter was computed and usedas a constraint on the solutions so that only those that result in a positive acceleration are considered. The analysiswas performed using the scalar-field velocity as the parameter of evolution. Finally the results and discussions arepresented in the last section of this work.This work is arranged as follows. In section II we present the model with the action and the corresponding EKGequations for our cosmological model under consideration and the Hamiltonain density as well. In section III weuse the Hamiltonian density to compute the corresponding WDW equation, which is solved by using the Bohmianapproach (a WKB-like formalism), from there we obtain a family of potentials which are suited to model inflation;we also obtain the exact solution to the WDW equation using the separation variable method with a general scalarpotential. In section IV a complete and exact classical representation of a canonical scalar field with exponentialpotential in a flat FRW metric is presented; in subsection IV A the inflationary observables are derived and theconditions such as the e-folding function N e and the acceleration parameter are also computed and imposed on thesolutions, the results are presented as well. Finally, in section V we present our conclusions for this work. II. THE MODEL
We begin with the construction of the scalar field cosmological paradigm, which requires a canonical scalar field φ .The action of a universe with the constitution of such field is L [g , φ ] = √− g (cid:18) M −
12 g µν ∇ µ φ ∇ ν φ + V( φ ) (cid:19) , (1)where R is the Ricci scalar, V( φ ) is the corresponding scalar field potential, and M P = 1 / πG denotes the reducedPlanck mass. The corresponding variation of Eq.(1), with respect to the metric and the scalar field gives the Einstein-Klein-Gordon field equationsR αβ −
12 g αβ R = − (cid:18) ∇ α φ ∇ β φ − g αβ g µν ∇ µ φ ∇ ν φ (cid:19) + 12 g αβ V ( φ ) , (2) (cid:3) φ − ∂ V ∂φ = g µν φ ,µν − g αβ Γ ναβ ∇ ν φ − ∂V∂φ = 0 . (3)From Eq.(2) it can be deduced that the energy-momentum tensor associated with the scalar field is T ( φ ) αβ M P = 12 (cid:18) ∇ α φ ∇ β φ − g αβ g µν ∇ µ φ ∇ ν φ (cid:19) − g αβ V ( φ ) . (4)The line element to be considered in this work is the flat FRWds = − N(t) dt + e (cid:2) dr + r (d θ + sin θ d φ ) (cid:3) , (5)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 ∈ ( −∞ , ∞ ).The classical solution to Einstein-Klein-Gordon Eqs.(2, 3) can be found using the Hamilton’s approach, so we needto build the corresponding Lagrangian and Hamiltonian densities for this cosmological model. In that sense, we useEq.(5) into Eq.(1) having L = e (cid:34) N − ˙ φ N + 1M NV( φ ) (cid:35) , (6)the momenta and the associated velocities areΠ Ω = 12e N ˙Ω , ˙Ω = N12 e − Π Ω , (7)and Π φ = − e M N ˙ φ , ˙ φ = − M Ne − Π φ . (8)The Hamiltonian density, written as L = Π q ˙q − N H , when performing the variation of this canonical Lagrangianwith respect to N, i.e. δ L /δ N = 0, results that it is weakly zero: H = 0. Hence the Hamiltonian density is H = e − (cid:20) Π − Π φ −
24 V( φ )M e (cid:21) . (9) III. QUANTUM APPROACH
