Integrable scalar cosmologies with matter and curvature

We show that some integrable (i.e., exactly solvable) scalar cosmologies considered by Fr\'e, Sagnotti and Sorin (Nuclear Physics B 877(3) (2013), 1028--1106) can be generalized to include cases where the spatial curvature is not zero and, besides a scalar field, matter or radiation are present with an equation of state $p^{(m)} = w \rho^{(m)}$; depending on the specific form of the self-interaction potential for the field, $w$ can be arbitrary or must be fixed suitably.


Introduction
Scalar fields in cosmology. The consideration of scalar fields in cosmological models has a long story, and arises from different motivations. On one hand the inflaton, i.e., the entity driving inflation, is often modeled as scalar field. This approach originates from the work of some scholars at the beginning of the 1980's: let us mention, in particular, Linde [17], Madsen and Coles [19]. On another hand, one can use a scalar field as a model for dark energy. This idea seems to have appeared in a 1988 paper by Ratra and Peebles [27]; Caldwell, Dave and Steinhardt [5] are credited for introducing, ten years later, the term "quintessence" to indicate a scalar model of dark energy. It is hardly the case to recall that the notion of dark energy was experimentally consolidated during the same years, thanks to the publication (between 1998 and 1999) of the observational results by the High-Z Supernova Search Team [28] and the Supernova Cosmology Project [23]. To our knowledge, Saini, Raychaudhury, Sahni and Starobinsky [31] were the first to set up a strictly quantitative connection between a scalar field model of dark energy and the observational data of [23,28]. More precisely, in [31] an unspecified self-interaction potential for the dark energy scalar field is assumed, and the most probable shape of this potential fitting the data of [23,28] on luminosity distance and redshift is determined for the epoch ranging from present time to the time when the spatial scale factor had half of its present value. Most of the papers cited before and in the sequel, as well as the present work, rely on a paradigm in which the universe is homogeneous and isotropic at each time, and the scalar field (modelling the inflaton or dark energy) is treated classically, so that its values are ordinary numbers rather than operators ( 1 ). Due to the assumption of homogeneity and isotropy, the spacetime metric has the form of Friedmann-Lemaître-Robertson-Walker (FLRW), possibly with non zero spatial curvature. For the same reason, the scalar field only depends on time. These cosmological models might involve, besides the scalar field, some other form of matter described as a perfect fluid; here and in the sequel the term "matter" is used in a broad sense, including especially the case of a radiation gas. The presence of a matter fluid is typical of models where the scalar field represents dark energy; these encompass all the story of the universe except for the very early stages, and include epochs in which the matter contribution is dominant with respect to that of dark energy. On the contrary, models for the very early, inflationary stage of the universe typically ignore the role of matter and focus the attention on the inflaton scalar field. All the above models give rise to systems of ODEs, describing the time evolution of the main actors which include, especially, the scale factor in the FLRW metric and the scalar field. Let us point out some additional features of these cosmological models. It is commonly assumed that the scalar field is minimally coupled to gravity, and that it does not interact directly with matter if the latter is present; correspondingly, the stress energy tensors of the scalar field and of the matter fluid are separately conserved. In absence of different indications, all papers cited in the sequel fit into the scheme just outlined (spatial homogeneity and isotropy, minimal coupling of the scalar field with gravity, no direct interaction between the field and matter).
Integrable scalar cosmologies. Since the late 1980's, the rising physical interest for cosmologies with scalar fields stimulated the search for integrable models, in which the evolution equations can be solved explicitly. It turned out that this is possible for models with certain features, like a special functional form for the self-interaction potential of the scalar field. Of course, the availability of exact solutions is an advantage with respect to numerical integration, since it allows to identify details and conceptual aspects that could be missed by a numerical approach. Since the very beginning of these investigations, it was understood that exact solutions can be obtained assuming an exponential form for the self-interaction potential of the scalar field; let us describe these results with the normalizations of the present work, borrowed from [13] which uses a suitable dimensionless version ϕ of the scalar field (up to a purely numerical factor, ϕ is the scalar field multiplied by the square root of the gravitational constant; for the precise definition see subsection 2.2, especially Eq. (2.2.9)). In 1987 Barrow [1] assumed a potential of the form V(ϕ) = const. e −λϕ (with λ another arbitrary constant), and a vanishing spatial curvature; he presented a particular exact solution of the evolution equations (but not the general solution) for the case of a scalar field alone. In the previously mentioned paper [27] of 1988, Ratra and Peebles considered the same exponential potential as in [1] (with λ > 0) and a vanishing spatial curvature; they presented some particular exact solutions of the evolution equations, both for the case of a scalar field alone and for a model with a scalar field and pressure-less matter (namely, dust) ( 2 ). In the same year, Burd and Barrow [4] considered again the potential V(ϕ) = const. e −λϕ (with λ > 0), with possibly non-zero spatial curvature in arbitrary spacetime dimension n + 1; they proposed a detailed stability analysis of the model and presented some new exact solutions exhibiting the transition to power-law inflation at late times. In 1990 de Ritis, Marmo, Platania, Rubano, Scudellaro and Stornaiolo [7] investigated a cosmology with a scalar field and no matter fluid in the case of zero spatial curvature, suggesting its use as an inflationary model. To analyze the evolution equations, they proposed a systematic use of the Lagrangian viewpoint. In this way they proved that the only potentials giving rise to a Noether symmetry for the system have the form (with the normalizations