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On exact solutions for quintessential (inflationary) cosmological models with exponential potentials

We first study dark energy models with a minimally-coupled scalar field and exponential potentials, admitting exact solutions for the cosmological equations: actually, it turns out that for this class of potentials the Einstein field equations exhibit alternative Lagrangians, and are completely integrable and separable (i.e. it is possible to integrate the system analytically, at least by quadratures). We analyze such solutions, especially discussing when they are compatible with a late time quintessential expansion of the universe. As a further issue, we discuss how such quintessential scalar fields can be connected to the inflationary phase, building up, for this class of potentials, a quintessential inflationary scenario: actually, it turns out that the transition from inflation toward late-time exponential quintessential tail admits a kination period, which is an indispensable ingredient of this kind of theoretical models. All such considerations have also been done by including radiation into the model.


I. INTRODUCTION
Scalar fields have been used extensively in cosmology since inflationary theories were first conceived in 1981 [1]. They are in fact useful to fill in the universe with a negative-pressure content, hence giving the typical inflationary behavior, with its exponentially increasing scale factor a(t) ∼ exp(H 0 t), where H 0 is the present value of the Hubble parameter. Scalar fields have been used both in minimal and non minimal coupling with geometry. (An overview of scalar-tensor theories is given in Refs. [2] [3]; a more updated review can be found in Ref. [4].) When observational results on Type Ia supernovae (SNeIa) have strongly supported the possibility that the universe is now in an accelerated stage of expansion [5] [6] [7] [8], it has become necessary to consider again a negative-pressure component. The late-time inflation so generated is however soft with respect to the earlier one and is again based on the dominance of at least one scalar field referred to as quintessence [9] [10] [11] [12] [13] or, earlier, as x-field [14] [15]. This gives rise to a sort of energy component which only appears measurable through gravitational effects. There is in fact a wide agreement that today a form of such a dark energy [16] [17] has to be taken into account in any realistic cosmological model. Seen as a constant Λ-term [18] [19] [20], dark energy density has been usually seen as the vacuum energy density [21], leading therefore to relevant problems due to huge discrepancies between theory and experiments [22] [19] [20] [23].
In this context, the scalar field in itself has gained new attention, and speculations about its nature and evolution have got new strength. Papers like Refs. [24] and [25] have been reconsidered again under a new light, as they show much of the role a scalar field could play when one wants to have an accelerated expansion today. Many models have thereof been constructed, first of all trying to consider more appealing kinds of potentials driving the dynamics of the scalar field. Among others, the potential has in fact to be seen as an important ingredient of all the involved theoretical models, and many are the trials to give prescriptions in order to reconstruct its form according to observational data (see Refs. [26] [27], for instance; but, for more theoretical considerations, see also Refs. [28] [10] [11] [12] [29] [30]). Here we take into account a specific class of potentials useful for the present evolution of the universe, later investigating all the relative involved cases so derived in the cosmological equations. The exponential-type potential has received much attention since the late 80's [31] [32] [24], (but see also Refs. [28] [33] [34] [35] [36] [37] [38] [39] [29] [40], [41], just to cite only some other old papers). In Refs. [42] [43] we have found general exact solutions for two classes of exponential potentials for a scalar field minimally coupled to gravity, in presence of another dust component (ordinary nonrelativistic pressureless baryonic matter), also showing [44] that such solutions seem to fit SNeIa data given in Ref. [6]. (See Ref. [45] for much more on confrontation of theoretical predictions with observational data.) Furthermore, in Ref. [46] other approximate and exact solutions for cosmology with exponential potentials can be found, while up to more recently it is still widely referred to (as, for example, in Ref. [47], where the Noether Symmetry Approach [48] [3] is used to probe the nature of dark energy, well underlining the role of the exponential potential in such a task). This seems to us a good motivation to investigate also on generalizations of this kind of potentials. (As a matter of fact, it is often studied also the case with the hyperbolic sine potential, which can easily be assimilated with some situations studied below).
Moreover, also for such a class of generalized exponential potentials it is possible to exhibit some special properties: it turns out, actually, that the Einstein field equations are integrable (in the Liouville sense) and separable, that is they can be analytically integrated, at least by quadrature. Finally, it is worth noting that our class of exponential potentials can be selected by finding the most general variables transformation which diagonalizes the scalar field kinetic-energy form, leaving the transformed Lagrangian simple.
Therefore, among other considerations, we then generalize the exponential potential to the extent allowed by the particular technique used in Refs. [42] [43] in order to integrate cosmological equations, still bearing in mind that, in order to describe the present universe, solutions should always allow a late-time acceleration. Such a technique is generally known as the Noether Symmetry Approach to cosmology [48], [3]. (Note that in Refs. [48], [3], such an application of the Noether Symmetry Approach already leads to prefer the exponentialtype potentials.) As in Refs. [42] and [43] we anyway do not completely adopt it here, limiting ourselves to import from it only the change of variables needed to solve equations.