The Wheeler-DeWitt equation has been treated in many different ways and there are a lot of papers that dealwith different approaches to solve it, for instance in [33], they asked the question of what a typical wave function forthe universe is. In [34] there is an excellent summary of a paper on quantum cosmology where the problem of howthe universe emerged from a big bang singularity can no longer be neglected in the GUT epoch. On the other hand,the best candidates for quantum solutions are those that have a damping behaviour with respect to the scale factor,since only such wave functions allow good classical solutions when using the WKB approximation for any scenario inthe evolution of our universe [35, 36].The Wheeler-DeWitt equation for this model is acquired by replacing Π q µ = − i (cid:126) ∂ q µ in Eq.(9) a . The factor e − may be factor ordered with ˆΠ Ω in many ways. Hartle and Hawking [35] have suggested what might be called asemi-general factor ordering, which in this case would order e − ˆΠ as − e − (3 − Q)Ω ∂ Ω e − QΩ ∂ Ω = − e − ∂ + Q e − ∂ Ω , (10)where Q is any real constant that measures the ambiguity in the factor ordering for the variable Ω. In the followingwe will assume such factor ordering for the Wheeler-DeWitt equation, which becomes (cid:126) (cid:3) Ψ + (cid:126) Q ∂ Ψ ∂ Ω − e U( ϕ )Ψ = 0 , (11)where the field was re-scaled as φ = √ ϕ , and (cid:3) = − ∂ ∂ Ω + ∂ ∂ϕ is the d’Alambertian in the coordinates q µ = (Ω , ϕ )and the potential is U = +24V( ϕ ). A. The Bohmian formalism
The main idea of this approach is the use of a WKB-like ansatz for the wave functionΨ = W( (cid:96) µ )e − S (cid:126)(cid:126) ( (cid:96) µ ) , (12)where S (cid:126) ( (cid:96) µ ) is known as the superpotential function and W is the amplitude of probability that is employed inBohmian formalism [37], which is then introduced into Eq.(11), obtaining (cid:126) (cid:20) (cid:3) W − (cid:126) W (cid:3) S (cid:126) − (cid:126) ∇ W · ∇ S (cid:126) + 1 (cid:126) W ( ∇ S (cid:126) ) (cid:21) + (cid:126) Q (cid:20) ∂ W ∂ Ω − (cid:126) W ∂ S (cid:126) ∂ Ω (cid:21) − U W = 0 , (13) a Only for convenience in this section we set M P = 1 which are ordered in power of (cid:126) , (cid:126) (cid:20) (cid:3) W + Q ∂ W ∂ Ω (cid:21) − (cid:126) (cid:20) W (cid:3) S (cid:126) + 2 ∇ W · ∇ S (cid:126) + QW ∂ S (cid:126) ∂ Ω (cid:21) + W (cid:104) ( ∇ S (cid:126) ) − U (cid:105) = 0 . (14)From this expansion we can see that the first term corresponds to the quantum potential in a Hamilton-Jacobi likeequation, for (cid:126) we have a constraint equation and for the linear (cid:126) we have the imaginary part, thus( ∇ S (cid:126) ) − U = 0 [ Einstein − Hamilton − Jacobi ] , (15a) (cid:3) W + Q ∂ W ∂ Ω = 0 [ constraint equation ] , (15b)W (cid:18) (cid:3) S + Q ∂ S (cid:126) ∂ Ω (cid:19) + 2 ∇ W · ∇ S (cid:126) = 0 [ imaginary part ] . (15c) B. Our approach to obtain a family of scalar potentials
The following approach has been proposed before (see [31]), so we use the same steps to obtain the correspondingscalar potential. First we solve Eq. (15a) − (cid:18) ∂ S ∂ Ω (cid:19) + (cid:18) ∂ S ∂ϕ (cid:19) = e U( ϕ ) , (16)and using the following ansatz for the superpotential function S = e g( ϕ ) /µ , then Eq.