of the present work) V(ϕ) = const. e ϕ + const. e −ϕ + const. . Moreover, they constructed the general solution of the evolution equations for this class of potentials. The same authors extended these results to the case of a field non minimally coupled to gravity in [8]. In 1998 Chimento [6] investigated cosmological models driven by two scalar fields, one of them self-interacting with an exponential potential of the form V(ϕ) = const. e −λϕ (as in [1,3,27]) and the other one free and massive. Exact general solutions were obtained and examined in detail; in particular these solutions show the transition from expansion dominated by the free scalar field to that dominated by the self-interacting field, yielding a power-law inflation. The potential V(ϕ) = const. e ϕ + const. e −ϕ + const. was reconsidered in 2002 by Rubano and Scudellaro [29], and in 2012 by Piedipalumbo, Scudellaro, Esposito and Rubano [25], again for zero spatial curvature but in presence of dust. These authors showed that the solvability of the evolution equations is preserved even with the addition of dust. They proposed this model for describing dark energy and dust up to the present time, and started an analysis of the physical significance of the solutions. All papers [1,6,7,8,25,27,29] considered a spacetime with the "physical" dimension 3 + 1. The integrable cosmologies of Frè, Sagnotti and Sorin [13]. In 2013 these authors considered the FLRW cosmologies with a self-interacting scalar field, no matter fluid and zero spatial curvature, in arbitrary spacetime dimension n + 1. Their analysis was based on the Lagrangian formalism and on the possibility of gauge transformations for the time coordinate. More precisely, the approach of [13] describes a cosmology of the above type as a Lagrangian system with two degrees of freedom plus the constraint of zero energy; the Lagrangian coordinates are, basically, the instantaneous values of the scale factor and the scalar field. The Lagrangian depends on the scalar field self-potential and on a gauge function (describing the choice of the time coordinate), to be specified according to convenience in the investigation of integrable cases. For nine classes of self-potentials individuated in [13], with a convenient choice of the gauge function and a suitable change of the Lagrangian coordinates, the Lagrange equations are solvable by quadratures (for arbitrary initial data) for one of the following reasons: i) the Lagrangian is quadratic, so it gives rise to linear evolution equations; ii) the Lagrange equations have a triangular structure, which essentially means that one of the equations involves only one of the (new) Lagrangian coordinates; the thesis [15], and somehow refines the investigation of the authors who discovered this integrable case [25,29]. The behavior of the solution of this model depends on the parameters in the potential V(ϕ) and on the integration constants; we show how to choose them so that the early universe is dominated by matter, the late universe by dark energy and the (dimensionless) energy densities of these two entities at present time have the values suggested by observational evidence (of course, in this computation we assume that the spacetime dimension is 3 + 1). Next, we consider a scalar field with a second class potential V(ϕ) to which, as already indicated, it is possible to add matter and curvature. Among the many integrable cases of this model, listed elsewhere in the paper, we choose that with zero spatial curvature and matter with arbitrary equation of state parameter w (and with the parameter γ in V(ϕ) fixed by the previously mentioned prescription γ = w, which ensures the triangular structure of the Lagrange equations). Again, there are many subcases of this model: we choose one with 0 < w < 1 and suitable signs of the coefficients in the potential V(ϕ), which exhibits a Big Bang and no Big Crunch. The asymptotic behavior of the relevant observables near the Big Bang and in the very far future is determined for arbitrary w ∈ (0, 1). Sticking to this subcase with a second class potential, we subsequently fix the spacetime dimension to be 3+ 1 and set w = 1/3 (radiation gas). With these choices, we individuate a solution of the Lagrange equations that, although being entirely built with elementary functions, has a rather complicated structure implying a stage in which the scale factor grows exponentially with the cosmic time, preceded and followed by epochs in which the scale factor behaves like a power of the cosmic time. We show that the free parameters of the model and the constants of integration in this solution can be adjusted so that the exponential growth occurs in the very early universe and and the scale factor is increased, say, by a factor 3 × 10 20 in a time interval between (1/2) × 10 −34 sec and 10 −34 sec after the Big Bang. This is the behavior postulated by inflationary theories: we think it can be of some interest to obtain such a behavior from an exact solution of Einstein's equations with the simultaneous presence of radiation and of a scalar field; clearly, the latter ought to be interpreted the inflaton in this model. The last case study considered in this paper is associated to the seventh class of potentials V(ϕ); the spatial curvature is zero and a type of matter is present with w = (ℓ − 1)/(ℓ + 1), where ℓ 2 is an integer. This case is discussed since it provides a rather interesting example of separable system (see item (iii) in the previous paragraph). Indeed, upon introducing a suitable pair (x, y) of Lagrangian coordinates, the Lagrangian is found to be the sum of two Lagrangians depending separately on x, y (and their time derivatives). The first Lagrangian describes a non-linear repulsor with potential energy proportional to −x 2ℓ , the second one described a non-linear oscillator with potential energy proportional to y 2ℓ . On account of energy conservation for these separate subsystems, we derive quadrature formulas for their motions and then return to the original variables of the model, i.e., the scale factor and the scalar field, ultimately performing a qualitative and quantitative analysis of their behavior. In this way we find, for example, that the system exhibits a Big Bang and an exponential growth of the scale factor (as a function of cosmic time) in the very far future; at intermediate times, there is a competition between the behaviors associated to the previously mentioned repulsor and oscillator, whose effects depend on the parameters in the potential V(ϕ) and on the values assumed for the constants of integration.