As a matter of fact, we have to point out that this transformation (together with the choice of a specific kind of potential) has to be considered as the main drawback of the approach above, since it leads to a point symmetry which probably applies only to the background evolution, having nothing to do with the actual symmetries of the full theory. We analyze the exact solutions we find in this way, discussing when they are compatible with a late time quintessential expansion of the universe.
As a further step, we also consider and illustrate a possible scenario in the framework of the quintessential inflation paradigm with scalar fields, where an inflationary potential drives also the quintessential phase of the scalar field evolution, by means of an exponential form of its late time tail. In this connection, we discuss how such an evolution mechanism for the scalar field potential can be compatible with the so-called inflation-kination transition, when the field energy density is dominated by the kinetic energy of the scalar field ϕ. All this clearly leaves apart the greater complexity involved by the more general forms of the potential studied above, and will be the subject of a forthcoming paper trying to discuss more in general the quintessential inflation. Even if in this second paper we should be able to overcome the dichotomy present in this work, where the two different parts of the paper could well live apart, we believe that it is nonetheless already interesting to offer both the issues in a single paper, trying to unify the whole subject a little forcibly.
In Sec. II we introduce the class of potentials and the general cosmological setting for further considerations. In Sec. III, we systematically derive, when possible, general exact solutions case by case. In Sec. IV we discuss connections between exponential potentials quintessence and inflation. Finally, Sec. V is devoted to a conclusive discussion.

II. COSMOLOGICAL MODELS WITH (GENERALIZED) EXPONENTIAL PO-TENTIALS
Motivated by the considerations above, let us continue to assume a Friedmann-Lemaître-Robertson-Walker (FLRW) metric and fix the curvature scalar k = 0, since the spatially flat situation appears to be the most appropriate one according to the CMBR observational data (see Refs. [49] [50] for recent information). In what follows we have to consider the period of life of the universe after the decoupling time, in order to be realistic when taking the two components like dust and scalar field ϕ into account, like in Refs. [42] [43].
If ϕ is minimally coupled to gravity, and considering no special choice for constants, the cosmological equations are written The fluids filling the universe have equations of state given by being ρ m = Da −3 , where the parameter D ≡ ρ m0 a 0 3 is determined by the current values of ρ m and a. Let us also recall that and The cosmological equations can then be rewritten as It can be shown [48], [3] that the last two equations can also be deduced from an action principle based on the point Lagrangian (2.10) In this way, cosmological dynamics can be considered on a space with two coordinates a and ϕ,ȧ andφ being the velocities [48] [3]. The fact that L has a constant additive term is understood by considering Eq. (2.7), which can be seen as E L = 0, where E L ≡ ∂L ∂ȧ + ∂L ∂φ − L is the so called energy function associated with L. We find the constant to be zero for physical reasons (i.e., induced by the homogeneous and isotropic limit of Einstein's general field equations). In this way, when Eqs. (2.8) and (2.9) are solved, Eq. (2.7) is nothing but a constraint on the integration constants involved.

On defining
Eq. (2.10) formally becomes L = T − U. This has been already noted in Ref. [51], where we found the most general transformation leading to a diagonalized and simpler form of the kinetic energy with α and β nonnegative real numbers. This simplification in T implies, of course, a complication in the transformed U ′ . However in Ref. [48] we also found that there is at least one class of potentials not giving such a complicated expression for U ′ , rendering, on the contrary, the transformed L ′ simpler than L. It turns out that such potentials are exponentials: with C ≡ (3πG)/c 2 and A, B real parameters. They in fact allow an exact integration of the cosmological equations [48] [3]. Moreover, it is worth noting that, for such a class of exponential potentials, being V 1 , V 2 , λ 0 all free parameters, the Einstein field equations admit alternative Lagrangians, which is a circumstance somehow exceptional for a dynamical system, with many meaningful consequences: actually, it turns out that it is integrable (in the Liouville sense) and it is separable, i.e. there is a suitable change of variables by which it is splitted into separated one-dimensional systems (which are integrable by quadratures [52]).
We thus feel motivated to continue considering exponential-like potentials, taking now the class of potentials where ǫ = ±1, σ ≡ 12πG/c 2 is a fixed constant, A 2 and B 2 are arbitrary nonzero parameters, and λ ⋚ 0. Eq. (2.16) generalizes the exponential potentials already considered in Ref. [42], which are These cases (where ǫ = +1 and λ = 0) are therefore omitted in this paper. Moreover, as noted, the case with ǫ = +1 and λ = −2AB has been already treated and discussed in Refs. [48] [51] [3], and it will not be touched upon again here.