(16) becomes an ordinarydifferential equation for the unknown function g( ϕ ) in terms of the scalar potential U( ϕ ), (cid:18) dgd ϕ (cid:19) − = µ U( ϕ ) , (17)this equations has several exact solutions, which can be generated in the following way when we consider thatU( ϕ ) = g ( ϕ )G(g), where g( ϕ ) and G(g) are functionals of the argument, yet to determine, then introducing this intoEq. (17), this one can be written in quadrature asd ϕ = ± dgg (cid:112) µ G , (18)in Table I appears the family of potentials and its corresponding generic functions found using this approach for theinflationary epoch,To solve the equation for the W function, Eq.(15c), we introduce the ansatz W = e u(Ω)+v( ϕ ) , by using the separationvariable method we obtain the following two equations for the functions (u,v),2 dudΩ − Q = 3k , (19)d gd ϕ + 2 dvd ϕ dgd ϕ = 3(3 + k)g , (20)where we have implemented a separation constant as 3k for simplicity. The corresponding solutions becomeu(Ω) = k + Q2 Ω + u , (21)v( ϕ ) = 3(3 + k)2 (cid:90) d ϕ∂ ϕ [ln(g)] − (cid:90) d gd ϕ ∂ ϕ g + v , (22)= 3(3 + k)2 (cid:90) d ϕ∂ ϕ [ln(g)] − µ (cid:90) d[U( ϕ )]( ∂ ϕ g) + v , (23) G(g) g( ϕ ) V( ϕ )0 g e ±√ ϕ g e ± λ ∆ ϕ V e ± λ ∆ ϕ G g g csch [3∆ ϕ ] V csch [3∆ ϕ ]G g − g sinh [3∆ ϕ ] V G g − n (n (cid:54) = 2) g (cid:2) sinh (cid:0) ϕ (cid:1)(cid:3) / n V (cid:2) cosh (cid:0) ϕ (cid:1) − (cid:3) − nn G ln g e v( ϕ ) V v( ϕ )e ϕ ) v( ϕ ) = (cid:0) ∆ ϕ (cid:1) G (ln g) e ω ( ϕ ) V ω ( ϕ )e ω ( ϕ ) ω ( ϕ ) = sinh(3∆ ϕ ) Table I:
The exact solutions to Eq. (18) are presented. Each row represents a different independent solutionand each column represents the form that each generic function must become for that specific solution and itscorresponding potential. and for the last equation, we use Eq. (17) for this transformation. Then, the function W has the following formW = W Exp (cid:20) k2 Ω + 3(3 + k)2 (cid:90) d ϕ∂ ϕ [ln(g)] (cid:21) Exp (cid:20)
Q2 Ω − µ (cid:90) d[U( ϕ )]( ∂ ϕ g) (cid:21) , (24)If we restrict the solutions in Eqs. (17, 24) to comply with the constraint Eq. (15b), this results in the nextconditions d vd ϕ + (cid:18) dvd ϕ (cid:19) − k − Q , dvd ϕ = 3(3 + k)2 1 ∂ ϕ [ln(g)] − µ d[U( ϕ )]d ϕ ( ∂ ϕ g) . (25)Considering the particular case for the function g = e − λ ϕ , and its corresponding scalar potential is U( ϕ ) = V e − λϕ ,yields a constraint between all constants (cid:18) µ V − λ (cid:19) − k − Q , (26)thus, the amplitude of probability W becomesW = W Exp (cid:26) k + Q2 [Ω + ± (k − Q) ϕ ] (cid:27) . (27)However, to study all the obtained potentials is an exhaustive work and as such we chose the exponential potentialas the first non trivial case to model inflation, and it remains as the object of interest through this work. IV. CLASSICAL SOLUTIONS USING AN EXPONENTIAL POTENTIAL OF THE FORM:
V = V e − λφ MP Using the gauge N = 24e , the Hamiltonian density is H = Π − Π φ −
24 V M e − λφ MP , (28)working with the Hamilton’s equations of motion we have the canonical velocities and momenta˙Ω = 2Π Ω , ˙Π Ω = 144 V M p e − λφMP , (29)˙ φ = − M P Π φ , ˙Π φ = − λ V M p e − λφMP , (30)from here one can obtain a relation between Π Ω and Π φ as˙Π Ω ˙Π φ = − P λ , (31)yielding a relation between the two momenta Π φ = − λ P Π Ω + P , (32)where P is an integration constant and remains a free parameter of the model to be adjusted with the observabledata. Substituting Eq.(32) in the Hamiltonian density, Eq.