Organization of the paper. Section 2 and the related Appendix A present some general facts on cosmologies with a scalar field minimally coupled to gravity and a matter fluid (not interacting directly with the scalar field, with a given equation of state p (m) = p (m) (ρ (m) )). After some generalities about the action functional and the stress-energy tensors of the field and of the matter fluid, we focus the attention on the homogeneous and isotropic case, in which the spacetime metric has the FLRW form, and the equation of state for matter is assumed to have the form p (m) = w ρ (m) ; this yields the Lagrangian setting with two degrees of freedom mentioned in the previous paragraphs. Section 3 considers the nine potential classes V(ϕ) of Frè, Sagnotti and Sorin, and lists the integrable cases that we have obtained adding matter or curvature. Section 4 and the related Appendices B, C, D present the explicit solutions for some integrable cases of Section 3, accompanied by a qualitative and quantitative analysis. Here we discuss the results mentioned in the previous paragraph, i.e.: a review of the Rubano-Scudellaro-Pedipalumbo-Esposito model with dust [25,29] (subsection 4.1); a general discussion of the class 2 potentials with the addition of matter (subsection 4.2), that includes the previously mentioned model for inflation (paragraph 4.2.3); an analysis of an integrable case with a class 7 potential and matter, yielding the previously mentioned model with a nonlinear repulsor and a nonlinear oscillator (subsection 4.3).
Final remarks. (a) One could wonder if the present integrability results (or those of [13]) could be extended to the case of non minimal coupling between gravity and the scalar field. We refer mainly to the case of a standard curvature coupling, in which the action functional for the system contains of a term proportional to R ϕ 2 (with R the scalar curvature). This problem certainly deserves further investigation. There is some hope to obtain these extensions for the purely scalar models of [13], using some formal transformations proposed in the literature [14,20] to connect minimally coupled theories to systems with curvature coupling. However, the cited transformations refer to systems with no type of matter fluid, so they cannot be used for the cosmologies with matter of the present work. (b) We already pointed out that no direct interaction between the matter fluid and the scalar field is ever considered throughout this paper. However, let us mention that some integrable FLRW cosmological models with such an interaction have appeared in the literature; we refer in particular, to the very recent work of Piedipalumbo, De Laurentis and Capozziello [24], where the scalar field represents dark energy and a possible interaction with (dark) matter is considered (see also the references cited therein). (c) In most of the integrable cases presented in this work, a finite particle horizon appears; this fact can be checked by hand noting that the reciprocal of the scale factor, viewed as a function of cosmic time, diverges in a non-integrable way at the Big Bang. In the case of non-positive spatial curvature, the deep reason for this fact was pointed out in [12]; therein it was shown that the particle horizon is finite for all homogeneous and isotropic cosmologies with non positive spatial curvature, a self-interacting scalar field minimally coupled to gravity and a matter fluid with equation of state p (m) = w ρ (m) , fulfilling the strong energy condition. As shown in [12], the particle horizon is absent if, instead of a canonical scalar field, one considers a phantom field whose action functional contains an anomalous term corresponding to a negative kinetic energy. It would be of some interest to search for FLRW integrable cosmologies with a phantom scalar field and matter; this subject is left to future investigations. (d) In the present work, in [13] and in most of the other previously cited papers, the attention is focused on a "direct problem": find for arbitrary initial data the solution of a cosmological model with a pre-assigned potential for the scalar field and, possibly, with matter having a suitable equation of state. On the other hand, there is also an "inverse problem": find the scalar field potential producing a time evolution with a prescribed feature in a FLRW cosmologies with a purely scalar content, or including a matter fluid. To our knowledge, the first examples of this inverse approach date back to 1980's and 1990's: we will mention, in particular, the papers by Lucchin and Matarrese [18], Barrow [2], Ellis and Madsen [11], Eashter [10]. More recently, nice "inverse" results have been obtained by Dimakis, Karagiorgos, Zampeli, Paliathanasis, Christodoulakis and Terzis [9], and by Barrow and Paliathanasis [3]; the same approach is also partly employed in [12], for the case of a phantom field. The feature specified in the cited papers to determine the scalar field potential is, for example, the dependence on cosmic time of one of the following observables: the scale factor, the Hubble parameter, the ratio between the pressure and the density produced by the scalar field alone, or jointly by scalar field and matter. The distinction between the "direct" and "inverse" problems mentioned above is essential to understand the difference between the present work and the ones we have just mentioned. (of course, n stands for the spatial dimension). Spacetime coordinates are typically indicated with (x µ ), and the line element reads ds 2 = g µν dx µ dx ν . The metric (g µν ) has signature (−, +, ..., +) and the corresponding covariant derivative, Ricci tensor and scalar curvature are respectively denoted with ∇ µ , R µν and R.