Since the theory is invariant under ϕ → −ϕ, the class of potentials in Eq. As (among others) in Ref. [42], then, let us use again the transformation This soon implies that A and B must have the same signs (i.e., AB > 0). In what follows, we choose to consider A > 0, B > 0, u > v, and u > 0, v > 0. We also disregard the absolute value sign in the expression for ϕ, which does not really limit the class of potentials we are studying. By virtue of Eq. (2.18), the potential in Eq. (2.16) becomes and we define for ǫ = +1, and for ǫ = −1.
In the case when A 2 = 0, anyway, we have to use the change of variables where the possibility of multiplying the Lagrangian by a costant, equal to 3 in our case, has been exploited In order to find equations and solutions for the cosmological model, at this point, it is better to consider the various different cases separately, and this is what we are going to do in the next Section.
In the following, we fix four conditions (see Ref. [43] for further details). First of all, we set the origin of time by choosing a(0) = 0, paying attention to the fact that, even if this can be done generally, it is not a real initial condition for a(t). The initial time t i n chosen for any specific model is in fact arbitrary, to some extent, and mathematical solutions can also present more than one zero, eventually infinitely many. In such cases, one must be careful in choosing only one physically meaningful period of time for the evolution of the universe, noting that we can accept no more than one zero at the beginning of time. An eventual second zero occurring in the future has then to be recognized as a time of recollapse of the universe. What happens later thus loses any meaning. We have thus to set u(0) = 0 or v(0) = 0 in evaluating initially Eq. (2.22), being it arbitrary for the moment to decide which one is better on solid ground. In Ref. [42], for instance, we decided to set to zero both, so selecting the tracking solution.
The second condition that we assume is that the present time (age of the universe) is the unit of time, t 0 = 1. Being t in unknown, this is not exactly the age of the universe, but the difference can be considered irrelevant for our purposes; anyway, even if it were be possible to avoid this condition, we thus get rid of a badly known quantity. Our third condition is to set a 0 ≡ a(t 0 ) = a(1) = 1, which fixes the normalization of a as standard, while the last condition is to set H(t 0 = 1) ≡ H 0 . After the choice of t 0 , this latter parameter turns out to be of order 1, even if it is not the same as the usual h. As a matter of fact, since we are using an arbitrary unit of time, such a parameter does not give any information on the observed value for H 0 [43].
As to λ = 0, as already said, some considerations are given in Ref. [42]. When λ = A 2 = 0 and ǫ = +1, for instance, Eq. (2.22) easily leads to general exact solutions for a(t) and ϕ(t) [42]. For λ = B 2 = 0, everything works the same, since there is symmetry in the potential in Eq. (2.16) with respect to a change of sign in ϕ. The situation where A 2 = 0, B 2 = 0, and ǫ = +1 has also been partially examined in Ref. [42].
In the next Section, we will thus focus our attention mainly on the case with A 2 = 0, B 2 = 0 and ǫ = −1, choosing not to consider at all the situation with λ = A 2 = 0 and ǫ = −1, since it involves an always negative potential and may also give a negative energy density, by virtue of Eq. (2.5). Most of all, let us note that it never allows the pressure to be negative, which forbids the possibility of describing the accelerated expansion we observe today.
We will also rule out the trivial case given by V ≡ λ, while the remaining situations will be dealt with systematically. A. The λ = 0 case As mentioned, this situation has already been partially treated in Ref. [42], where we consider only three cases: i) A 2 = 0 and ǫ = +1, ii) B 2 = 0, and iii) A 2 = 0, B 2 = 0, and ǫ = +1. The case with A 2 = 0 and ǫ = −1 has already been touched upon in Sec.
II, noting that it is not so interesting here in our considerations on present cosmology. On the other hand, when both A 2 and B 2 vanish, the potential vanishes and ϕ is free, giving p ϕ ≡ ρ ϕ ≡φ 2 /2 > 0, and w ϕ = 1. This means that the scalar field behaves like stiff matter and introduces a term ∝ a −6 in the cosmological equations, thus becoming soon negligible in the expanding evolution of the universe. In this way, ϕ would never produce a late-time inflation, for instance, as instead recent observations seem to require.