(28) ( H = 0), we arrive to the following relation˙Π Ω + 2 (cid:0) λ − (cid:1) Π − P λ P Π Ω + 72M P = 0 (33)which solution is 124 √ P P log (cid:34) ( λ − Ω − λ + √ P P ( λ − Ω − λ − √ P P (cid:35) = − t + P , (34)where P is a time-like integration constant. With Eq.(34), the canonical momentum Π φ and the rest of variables canbe solved. Hence the solutions areΠ Ω (t) = 6 λM P P λ − √ M P P λ − (cid:104) √ M P P ( t − P ) (cid:105) , (35)Π φ (t) = − P λ − − √ λP λ − (cid:104) √ M P P ( t − P ) (cid:105) , (36) φ (t) = φ + 72 M P P tλ − λM P λ − (cid:104) sinh (cid:104) √ M P P ( t − P ) (cid:105)(cid:105) , (37)Ω(t) = Ω + 12 λM P P tλ − λ − (cid:104) sinh (cid:104) √ M P P ( t − P ) (cid:105)(cid:105) . (38)This set of solutions is a complete and exact classical representation of a canonical scalar field with exponentialpotential in a flat FRW metric. Also from Eqs. (37, 38) we find the following relation between the coordinates fields(Ω , φ ): ∆ φ = 2 λ M P ∆Ω − P t. Moreover, V isV = −
32 P ( λ −
3) exp (cid:20) λφ M P − (cid:21) M , (39)for V > λ <
3; constricting the domain of λ we subsequently work with. A. Observables
Inflation is characterised by the number of e-folds it expands during such period, that corresponds to A (cid:48)(cid:48) phys > phys . The e-folding function N e = (cid:82) 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 timewhen the relevant cosmic microwave background (CMB) modes become superhorizon at 50-60 e-folds before inflationends at t phys end ; and H(t phys ) = H phys = A (cid:48) phys / A phys is the Hubble parameter. Although, in our prescription we usea proper time t, we can evaluate the Hubble function in the corresponding gauge as H phys = ˙Ω / N. Moreover, in orderto compute the function N e , it is convenient to use another variable to describe such quantity; let us use ˙ φ , the scalarfield velocity, instead of the proper time t as an evolution parameter. Hence, taking the time derivative of Eq.(37),we have t = 112 √ P P arccoth(x) + P , (40)where x = λ − √ λ M P ˙ φ − √ λ , (41)this variable x must satisfy that | x | > (cid:15) (cid:60) . At the end of inflation the expansion rate of the scalefactor must be null which translates to − H (cid:48) phys = H or ¨Ω = 2 ˙Ω . From here we can compute the time wheninflation ends, which is t end = 112 √ P P arccoth (cid:34) ∓ √ (cid:20) λ ± λ ± (cid:21)(cid:35) + P . (42)Then comparing above result and Eq.(40), at the end of inflation ˙ φ end is˙ φ end = ± P λ ∓ . (43)Henceforth we will use ˙ φ as the evolution parameter. The number of e-folds isN e = √ λ − (cid:26) √ λ M P P P + ( √ − λ ) log (cid:20) x ∗ + 1x end + 1 (cid:21) + ( √ λ ) log (cid:20) x ∗ − end − (cid:21)(cid:27) , (44)where x ∗ , implying ˙ φ ∗ , is to be determined at 50-60 e-folds before inflation ends, andx end = − √ λ ± √ λ (cid:18) λ − λ ∓ (cid:19) . (45)For the standard Bunch-Davies vacuum phase space distribution at the time when observable CMB scales leave thehorizon during inflation, t phys ∗ , the scalar amplitude of the primordial perturbations ∆ R has the form [8]∆ R = H ∗ π H ∗ φ (cid:48) ∗ , (46)where H phys ∗ = ˙Ω ∗ / N ∗ and φ (cid:48) phys ∗ = ˙ φ ∗ / N ∗ ; therefore in our proper time we have∆ R = 14 π ˙Ω ∗ N ∗ ˙ φ ∗ . (47)We need to have a relation of Ω ∗ = Ω( ˙ φ ∗ ) (given that N = 24e ) and ˙Ω ∗ = ˙Ω( ˙ φ ∗ ). First we have˙Ω ∗ = (cid:16) ˙ φ ∗ + 24M P (cid:17) λ M P , (48)and then Ω ∗ = √ λ − (cid:110) √ λ − + 24 √ λ M P P P − log (cid:104) (x ∗ + 1) √ − λ (x ∗ − √ λ (cid:105)(cid:111) , (49)where the term (x ∗ + 1) √ − λ (x ∗ − √ λ can be found when inverting Eq.