We assume that the content of the universe consists of: (i) a scalar field φ (of dimension L −n+1 ), minimally coupled to gravity and self-interacting with potential V (φ) (of dimension L n+1 ); (ii) some kind of matter which can be described as a perfect fluid with mass-energy ρ (m) and pressure p (m) , fulfilling an assigned equation of state p (m) = p (m) (ρ (m) ). Such matter does not interact directly with the scalar field. Besides, let us remark that here and in the sequel the term "matter" is used in a very generic sense (e.g., it possibly refers to a radiation gas).
The action functional SS for the above model depends on the spacetime metric, on the scalar field history and on the matter history (defined as in [16]) with the law where g := det(g µν ) and κ n (of dimension L (n−1)/2 ) is, essentially, the square root of the universal gravitational constant. Note that SS is dimensionless in our units with = 1. Demanding SS to be stationary under variations of the metric (g µν ) entails the Einstein's equations where the r.h.s. contains the stress-energy tensors of the scalar field and of the matter fluid: with U µ indicating the (n + 1)-velocity of the fluid. The stationary condition for SS with respect to variations of the field φ gives the Klein-Gordon-type equation Finally, the stationarity of SS under variations of the matter history gives the conservation law for stress-energy tensor of the matter fluid, namely for some suitable real constant w, in principle arbitrary. When w = 0, the fluid is a dust; if w = 1/n the trace T (m) µ µ vanishes, as typical of a radiation gas. Besides, let us mention that the weak, dominant and strong energy conditions for T (m) µν are respectively equivalent to (see, e.g., [16,21]) Comparison with [13]. As already stressed, [13] considers a scalar field as the only content of the universe; thus, any statement of ours involving the matter fluid has no counterpart in the cited work.
Here and in the sequel, we employ notations as close as possible to those of [13]; however there are minor differences, to be pointed out step by step. For the moment, let us mention that our convention (−, +, +, ..., +) for the metric signature is opposite to the convention (+, −, −, ...) employed in [13]. Following [13], we indicate with d the spacetime dimension; however, differently from [13] we often refer to the space dimension n and write d = n + 1. In particular, our constant κ n coincides with the quantity k d of [13] (with d = n + 1).
In the sequel, as in [13] we restrict our attention to the case of a FLRW geometry that we describe using similar notations, apart from the symbol τ for cosmic time replacing the notation t c of [13]. In addition, let us stress that we admit arbitrary values for the constant, spatial sectional curvature, while [13] discusses only the case of zero curvature.

The homogeneous and isotropic case
From here to the end of the paper, the general model of the previous subsection is specialized to the case of a spatially homogeneous and isotropic universe.
The spacetime and its metric. To implement the above assumptions we consider a FLRW spacetime, given by the product of the time line and of a (simply connected) Riemannian manifold M n k of constant sectional curvature k (of dimension L −2 ). Using cosmic time τ and any system of coordinates x = (x i ) i=1,...,d for M n k , we have where dℓ 2 = h ij (x) dx i dx j is the line element of M n k and a(τ ) > 0 is the dimensionless scale factor; typically, the latter is fixed so that a(τ * ) = 1 at some reference time τ * . For our purposes it is convenient to use in place of τ a dimensionless "time" coordinate t, implicitly defined by the identity dτ = θ e B(t) dt , where B(t) is a dimensionless "gauge function" to be determined and θ is a constant of dimension T. This re-parametrization of time is suggested in [13] where, however, the analogue of Eq. (2.2.2) contains no dimensional constant θ and reads dτ = e B(t) dt; due to this, the coordinate t of [13] has dimension T.
Having θ at our disposal, we re-express the scalar curvature in terms of a dimensionless coefficient k, setting to compare with [13], let us recall that k = 0 therein. Having introduced the time coordinate t, we can regard the scale factor as a function of it, i.e., a = a(t); inspired again by [13], we write which coincides with the one given in [13] apart from the presence of the constant θ and from the extension to non-zero values for the curvature k of dℓ 2 . From here to the end of this work we will use a spacetime coordinate system where, as above, x = (x i ) are coordinates on M n k ; Greek indexes always range from 0 to n, Latin indexes from 1 to n. Moreover, we indicate derivatives with respect to t with dots, namely, In Appendix A we report the explicit expressions of the Ricci tensor R µν and of the scalar curvature R for the metric (2.2.5).
Let us indicate with U µ the (n + 1)-velocity of the FLRW frame (i.e., the future-directed, timelike vector field tangent to the lines with x = const., normalized so that U µ U µ = −1); we have The scalar field and the matter content. Let us now introduce rescaled versions ϕ, V of the field and of the potential, defined so that (2.2.9) Both ϕ and V(ϕ) are dimensionless. In the sequel, the terms "scalar field" and "potential" will be generally employed to indicate these rescaled quantities. Let us also remark that in [13] there are similar rescaled objects ϕ [13] = ϕ and V [13] (ϕ [13] To comply with the hypothesis of spatial homogeneity, we assume that the field and the matter density depend only on time: (2.2.10) In addition, we assume the matter fluid to be at rest in the FLRW frame, which entails that its (n + 1)-velocity U µ is fixed as in Eq. (2.2.8). Let us also recall that we are considering an equation of state of the form (2.1.10), so that p (m) = w ρ (m) .