There only remains one case to discuss, i.e., With such assumptions, setting λ = 0 in Eq. (2.24) yields so that the related Euler-Lagrange equations arë where It can be shown that the solution reads as so that we get two functions of time t depending on four arbitrary parameters (u 1 , u 2 , t 1 , and t 2 ). By virtue of the energy constraint E L − = 0, which is writteṅ Eq. (3.4) yields The initial scale factor at t = 0 is such that the volume a 3 (t) is at the beginning a 3 (0) = (c 1 2 − c 2 2 )/4, in light of Eq. (2.18). For sake of simplicity, we can fix the origin of time in such a way as to get a vanishing scale factor (as in Ref. [42]). This, of course, is completely arbitrary, and we must always remember that the real beginning of time for the model under examination is actually afterwards, i.e. , at an after-decoupling moment which can be arbitrarily delayed for the kind of considerations we are making here. This also means, then, that we are setting c 1 = c 2 or, possibly, c 1 = c 2 = 0 (since we have put a(0) = 0). Anyway, both choices still allow D to be nonvanishing and positive, which we certainly require. Of course, this would be gained even taking c 2 = 0 simply. But let us here put c 1 = c 2 = 0, so that we can write from which we find that the two constants k 3 and k 4 have the same sign, i.e., k 3 k 4 > 0, Let us also note that, from Eqs.  after some algebra we find the cosmological solutions The scale factor in Eq. (3.11) yields a sort of cyclic universe with spatially flat hyper-surfaces (being k = 0). Since, in practice, it is nonvanishing only in an infinite number of limited intervals, this means that the solution describing expansion should be considered only in one stage (in its first stage, for example).
On the other hand, the present value of the matter density parameter is given by Here, first of all, we have to distinguish between the two cases with ǫ = +1 or ǫ = −1, i.e., between Lagrangian functions in Eq. (2.23) or Eq. (2.24), respectively. Then, as we can see, our considerations must also take the relative values of AB and λ into account.
With respect to this, note that the situation given by A 2 = B 2 = 0 describes a universe filled in with dust and free scalar field in presence of a positive cosmological constant, since the potential V (ϕ) reduces to λ only. This means we shall limit our considerations here to First of all, let us note again that the case with ǫ = +1 and λ = −2AB has been already studied elsewhere [48] [51] [3]. Thus, we do not comment on it here. Now, on the other hand, bearing in mind that u 2 > v 2 and AB > 0, the choice ǫ = +1 always gives a positive potential for the scalar field and leads to two different cases, since we can write (for We thus deduce the equationsü where the plus (minus) sign corresponds to the minus (plus) sign in the Lagrangian L + . In both cases we find α and β being integration constants. Furthermore, we find when the plus sign (2AB > λ) is taken in the Lagrangian, or when the minus sign (2AB < λ) is chosen there. (v 1 and v 2 are integration constants.) The integration constants are constrained by means of the equation E L + = 0, which iṡ This gives a relationship that makes it possible to determine the more physically meaningful constant D in terms of the other constants, i.e., when the minus sign is taken in Eq. (3.21), and with the plus sign. Focusing on the situation described by Eq. (3.19), i.e. choosing 2AB > λ, we see that the inequality (3.22) implies that −4αβω , giving αβ < 0. So, when β = −α, one finds u(t) = 2α sinh (ω 1 t), and we may write the volume a 3 (t) as the difference between the squares of a hyperbolic sine and a sine, where now 4α 2 ω 1 2 > v 1 2 ω 2 2 . On the other hand, choosing v 2 = 0 does not change the sign of the value of D and implies a(0) = 0, i.e., we can fix the origin of time as before, as well as in Ref. [42]. Of course, putting v 1 = 0 also leads to a(0) = 0, but then we always have v(t) = 0, so that a 3 (t) = 4α 2 sinh 2 (ω 1 t), being u 2 (0) = 0. Anyway, let us stress that, in the case under examination, it must be a(0) = 0 in order to avoid an initial negative value for the scale factor.
At the same initial time t = 0, the scalar field generally has the constant value which gives rise to an undetermined form for β = −α and v 2 = 0, unlike what was found in the case studied above, when λ = 0.
The cosmological solutions can be then written as , (3.26) so that the asymptotic behavior of the scale factor is an exponential one. As before, it thus expresses a late-time nonsoft accelerated expansion of the universe, but a better agreement with observations demands other considerations as to a more refined comparison with astrophysical data.
On the other hand, discussing the other allowed situation, with 2AB < λ, the condition , and now we can only say that v 1 v 2 < 0 gives αβ < 0, while v 1 v 2 > 0 leaves the possibility of having also αβ > 0. But let us note that a(t) > 0 involves u − v > 0 and, therefore, u = 0, which means that αβ = 0. If we then put β = −α (i.e., αβ < 0), we get u(t) = 2α sinh ω 1 t, as before; this, in turn, requires that Thus, we can also choose v 2 = −v 1 (implying α 2 > v 1 2 (λ − 2AB)/(2AB + λ)) and get v(t) = 2v 1 sinh (ω 2 t), so that we may write the volume a 3 (t) as the difference between the squares of two hyperbolic sines. As before, this choice implies a(0) = 0 for every value of α and v 1 . Moreover, at t = 0, the scalar field has in general the constant value which is again undetermined when we choose β = −α and v 2 = −v 1 . Now, the cosmological solutions are written as Of course, the expression of a(t) is non negative only when the constants α, ω 1 , v 1 , and ω 2 are such that the difference present in a(t) is positive. In such a case, when ω 2 < ω 1 we have an asymptotic exponential regime for the scale factor, as the one we already found before, with the other choice for the signs of the constants. The case with ω 2 > ω 1 , on the other hand, has to be discarded since it never happens. It asymptotically leads to a non physical situation, with a negative infinite value of the scale factor.