(44); as a result we have thatΩ( ˙ φ ∗ ) = √ λ − (cid:110) √ λ − − N e ) + 48 √ λ M P P P − log (cid:104) (x end + 1) √ − λ (x end − √ λ (cid:105)(cid:111) , (50)therefore ∆ in terms of ˙ φ ∗ reads as∆ R = 14 π λ M P (cid:16) ˙ φ ∗ + 24 M P P (cid:17) ˙ φ ∗ ×× (cid:110) ( x end + 1) √ − λ ( x end − √ λ ×× exp (cid:104) − √ λ − − N e ) − √ λM P P P (cid:105)(cid:111) √ λ − , (51)once we fix ∆ R = 2 . × − [9] one can obtain Ω at 50-60 e-folds before inflation ends.Since we want to contrast the observables parameters with Planck data, we need to evaluate the scalar spectralindex n s at horizon crossing, which is defined asn s − ≡ d ln ∆ R d ln k (cid:12)(cid:12)(cid:12) ∗ = d ln ∆ R dN e dN e d ln k (cid:12)(cid:12)(cid:12) ∗ (cid:39) d ln ∆ R dN e (cid:12)(cid:12)(cid:12) ∗ , (52)where dN e / d ln k (cid:39)
1, since at horizon crossing (k = A phys ∗ H phys ∗ ) we have that ln k = N e + ln H phys ∗ andd ln H phys ∗ / dN e (cid:28) − H (cid:48) phys ∗ / H ∗ (cid:28)
1. In order to write down n s = n s ( ˙ φ ∗ ); we haven s − (cid:39) d ln ∆ R dN e (cid:12)(cid:12)(cid:12) ∗ = 1H phys ∗ ∆ R d∆ R dt phys (cid:12)(cid:12)(cid:12) ∗ = 1˙Ω ∗ ∆ R d∆ R dt (cid:12)(cid:12)(cid:12) ∗ , (53)then the time derivative of the scalar amplitude isd∆ R dt (cid:12)(cid:12)(cid:12) ∗ = 2∆ R (cid:34) ∗ ˙Ω ∗ − ¨ φ ∗ ˙ φ ∗ − ∗ (cid:35) , (54)thus the spectral index is n s (cid:39) − ∗ ˙Ω ∗ − φ ∗ ˙ φ ∗ ˙Ω ∗ . (55)0Once again we need to have a relation of ¨Ω ∗ = ¨Ω( ˙ φ ∗ ) and ¨ φ ∗ = ¨ φ ( ˙ φ ∗ ). First we have that ¨Ω = ¨ φ/ (2 λ M P ), so weneed only one relation ¨ φ ∗ = ¨ φ ( ˙ φ ∗ ); hence¨ φ ∗ = − λ M P (cid:104) ( λ −
3) ˙ φ ∗ − (cid:16) ˙ φ ∗ + 12M P (cid:17) M P (cid:105) . (56)Now we can finally compute the spectral index in terms of the desired parameter, thusn s (cid:39) − −
2( ˙ φ ∗ − P )˙ φ ∗ ( ˙ φ ∗ + 24M P ) (cid:104) ( λ −
3) ˙ φ ∗ − (cid:16) ˙ φ ∗ + 12M P (cid:17) M P (cid:105) . (57)The next observational parameter is the tensor-to-scalar ratio r, which isr = ∆ ∆ R , (58)where ∆ = 2H ∗ / ( π M ) is the tensor power spectrum. Then∆ = 2H π M = 2 π ˙Ω ∗ N ∗ M , (59)hence r = 2 λ ˙ φ ∗ (cid:16) ˙ φ ∗ + 24M P (cid:17) . (60)Finally, combining Eqs.(57,60) we obtainn s = 1 − r8 + (cid:34) √ − λ √ r (cid:35) . (61)Eq.(61) turns out to be a very interesting result since in the slow-roll approximation n s and r are related asn s = 1 − r /
8, which means that the exact solutions of this model yield a general relation between the scalar spectralindex and the tensor-to-scalar ratio that includes λ , which parametrises the slope of the potential.Furthermore, as mentioned before, inflation happens in a period such that A (cid:48)(cid:48) phys >
0, then H (cid:48) phys + H >
0; wherein our gauge or proper time, inflation occurs only when ¨Ω − >
0. Taking the time derivatives to Eq.(38) yields( λ −