In Appendix A we compute the stress-energy tensors of the scalar field and matter fluid starting from the general expressions (2. Here we only mention that T (φ) µν has the form of the stress-energy tensor for a perfect fluid with the (n + 1)-velocity U µ of the FLRW frame, and with density and pressure respectively given by In the sequel, we often refer to the "equation of state coefficient" depending on t and defined whenever ρ (φ) (t) = 0.
An anticipation. For the moment, B is treated as an unspecified function of t (the same viewpoint is assumed in Appendix A). Starting from the forthcoming subsection 2.4 to the end of the paper, following [13] we will assume B(t) = B(A(t), ϕ(t)) for some assigned function B, and refer to this procedure as a gauge fixing. Thus, A and ϕ will be ultimately recognized as the true degrees of freedom of the model.  7)) in the spacetime region where the scalar field has a non-vanishing differential. As a matter of fact, in the present setting it can be checked by direct computations thatφ (ii) As a partial converse, let us consider the relations A = 0, F = 0 supplemented with the initial condition E(t 0 ) = 0 (namely, E is required to vanish at some given time t 0 ); we claim that To prove this, let us reconsider the identity in the text line before Eq. (2.2.17). If A = 0, F = 0 (at all times), this impliesĖ + (Ȧ − 2Ḃ) E = 0 whence (d/dt)(e A−2B E) = 0; the latter identity, supplemented with the initial condition E(t 0 ) = 0, gives E = 0 at all times.
In the sequel we stick to the viewpoint expressed in item (ii): we regard A = 0 and F = 0 as the authentic evolution equations for the model, and E = 0 as a constraint that is fulfilled at all times as soon as it is fulfilled by the initial data at some fixed time t 0 .
Maximal domains for the solutions; Big Bang and Big Crunch. Of course, each solution (A(t), B(t), ϕ(t)) of the system A = 0, F = 0 (E = 0) is well defined for t in a suitable interval I ⊂ R. From now on, when we speak of a solution we always assume I to be maximal (i.e., that the solution cannot be extended to a larger interval).
Recall that a(t) := e A(t)/n is the scale factor and that t, τ are related by Eq. (2.2.2), which is equivalent to (here t r ∈ I is chosen arbitrarily). If a(t) → 0 (i.e., A(t) → −∞) for t → t + in and e B(t) is integrable in a right neighborhood of t in (initial singularity at a finite cosmic time), we say that the model has a Big Bang at t = t in . If a(t) = e A(t)/n vanishes and e B(t) is integrable for t → t − f in (final singularity at a finite cosmic time), we say that the model has a Big Crunch at t = t f in .
The particle horizon. Suppose the model has a Big Bang at τ in = τ (t in ). The lapse of conformal time that has passed from the Big Bang to any cosmic time τ = τ (t) is The above integral can be finite or infinite. The interpretation of Θ(τ ) is well known, and can be summarized as follows, writing p 0 , p, etc. for the points of M n k and dist for the distance on M n k corresponding to the metric dℓ 2 (see Eq. (2.2.1)); correspondingly, for each p ∈ M n k , the ball B(p, τ ) := {p 0 ∈ M n k | dist(p 0 , p) ≤ Θ(τ )} is the subset of M n k formed by the points p 0 which had the time to interact causally with p from the Big Bang up to τ ( 4 ). This subset is the whole M n k if and only if Θ(τ ) δ k , where δ k := sup{dist(p 0 , p) | p 0 ∈ M n k } is the diameter of M n k and is in fact independent of p. One has δ k = +∞ if k 0 and δ k = π/ √ k = θπ/ √ k if k > 0. Of course the situation where Θ(τ ) δ k is of special interest, since it allows to explain the homogeneity of the universe at time τ . As well known, many FLRW cosmologies violate this condition; this will happen, in particular, for many of the integrable cosmologies presented in this work.
which entail for the matter stress-energy tensor the expression Moving this term from the left-hand side to the right-hand side of the Einstein's equations (2.1.3) we get In this case, the field stress-energy tensor becomes (see Appendix A) (2.2.27) Bringing this term to the right-hand side of the Einstein's equations (2.1.3) we obtain where the second equivalence holds under the complementary assumption ρ (φ) = 0, or V(ϕ 0 ) = 0.
In agreement with the remark after Eq.

A Lagrangian viewpoint
Let us return to the general expression (2.1.2) for the action functional, and evaluate it on a history of the type considered in subsection 2.2. A computation sketched in Appendix A yields

Gauge fixing and the energy constraint
From here to the end of the work it is assumed that where B : R 2 → R is a suitable function, referred to as the gauge function in the sequel (the same attitude is proposed in [13] for the special case Ω (m) * = k = 0). Of course, the evolution equations are still A = 0, F = 0, E = 0. Besides, the results of the previous paragraphs continue to hold, with B fixed according to Eq. (2.4.1) anḋ Note that L is a non-degenerate Lagrangian of mechanical type, whose kinetic part is induced by a metric of signature (−, +) on the (A, ϕ) configuration space.  Of course, E is a constant of motion for the Lagrange equations δL/δq = 0 (q = A, ϕ). Moreover, it can be easily checked that Summing up: after gauge fixing, the evolution equations A = 0, F = 0, E = 0 are equivalent to the Lagrange equations δL/δq = 0 (q = A, ϕ), supplemented with the condition E = 0 (the latter condition is satisfied at all times if and only if it is fulfilled by the initial datum A(t 0 ),Ȧ(t 0 ), ϕ(t 0 ),φ(t 0 ) ).