Let us consider, now, the Lagrangian in Eq. (2.24), resulting from the potential V − expressed in Eq. (2.21). Since V − has to be positive, as we have chosen, it must be being u 2 > v 2 , hence allowing uv ≷ 0. (Note, in fact, that when uv = 0, the potential simply reduces to V − = λ, therefore becoming negative.) Putting so that the resulting Euler-Lagrange equations arë These equations constitute a linear system of homogeneous coupled ordinary differential equations of second order, which can, more appropriately, be rewritten as: Actually, if we introduce the rotation matrix R (whose column are the eigenvectors ψ 1 , ψ 2 of A), the transformation (where we have posed y = (y 1 (t), y 2 (t)) and x = (u(t), v(t))) decouples our starting system.
One has¨ where RAR −1 is the eigenvalues' diagonal matrix. Such eigenvalues are and the eigenvectors associated with E 1 and E 2 are written as In terms of the transformed vector y, the system in Eq. (3.33) reduces to a linear system of homogeneous decoupled ordinary differential equations of second order, which can now be integrated exactly:  The solution, in its general form, is The transformation R −1 y provides the analytical expressions of u(t) and v(t): Through an appropriate setting of the integration constants it is possible to see that the imaginary part of the scalar field slightly oscillates around zero, and the real part dominates, as shown in Fig. (1). In Eq. (3.47), on the other side, the scale factor exhibits an oscillation around an exponential growth for large t, hence offering the possibility to describe the now observed stage of dark energy dominance.
C. The λ < 0 case As above, we have again to discuss separately the two cases with ǫ = +1 and ǫ = −1, also considering the relative value of AB with respect to λ. First of all, let us note that, when A 2 = B 2 = 0, we are once again in a universe with a free scalar field plus dust, but, now, in presence of a negative cosmological constant. Thus, we shall once again limit our following discussion to the situation with AB = 0, still describing the analysis as systematically as we can, in order to offer a useful tool in the whole issue.
The Lagrangian in Eq. (2.23) is written as giving rise to the equationsü where the plus (minus) sign corresponds to the minus (plus) in Eq. (3.49). The solution for for the minus sign in Eq. (3.49) (i.e., with 2AB < λ), and for the plus sign (i.e., with 2AB > λ). α and β being integration constants.
The constraint equation E L + = 0 now becomeṡ leading to for the plus sign in L + (corresponding to the minus one in Eq. (3.54)), and to for the alternative case with the minus sign in L + (and the plus sign in Eq. (3.54)).
With such choices the cosmological scale factor and the scalar field are But, even if this solution would be asymptotically good for describing an inflationary stage at present in the universe, we must rule it out because of the above considerations..
Turning, thus, to the situation leading to Eq. (3.56), we see that it has to be u 1 2 < α 2 (ω 2 2 /ω 1 2 ) in order to get D > 0. This does not imply special restrictions on the constants involved, and, without making any particular choice for them, we can now write the cosmological solutions as As to the time behavior of the scale factor, we here limit ourselves to note that it has to be a(0) = u 1 2 sin 2 (u 2 ) − α 2 sin 2 (β) which soon involves a constraint on the integration constants, that is, u 1 2 sin 2 (u 2 ) = This situation has to be considered, now, starting from the Lagrangian in Eq. (2.24).
Putting ω 2 ≡ σ 2 AB then yields from which we deduce the equations It turns out that this system is of the same kind as the one in Eqs.

PONENTIAL POTENTIAL MODELS
During inflation, the scalar field ϕ is supposed to be highly excited and slowly evolving ("rolling down") to the minimum of the potential. In this phase the potential energy of the inflaton field dominates the energy density of the universe. After the period of inflation ϕ rapidly oscillates around the minimum of the potential and, by virtue of coupling with other scalar and spinor matter fields, massive and massless particles are created and the universe reheats, starting the standard post-Big-Bang evolution. Simple models of inflation use a scalar field with a potential of the form V (ϕ) = 1 2 m 2 ϕ 2 , where m is the mass of the inflaton field. In the standard scenario the energy of the inflaton field is then transformed into mass-energy of created particles. Another possibility is offered by a quartic potential  [56]). Actually, in such scenarios the scalar field responsible of late time acceleration is nothing else but the remnant of the one which caused inflation at early time.