1) coth (cid:104) √ P P (t − P ) (cid:105) + 4 √ λ (cid:104) √ P P (t − P ) (cid:105) + (cid:18) − λ (cid:19) > . (62)We have that for − < λ < ξ − < coth (cid:104) √ P P (t − P ) (cid:105) < ξ + , ξ ± = √ (cid:2) − λ ± ( λ − (cid:3) λ − . (63)Above condition, Eq.(63), must be fulfilled during the whole dynamics. Once again, it is convenient to parametrisethe evolution with ˙ φ : we substitute Eq.(40) into Eq.(63) having1 ξ − < λ − √ λ M P ˙ φ − √ λ < ξ + , (64)after giving a massage to above equation, we have − λ + 1 < ˙ φ M < λ − . (65)This is a very important outcome since we can relate previous results. For instance, the maximum limit of Eq.(65)corresponds to the exact expression of Eq.(43) (with the ˙ φ end+ root), which precisely indicates when inflation endsregarding any value of λ . Moreover, having the minimum value ˙ φ we can compute the number of e-folds. Followingthe same notation; substituting the maximum value as x end ( ˙ φ end = 24P0 / ( λ − ∗ ( ˙ φ ∗ = − / ( λ + 1)) in Eq.(41) such asx end = √ λ (cid:18) λ − λ − (cid:19) − √ λ , x ∗ = − √ λ (cid:18) λ − λ + 1 (cid:19) − √ λ , (66)then substituting above equations in Eq.(44), we haveN e = √ λ − (cid:26) √ λ M P P P + √ (cid:20) ( λ − ( λ + 1) (cid:21) − λ log[7 − √ (cid:27) . (67)With this Eq.(67) one can solve the number of e-folds without the dependency of ˙ φ ∗ . Besides, with this choice of˙ φ ∗ we find that the tensor-to-scalar ratio is exactly r = 32. This result implies that r is very large by contrasting tothe latest Plank survey 2018 [9], where r (cid:46) . < λ < √ −√ < λ < − (cid:104) √ P P (t − P ) (cid:105) < ξ − and coth (cid:104) √ P P (t − P ) (cid:105) > ξ + , (68)then performing the same previous procedure we have that˙ φ M < − λ + 1 and ˙ φ M > λ − . (69)Hence Eqs.(65, 69) are the constraints that must be satisfied in order to have an accelerated expansion. The twodistinct domains for λ , result in two different parameter spaces of { P , P , λ, ˙ φ ∗ } , in our analysis we restricted thisparameters to those regions that are the most relevant in regards to the observable functions and at the same timeensuring that the inflationary constrictions are satisfied. Besides, given that negative values of λ correspond to amirror-like parameter space with an equivalent outcome, thus we only report the region where λ >
0. Therefore for0 < λ <
1: P (cid:15) [ − ,
0) and P M P (cid:15) [13 , < λ < √
3: P (cid:15) [ − ,
0) and P M P (cid:15) [0 , . φ ∗ must satisfy Eqs.(65, 69).2 Figure 1:
Observational predictions with an exponential potential for 50-60 e-folds of inflation. The plot showstwo distinct regions, when fixing the primordial tilt 0 . ≤ n s ≤
1, for the tensor-to-scalar ratio by considering twodifferent domains of λ . The red contour represents the computed values when 0 < λ <
1, where 32 ≤ r (cid:46)
48; onthe other hand the blue region corresponds to the set of parameters when 1 < λ < √
3, having that65 . (cid:46) r (cid:46) . In Figure 1 we present the results when evaluating numerically the observables n s and r at 50-60 e-folds beforeinflation ends. By fixing the primordial tilt at 0 . ≤ n s ≤ λ , this yieldstwo well defined regions in the plot for the tensor-to-scalar ratio. The red contour represents the computed valueswhen 0 < λ <
1, where 32 ≤ r (cid:46)
48; on the other hand the blue region corresponds to the set of parameters when1 < λ < √
3, having that 65 . (cid:46) r (cid:46) .