From now on, to analyze the dynamics of our cosmological model we systematically refer to the Lagrangian L of Eq. (2.4.3) and to the energy constraint E = 0. Whenever we speak of a solution of (one or all) these equations, we always tacitly assume that the interval of definition is maximal; this convention is consistent with the domain prescriptions of subsection 2.2, and will be applied also to the solutions obtained using Lagrangian coordinates different from (A, ϕ) (say, the coordinates (x, y) of the next sections). The plan for the sequel is to consider specific choices for V, allowing to solve explicitly the corresponding Lagrange equations.

Adding matter and curvature to the integrable models of Fré, Sagnotti and Sorin
Let us repeat once more that [13] considers purely scalar, spatially flat cosmologies, i.e., models with no matter content and zero spatial curvature. Referring to this framework, Fré, Sagnotti and Sorin identified nine classes of self-interaction potentials V(ϕ) for the scalar field that, after an appropriate gauge fixing B = B(A, ϕ) and a suitable coordinate transformation for the Lagrangian L(A, ϕ,Ȧ,φ), produce solvable Lagrange equations. The gauge function B(A, ϕ) and the coordinate transformation just mentioned are given explicitly in [13] (together with the energy constraint) for each one of the nine potential classes; these results are summarized in Table 1 on page 1048 of [13].
In this section we show that, for all classes of potentials in the cited Table 1, extended cosmological models including matter and possibly curvature can be introduced and solved explicitly using the same coordinate transformations employed in [13] for the corresponding, purely scalar cosmologies. In these extended cosmologies the matter fluid has an equation of state of the form p (m) = w ρ (m) , where the coefficient w has some fixed specific value or remains arbitrary; in the cases with arbitrary w (occurring for three of the nine potential classes), some free parameter γ labeling the potentials becomes a prescribed function of w.
To the best of our knowledge, the possibility to build integrable extensions with matter or curvature was previously unknown for all the cosmologies in Table 1 of [13], with the notable exception of the first class of potentials which was analyzed in [25] a short time before the publication of [13] in the case of matter with w = 0 (dust), zero curvature and space dimension n = 3. The following subsections 3.1-3.9 present extended cosmologies for the nine potential classes in Table 1 of [13], starting from the case of [25] (here generalized to an arbitrary space dimension).
In each subsection we indicate how the Lagrangian function can be reduced by a proper gauge fixing and a suitable coordinates transformation to one of the canonical forms analyzed in the forthcoming paragraph preceding subsections 3.1-3.9. Following the strategies outlined in the said paragraph, the Lagrange equations can be systematically reduced to quadratures in all cases of interest; in particular, explicit expressions for the corresponding solutions can always be derived. These expressions can be used to investigate the chief qualitative features of each specific model: presence of a Big Bang and corresponding asymptotic behavior; presence of a Big Crunch or, in absence of it, long time behavior of the system; comparison of the density parameters. We will exemplify these issues for some subcases of the nine classes in the forthcoming Section 4. To conclude this prologue to the nine cases, let us repeat what we already stated in the Introduction: after the nine potential classes of Table 1, [13] considers other 26 "sporadic" potentials, also producing purely scalar integrable cosmologies; the possibility to maintain the integrability features of these sporadic models with the addition of matter or curvature is not discussed in this work, and left as an open problem for future investigations.
Solvable Lagrangian systems met in the analysis of the nine potential classes. In the subsequent developments we will replace the Lagrangian coordinates A, ϕ with either a new pair of real coordinates or with a complex one, with the rationale of obtaining simple canonical forms for the Lagrange equations. Under these coordinate transformations, the Lagrangian function L assumes the form of one of the types described below (which are actually the same types occurring in [13] in the case of zero spatial curvature and no matter content). Let us point out a fact that will never be mentioned again in the sequel: like the quantities A, ϕ, V(ϕ), the new Lagrangian coordinates x, y, ξ, η, etc. introduced in the sequel to treat the nine potential classes are all dimensionless. a) Quadratic Lagrangians. Assume that there exist two real Lagrangian coordinates x, y such that L(x, y,ẋ,ẏ) is the difference between two quadratic functions of the variables (ẋ,ẏ) and (x, y). In this case the Lagrange equations are linear and can be decoupled via further, linear coordinate transformations. It is unnecessary to give further details on this elementary case, that will appear in subsection 3.1.