Indeed, a successful model of quintessential inflation is subject to the constraints of both inflation and quintessence simultaneously. For example, the minimum of the potential must not have been yet reached by the scalar field (generally, this requirement is satisfied by assuming the presence of a quintessential tail, that is assuming potentials with the minimum displaced at infinity). Moreover, it is needed that ϕ should not decay completely into a thermal bath of particles in order to survive until today, just to drive the late phase of accelerating expansion. As a consequence of this, the universe undergoes a period of kination expansion, when its energy density is dominated by the kinetic energy of ϕ. In this context, the standard reheating mechanism usually assumed to generate the primordial plasma does not work; however, the mechanism of gravitational particle production can still reheat the universe in the framework of quintessential inflation. Actually, even if the amount of radiation produced by such a mechanism is largely sub-dominant when compared with the energy field contribution during kination, the energy density of kination is redshifted by the cosmological expansion much faster than the radiation density and will start to dominate at some temperature.
Generally, the constraints and requirements which are to be satisfied by quintessential inflation are satisfied by using a multi-branch scalar potential, where the change of the potential, when the field moves from the inflationary to the quintessential frame of its evolution, is fixed by hand [53] or is the outcome of a phase transition arranged by the interaction with other scalar fields (see, for instance, Ref. [57]). Recently, the possibility has been investigated to connect the inflationary and quintessential expansion of the universe within the theoretical framework of particle production, usually developed in the very early universe. (Even if the quantum aspect of the creation mechanism is not yet very well known, some classical aspects due to kinetic collision in the hot dense regions of the early universe have been discussed in the literature.) This suggests that the same mechanism may occur in the late universe, also leading to late time cosmic acceleration (see, for instance, Refs. [58] [59]). In other approaches, a large variety of quintessential inflationary potentials are derived from theories of non-minimally coupled gravity (see, for instance, Ref. [60]). In our model the quintessential inflation is formulated in terms of a multi-branch scalar field, driving both the inflationary and the quintessential phases of the evolution of the universe. The quintessential tail is realized through an exponential potential (hence choosing the latter as our working potential for quintessence); on the contrary, to describe the inflationary plateau we do not fix any inflationary potential, but we propose a parametrization of the inflationary scalar field equation of state, and implement the transition from an inflationary stage to a kination evolution, which is characterized by the value w ϕ = 1 of the equation of state, and corresponds just to the asymptotic in the past value for the equation of state of the exponential potential scalar field.

A. The fiducial cosmological model
In order to illustrate our paradigm, we use, as fiducial cosmological model, the one considered in Refs. [43] [45]. This model is based on the simplest form of exponential potential of the quintessence field 2) and the assumption that the universe is spatially flat and filled in with dark matter and scalar field. We postpone to a forthcoming paper the detailed investigation of the impact of a kination-dominated phase generated by our class of potentials. The equations that determine the dynamics of this model are the Friedmann equation (where we use units in which 8πG = c = 1, H =˙a a is the Hubble constant, ̺ m ∼ a −3 is the density of matter, and ̺ ϕ = 1 2φ 2 + V (ϕ) is the energy density of the scalar field), the Raychaudhury equation   parameter and H 0 the present value of the Hubble constant, we thus obtain the following set of equations that contain only dimensionless variables with prime denoting derivative with respect to u. Such equations can be solved analytically but we here limit ourselves only to present the plots of some important quantities of the model, such as w ϕ (u), Ω m (u) and Ω ϕ (u). (See Fig. 2 and Fig. 3 for their behaviors.) This model of the universe is described by an exact solution of the dynamical equations.
The arbitrary parameters that appear in the solution are determined by specifying the initial conditions; we in fact require that at the present time, e.g. at u = 0, one should have Ω m (u = 0) = 0.3 and Ω ϕ (u = 0) = 0.7. The variable u is such that it decreases as time increases, and, at early times, the scalar field is almost constant and only recently it starts to increase. On the other hand, the potential of the scalar field at early times is constant and only recently is rapidly decreasing, while in the future it assumes a constant value again.
The dark energy equation of state w ϕ parameter in the far past is equal to −1, so that at the early stage of the evolution of the universe dark energy behaves as a cosmological constant, but (as is shown in Fig. 3, through the behaviors of the Ω parameters) it only recently has started to dominate the expansion rate of the universe.
The quintessential exponential potential admits, as said, exact solutions of the Einstein field equations. Actually, from them we find [42] which was obtained using as suitable unit of time the age of the universe, i.e. t 0 = 1. Thus, at the present time ϕ 0 = 0. At the time of reheating t is virtually zero, so that we may set (2). We now shift ϕ to x ≡ ϕ − ϕ in , so that the potential may be written as where the prefactor 4 is due to the particular choice of units (and, of course, is also the initial value for V ) and represents the value of the effective cosmological constant at that time, so that it has the dimension of a [length] −1 . In our units, the unit length is of the order of Hubble length, just as the current estimate for Λ. An effective cosmological constant near to 4 is thus slightly greater than this and slowly evolves towards a smaller value of 2 nowadays.