85. Indeed, since the contribution of the tensor fluctuations in the tensor-to-scalar ratio is rather small [9], these results picture a disappointing outcome in regards to the phenomenologicalaspect of this inflationary scenario. Thus, once and for all ruling out this potential when studying the primordialinflation. However, such potential could be relevant in the description of the late time acceleration of the universe interms of quintessence [38].
V. CONCLUSIONS
The Quantum approach with a WKB-like ansatz for the wave function, was employed in a Bohmian formalism[37], where the proposal [31] was followed in order to find a family of canonical potentials. For the first non trivialcase to model inflation, we selected an exponential potential of the form V = V e − λφ Mp . With this concrete shape ofthe potential we computed the Hamiltonian’s equations of motion using a particular gauge; having thus the exactset of classical solutions of the relevant dynamical parameters. We found that V > λ <
3, thus3restricting the value of λ purely from dynamics. We computed the observable constraints: ∆ R , n s and r, in ourproper time; and we were able to evaluate them when the relevant cosmic microwave background (CMB) modesbecome superhorizon at 50-60 e-folds before inflation ends. We constrained the parameter space: { P , P , λ, ˙ φ ∗ } , inorder to have an accelerated expansion, following a very restrict set of conditions. To our knowledge, for this particularmodel of inflation, the observables have not been rigorously evaluated at horizon crossing, since it was believed thatthis scenario exhibited eternal acceleration when λ < λ <
3. However, the observable parameters present a very discouraging behaviour; for instance by fixing thescalar spectral index within the observational window (0 . ≤ n s ≤ ≥ (cid:46) .
064 (Planck [9]), there is rather large discrepancy with the latest Plank 2018 data.Even though the model is not as phenomenological fitting as expected, the employed method exhibits a remarkablesimplicity with rather interesting applications in the near future, perhaps it would require more considerations orfurther refinement; nevertheless, more potentials or specific models could be analyzed under such procedure.
Acknowledgments
RHJ acknowledges CONACyT for financial support. This work was partially supported by CONACYT 167335,179881 grants. PROMEP grants UGTO-CA-3. This work is part of the collaboration within the Instituto Avanzadode Cosmolog´ıa and Red PROMEP: Gravitation and Mathematical Physics under project
Quantum aspects of gravity incosmological models, phenomenology and geometry of space-time . Many calculations where done by the programminglanguage FORTRAN, Symbolic Program REDUCE 3.8. and Wolfram Mathematica 10.0 [1] A. H. Guth
Inflationary universe: A possible solution to the horizon and flatness problem, Phys. Rev. D , 347 (1981).[2] A. D. Linde A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy andprimordial monopole problems, Phys. Lett. B , 389-193 (1982).[3] J. D. Barrow and M. S. Turner
Inflation in the Universe, Nature , 35-38 (1981) [doi:10.1038/292035a0].[4] A. A. Starobinsky
A new type of isotropic cosmological models without singularity, Phys. Lett. B , 99 (1980).[5] A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. , 682 (1979)[Pisma Zh. Eksp. Teor. Fiz. , 719 (1979)].[6] V. F. Mukhanov and G. V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett. , 532 (1981)[Pisma Zh. Eksp. Teor. Fiz. , 549 (1981)].[7] H. Kodama and M. Sasaki Cosmological Perturbation Theory, Progress of Theoretical Physics Supplement , 1-166 (1984)[doi.org/10.1143/PTPS.78.1][8] B. A. Bassett, S. Tsujikawa and D. Wands, Rev. Mod. Phys. , 537 (2006) [arXiv:0507632].[9] P. A. R. Ade et al. (Planck Collaboration), Planck 2018 results. X. Constraints on inflation , [arXiv:1807.06211].[10] J. D. Barrow