b) Triangular Lagrangians. Assume that there exist two real Lagrangian coordinates x, y such that for some µ ∈ R\{0} and some pair of smooth functions u, h. The corresponding energy function is while the Lagrange equations δL/δy = 0 and δL/δx = 0 are, respectively,  For any such solution, let t 0 < t 1 be fixed instants in its domain of definition and assume that which entails µ 2 Next, let t → y(t) be a map forming, together with the previous function t → x(t), a solution of the system (3.0.3) (3.0.4) and consider the total energy E(x(t), y(t),ẋ(t),ẏ(t)) ≡ E. Sinceẋ(t) does not vanish and has constant sign for t ∈ (t 0 , t 1 ), there exists a smooth function Inserting the latter expression forẋẏ and the relation (3.0.10) for y into Eq. (3.0.2) we obtain that, for Noting that the latter is a linear inhomogeneous ODE for Y , by elementary manipulations we get can be treated in a similar way, interchanging the roles of x and y.
c) Special triangular Lagrangian. A very simple subcase of the previous framework occurs if the Lagrangian has the triangular form (3.0.1) with u(x) = λ x − ν for some λ, ν ∈ R, so that  Depending on the sign of λ/µ, Eq. (3.0.16) describes a harmonic oscillator, a "free particle" or a harmonic repulsor with a constant external force ν. In the sequel, for the sake of brevity we will always use the term "oscillator" to indicate a system of any one of the three kinds just mentioned. Regarding x = x(t) as a known function, Eq. (3.0.17) describes an oscillator with a time-dependent external force h ′ (x(t)).
As an example, assume that λ/µ > 0 and set where A ∈ R is an arbitrary constant. If t 0 ∈ R and J is any open real interval such that t 0 ∈ J and the integral appearing in the forthcoming Eq.
where B, C ∈ R are arbitrary constants. Of course, also in this case we obtain a system with similar solvability features interchanging the roles of x and y in Eq. (3.0.15).
d) Separable Lagrangian. Assume there exist two real Lagrangian coordinates x 1 , x 2 such that ( 5 ) where C ∈ R, µ i ∈ R\{0} and U i is a smooth real function for i = 1, 2. In this case the Lagrange equations 0 = δL/δx i (i = 1, 2) describe two decoupled subsystems admitting as constants of motion the energies of course, the total energy corresponding to the Lagrangian (3.0.21) is Any pair of motions x i (t) (i = 1, 2) of the separate subsystems with energies E i are confined within connected regions where sign(µ i )(E i − U (x i )) 0, and can be reduced to quadratures via the relations µ i 2 for any t such that σ i := signẋ i (t) = const. ∈ {±1} on (t 0 , t).
e) One-dimensional, holomorphic conservative system. From here to the end of the paper, ℜz, ℑz and z indicate the real part, the imaginary part and the conjugate of any complex number z. Assume that there exist an open subset D ⊂ C and a complex Lagrangian coordinate z ∈ D (withż ∈ C) such that ( 6 ) starting from here, we derive a quadrature formula by a natural adaptation to the complex framework of the approach usually employed for real, one dimensional conservative systems. More precisely, consider an open set P ⊂ C\{0} such that the mapping P → z(t 0 ) indicates the integration along any path in O with initial point z(t 0 ) and final point z(t) (the integral is independent of the chosen path).
To proceed, let us remark that the usual energy function E :=ż ∂L/∂ż +ż ∂L/∂ż − L associated to the Lagrangian (3.0.25) is  and we refer to this system as a "complex oscillator".

Class 1 potentials
The first class of potentials in Table 1 of [13] has the form For these potentials, the cited reference suggests to introduce the gauge function B and the coordinates x, y, defined by In the case of no matter and zero curvature (Ω (m) * = 0, k = 0), the above positions give rise to a quadratic Lagrangian and, consequently, to linear evolution equations for x and y. Let us implement the same positions in our framework with matter and curvature, searching for additional cases with a quadratic Lagrangian. Eqs.
The Lagrangian (3.1.4) is still quadratic, up to additive constants, in the following cases with matter or curvature (i.e., with (Ω (m) * , k) not constrained to be (0, 0)): In each one of the above cases, the Lagrange equations in the coordinates x, y form a linear system, and can be decoupled via further coordinate changes of linear type. Correspondingly, let us stress that the admissible solutions must fulfill the energy constraint E = 0, as well as the conditions x(t), y(t) > 0 (cf. Eq. (3.1.3)).
As an example, let us consider case (i), providing (at least if n = 3) a rather realistic model of the physical universe for most of its history; this is just the case considered (for n = 3) in [25]. The Lagrangian (3.1.4) and the energy (3.1.5) reduce, respectively, to The Lagrange equations decouple under a further, linear change of coordinates ( which transforms the Lagrangian (3.1.6) and the energy (3.1.7) into (3.1.10) The (separable) Lagrangian (3.1.9) gives rise to the system of uncoupled equations whose solutions can be determined by elementary means. Let us point out that in the present case, the admissible solutions are those fulfilling u(t) > |v(t)| (a relation which is equivalent to x(t), y(t) > 0). More details about the qualitative behavior of such solutions will be given in subsection 4.1 of Section 4.

Class 2 potentials
The second class of potentials in Table 1 of [13] is formed by the functions Ref. [13] suggests to study these potentials fixing the gauge function B and introducing new coordinates x, y as follows: In the case of no matter and zero curvature, the Lagrangian obtained via these prescriptions is of the special triangular kind ( The Lagrangian (3.2.4) has a triangular structure in the cases with matter or curvature listed below.