Let us now consider more realistically the inclusion of radiation, too, into such a model.
In this case the dynamical equations as far as we know do not have analytical solutions, and therefore we will rely on numerical computations. Following the procedure used above, we use the variable u instead of time t together with the rescaled variables, so that the equations contain only dimensionless variables and can now be written in the form where̺ r ∼ a −4 is the rescaled energy density of radiation.
We can numerically solve this system of coupled equations, specifying the initial condition at u = 60, for example, and assuming that ϕ (30)

B. Considerations about the inflation-kination transition
On the other side, as remarked above, an indispensable ingredient of quintessential inflationary scenarios is the existence of an early kination-dominated (KD) era, where the universe is dominated by the kinetic energy of the quintessence field. During this era, the expansion rate of the universe is larger compared to its value during the usual radiationdomination (RD) epoch. A generic reasonable inflation-kination period can usually be described by two main parametrizations of the transition, without specifying the details of the modelization. One is a polynomial parametrization for a 2 (η) (η being the conformal time) [62], and the other is a parametrization of the equation of state w, smoothly connecting the inflation (w = −1) and kination (w = 1) regimes. (See, for instance, Ref. [63], where a hyperbolic tangent parametrization is used to study gravitational reheating in quintessential inflation). Moreover, the impact of a kination-dominated phase generated by a quintessential exponential model or by quintessential power law, or also by a running kinetic inflation model, has been already investigated in literature (see for instance [64], [63], [65]). Here our approach is quite different: actually, we are not fixing a single-branch scalar field potential which drives both the early-time inflationary and the late-time quintessence evolution of the universe; we instead want to show that a quintessential exponential potential tail can be naturally connected to the inflation through a kination dominated era, since the asymptotic value (in the past) of the equation of state, w ϕ , is just w ϕ = 1, which characterizes the kination phase. We illustrate such a mechanism by using a new parametrization of the scalar field equation of state, which could also be used to study statistical properties of massive non relativistic bosons arising at the first stage of reheating as a result of a quantum decay of a classical quantum field in inflationary cosmological model without slow rolling [66].
The FRW evolution of the universe in the inflationary epoch is described by the equations where we are using dimensionless units and a 0 = 3 8π l P l E P l E GU T 2 ≃ 10 −25 cm. The quantities ρ φ and P φ are related by the equation of state P ϕ = w ϕ ρ ϕ = (γ ϕ − 1) ρ ϕ , and their evolution is described by the continuity equatioṅ (4.14) From Eqs. (4.12), (4.13), and (4.14) it turns out that It is straightforward to see that for γ ϕ = 0 we have an exponential behavior for a(t); if γ ϕ = γ 0 is constant, there is a power-law expansion, and we have inflation iff γ 0 < 2 3 , the more de Sitter-like, the closer γ 0 to zero.
All this suggests parametrizing the dynamics of the universe during such a phase in terms of γ ϕ (t). Indeed, we have that dt . (4.20) The above Eqs. (4.16) and (4.20) are parametric equations for the potential V , and can allow to reconstruct the scalar field potential. Thus, let us consider a scalar field which at the beginning describes an inflationary stage, and undergoes a phase transition into a kination phase. This can be achieved by postulating a special form for γ ϕ (and, therefore, w ϕ ), and reconstructing the potential.
Our choice for γ ϕ is , (4.21) where β and t f are parameters giving, respectively, the rate of the phase transition and the time at which inflation ends. On the other hand, the parameter α is related to the asymptotic value of w ϕ , which in our model is w ϕ = 1. (For different values of α, it is possible to obtain a transition from vacuum into dust or radiation. Moreover, by using a slightly different parametrization of the equation of state, as ) it is also possible to obtain a double transition from vacuum into kination and into radiation, as illustrated in Fig. (9). Such a form, even if arbitrarily assigned, can easily implement the transition from an inflationary stage to a kination one, as we see in Figs. 6 and 7, where we plot the behavior of the equation of state and the reconstructed scalar field potential, respectively. We have to note that the potential changes rather roughly from its value during inflation to its final values, and that the asymptotic in the future value for the equation of state w ϕ = 1 corresponds to the asymptotic in the past value for the quintessential exponential potential. Thus, these behaviors can be naturally connected, as shown in Fig. 8, where the time-scale has been arbitrarily taken in order to show all the evolution from inflation into kination, toward the late-time exponential quintessential stage, and the values of the parameters α, β and t f are fixed as just optimizing the link between the early and late time evolutions. Finally, in Fig. 10 we compare the behavior of the scalar field potential (with radiation included) during the two stages: it turns out that the quintessential exponential potential asymptotically behaves like the inflationary one.