Slow-roll inflation in scalar-tensor theories, Phys. Rev. D , 2729 (1995).[11] A. R. Liddle and R. J. Scherrer Classification of scalar field potential with cosmological scaling solutions, Phys. Rev. D ,023509 (1998) [doi.org/10.1103/PhysRevD.59.023509].[12] P. G. Ferreira and M. Joyce Cosmology with a primordial scaling field, Phys. Rev. D , 023503(1998) [doi.org/10.1103/PhysRevD.58.023503].[13] E. J. Copeland, A. Liddle and D. Wands Exponential potentials and cosmological scaling solutions, Phys. Rev. D Quintessence arising from exponential potentials, Phys. Rev. D Stability of multifield cosmological solutions, Phys. Rev. D Scalar-Tensor theories and current Cosmology, Problems of Modern Cosmology (2008) [arXiv:0812.1980].[17] M. Capone, C. Rubano and P. Scudellaro
Slow rolling, inflation and quintessence, Europhys.Lett The Early Universe, (Addison-Wesley publishing co., Illinois, 1998).[19] F. Lucchin and S. Matarrese,
Power Law Inflation, Phys. Rev. D , 1316 (1985) [doi:10.1103/PhysRevD.32.1316]. [20] D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys.Rev. D , 3936 (1990) [doi:10.1103/PhysRevD.42.3936].[21] B. Ratra, Quantum mechanics of exponential-potential inflation, Phys. Rev. D Inflation in an exponential-potential scalar field model, Phys. Rev. D Exact solution of scalar field cosmology with exponential potentials and transient acceleration, Phys. Lett. B
General solution of scalar field cosmology with a (piecewise)exponential potential,
JCAP , 004 (2011) [arXiv:1105.4515].[25] E. Piedipalumbo, P. Scudellaro, G. Esposito and C. Rubano,
On quintessential cosmological models and exponential po-tentials,
Gen. Rel. Grav. , 2611 (2012) [arXiv:1112.0502].[26] P. Fr´e, A. Sagnotti and A. S. Sorin, Integrable Scalar Cosmologies I. Foundations and links with String Theory,
Nucl. Phys.B , 1028 (2013) [arXiv:1307.1910].[27] O. Obreg´on, J. J. Rosales, J. Socorro and V. I. Tkach,
Supersymmetry breaking a normalizable wavefunction for the FRW(k=0) cosmological model, Classical and quantum gravity Inflation from supersymmetric quantum cosmology, Phys. Rev. D (4) 044008,(2010) [doi.org/10.1103/PhysRevD.82.044008].[29] J. Socorro, M. Sabido and W. Ram´ırez and M. G. Ag¨uero, Inflaci´on cosmol´ogica vista desde la mec´anica cu´antica super-sim´etrica , Ed. Notabilis Scientia (2013).[30] J. Socorro, M. D’oleire and L. O. Pimentel,
Time-varying cosmological term, J. Phys. Conf. Ser. , no. 1, 012007(2015) [doi:10.1088/1742-6596/654/1/012007].[31] W. Guzm´an, M. Sabido, J. Socorro and L. A. Ure˜na-L´opez,
Scalar potentials out of canonical quantum cosmology, Int. J.Mod. Phys. D (4), 641-653 (2007).[32] J. Socorro and O. E. N´u˜nez, Scalar potentials with multi-scalar fields from quantum cosmology an supersymetric quantummechanics , E ur. Phys. Journal Plus : 168 (2017) [arXiv:1702.00478].[33] G. W. Gibbons and L. P. Grishchuk, Nucl. Phys. B , 736 (1989).[34] L. Z. Fang and R. Ruffini,
Quantum Cosmology, Advances Series in Astrophysics and Cosmology Vol. 3, (World Scientific,Singapore, 1987).[35] J. B. Hartle and S. W. Hawking,
Phys. Rev. D , , 2960 (1983).[36] S. W. Hawking, Nucl. Phys. B , 257 (1984).[37] D. Bohm,
Suggested interpretation of the quantum theory in terms of ”Hidden” variables I, Phys. Rev. (2), 166 (1952).[38] F. Cicciarella and M. Pieroni, Universality for quintessence , JCAP08