V. DISCUSSION AND CONCLUSIONS
After the historical role of scalar fields in inflationary cosmology, the quite recent reconsideration of their crucial importance relies upon the fact that they can improve a dynamical mechanism for giving rise today to a repulsive component in cosmic energy, the dark energy. The interest for a scalar field in the universe, otherwise usually thought of as a  matter-dominated one, is in fact tied to the very recent cosmological history revealed by astrophysical observations.
In this paper, our attention has been generally focused on the role the exponential-like potential has still to play in nowadays cosmology. At first, we have discussed some models generalizing the simple exponential form of the potential, in order to derive general exact solutions. The technique used has been very simple, being it based on a given ad hoc  apply to the variables a and ϕ involved, but also on the natural kind of potential V (ϕ) to study. On the other hand, the results we have found in this first part of the paper are many and not always easily discernable. Basically, they have been mathematically derived but not yet appropriately discussed on physical ground. This deserves, of course, further investigations in a forthcoming paper, but we can however try here to sum up what has come out as more interesting for nowadays cosmology.
When λ = 0 in the potential, first of all, the only case we have discussed here is for A 2 = 0 , B 2 = 0 , ǫ = −1. The universe then displays infinitely many periods of closed evolution, and one should choose which branch to investigate, anyway without succeeding in describing the currently observed behaviour. The situation with λ > 0, on the other hand, depends on the sign of ǫ, even if it presents accelerated asymptotic evolutions both for ǫ = +1 and ǫ = −1. In the latter case, such an acceleration is however obtained together with an oscillating behavior. This is also true for λ < 0 when ǫ = −1, while for ǫ = +1 we again find a closed universe.
It remains to be seen that the only physically acceptable situation (and solution) is given for λ > 0 and ǫ = +1. In a forthcoming paper we are going into details of the cosmological evolution, with regard to each specific solution discussed above. In particular, we are going to perform the necessary confrontation of this theoretical output with the observational data sets.
The second part of the paper is still connected with the exponential potential. But, now, we instead focus on considering and illustrating a possible quintessential inflationary scenario, formulated in terms of a multi-branch scalar field, driving both the inflationary and the quintessential phases of the evolution of the universe. The quintessential tail is realized through an exponential potential (choosing, thus, the latter as our working potential for quintessence); on the contrary, for describing the inflationary plateau we do not fix any inflationary potential, but we propose a parametrization of the inflationary scalar field equation of state, and implement the transition from an inflationary stage to a kination evolution, which is characterized by the value w ϕ = 1 of the equation of state, which is needed in any reasonable quintessential inflationary model and corresponds just to the asymptotic in the past value for the equation of state of the exponential potential scalar field. It turns out that the reconstructed potential changes rather roughly from its value during inflation to its final values, and the asymptotic in the future value for the equation of state w ϕ = 1 corresponds to the asymptotic in the past value for the quintessential exponential potential.
Thus, we want to stress again that these behaviours can be naturally connected, and the exponential form of the scalar field potential driving the late stage of the universe could indeed be the asymptotic late time behaviour of the inflationary scalar field, which transits from inflation into kination, toward the late quintessential stage. It is worth noting that such a conclusion is somehow independent of the mechanism proposed for the evolution of the scalar field potential, being only based on a parametric description of the very early inflationary dynamics of the universe, and on the property of the equation of state w ϕ = 1, which characterizes our exponential potential. In our cited forthcoming paper we are also going to select, among all solutions, the cases which preserve such asymptotic behaviour w ϕ = 1.
Of course, this mechanism for driving the transition from the inflationary evolution toward the late time accelerated expansion has to be considered as mainly exploratory, and some topics need more investigation. For example, the particles production, that is the gravitational production during the reheating or the preheating, in which particles are produced by virtue of the variation of the classical inflaton field, needs to be investigated in a forthcoming paper, considering the scalar field potential given by Eq. (4.19). (As to this, we have to remind that, in order to study preheating, it is necessary to couple the classical inflaton field ϕ to a massless quantum scalar field χ, as for instance in the Lagrangian where R is the scalar curvature, ξ is the gravitational coupling and g is a quartic coupling constant between φ and χ.) To conclude, let us note that considering radiation in the model only changes the evolution of the Ω parameters. Radiation initially dominates on matter, while later on the energy densities of matter and radiation become equal; after that, for some time matter dominates in the universe, while dark energy starts dominating the expansion rate of the universe only afterwards. These results seem to confirm other investigations of effects of radiation on the evolution of the quintessence field, according to which the energy density of the scalar field during the epoch of nucleosynthesis (z ∼ 10 9 ) is much smaller than the energy density of radiation. While the kinetic term in the scalar field energy density vanishes and the potential term becomes constant, the dark-energy term behaves like an effective cosmological constant, not affecting the process of primordial nucleosynthesis.