In each one of these cases, the admissible solutions are those fulfilling the energy constraint E = 0 and the conditions x(t), y(t) > 0 (cf. Eq. (3.2.3)).
ii 0 ) n = 3, γ = w = 1 3 . In this particular subcase of case (ii), the Lagrangian (3.2.4) and the energy function (3.2.5) read The Lagrangian (3.2.11) is of the special triangular form (3.0.15) and the related equations δL/δy = 0, δL/δx = 0 entail, respectively, The first relation in Eq. (3.2.13) describes an oscillator with a constant "curvature force"; once x(t) has been determined, the second relation in Eq. (3.2.13) describes another forced oscillator. Also in this case, we have a system of equations which can be solved by elementary means.
iii) γ = w = 2 n − 1 . ( 9 ) The Lagrangian (3.2.4) and the energy (3.2.5) reduce to The Lagrangian (3.2.14) has the special triangular form (3.0.15) and the equations δL/δy = 0, Again, the equation for x(t) describes a free oscillator and, once this function has been determined, the equation for y(t) describes a forced oscillator.
. This is the subcase of case (iv) corresponding to n = 2. The Lagrangian (3.2.4) and the energy function (3.2.5) reduce to The Lagrangian (3.2.19) has the special triangular form (3.0.15). The related equations δL/δy = 0, δL/δx = 0 read, respectively, respectively describe a free oscillator and another oscillator with an external force proportional to x(t). Let us remark that, up to a constant, the Lagrangian (3.2.19) also belongs to the class of the quadratic Lagrangians.

.4) and the energy (3.2.5) become
3+w . 9 The position w = 2/n − 1 makes this case perhaps less interesting than the previous ones, since it gives w < 0 for n 3 (for n = 2 one has w = 0, typical of a dust fluid). Nonetheless, if w = 2/n − 1 and we assume Ω . This is a subcase of case (v), corresponding to w = −1/3. The Lagrangian (3.2.4) and the energy (3.2.5) reduce to The Lagrangian (3.2.24) has the special triangular form (3.0.15). The equations δL/δy = 0, δL/δx = 0 entailẍ and describe two forced oscillators, the first one with a constant external force and the second one with an external force depending on x(t).
The Lagrangian (3.2.27) has the triangular structure (3.0.1), and can be treated with the corresponding methods; in particular, Eqs.
. This is the subcase of case (vi) corresponding to n = 3. The Lagrangian (3.2.4) and the energy (3.2.5) reduce to The Lagrangian (3.2.29) has the special triangular form (3.0.15) and the equations δL/δy = 0, δL/δx = 0 implyẍ Again, we have an oscillator with a constant external force and another oscillator with an external force depending on x(t).

Class 3 potentials
We consider potentials of the form Ref. [13] treats these potentials fixing the gauge function B and introducing the Lagrangian coordinates x, y, defined as follows:

3.2)
In the case of no matter and zero curvature, the Lagrangian obtained with the above positions has the special triangular form (3.0.15). Let us make the same positions in our framework with matter and curvature, and search for other triangular cases. Eqs.
We aim to classify the cases with matter or curvature in which the Lagrangian (3.3.4) has a triangular form. It is evident that a triangular structure cannot be attained when k = 0; thus, we set k = 0 and search for triangular cases with matter (i.e., with Ω (m) * = 0). Below we report a list of such cases; in each one of these cases, the admissible solutions are those fulfilling the energy constraint E = 0 and the condition The Lagrangian (3.3.6) has the special triangular form (3.0.14). The related equations δL/δx = 0, δL/δy = 0 imply, respectively,ÿ their general solution is readily found to be where α, β, γ, δ are arbitrary integration constants and Erfi is the imaginary error function.
As an example, let us mention that Eqs. ii) k = 0, w = −3 . The Lagrangian (3.3.4) and the energy (3.3.5) become, respectively, iii) k = 0, V 2 = 0, w = 1 . Let us first remark that in this case the potential does also belong to the class discussed in subsection 3.2 (since V(ϕ) is of the form (3.2.1) with V 2 = 0 and γ = 1). The Lagrangian (3.3.4) and the energy (3.3.5) read, respectively, We have a special triangular Lagrangian, of the form (3.0.15) with λ = σ = 0. The corresponding equations δL/δy = 0, δL/δx = 0 entail Again, the solutions can be easily derived and read

Class 4 potentials
Let us consider the potential Ref. [13] suggests to treat these potentials using the gauge function B and the coordinates x, y, defined by In the case with no matter and zero curvature, these prescriptions yield again a special triangular Lagrangian of the form (3.0.15). Also in this case, we try to extend the treatment of [13] In presence of matter (Ω (m) * > 0), the only case where the Lagrangian (3.4.4) has a triangular structure is the following.
The Lagrangian (3.4.6) has the special triangular form (3.0.15). The related equations δL/δy = 0, δL/δx = 0 entail, respectively, their general solution is given by where α, β, γ, δ are integration constants and ∆ := , which shows that the energy constraint E = 0 holds if and only if Again, finding a (maximal) interval where x(t) > 0 is elementary, once the integration constants in Eq. (3.4.9) have been assigned.

Class 5 potentials
This class is formed by potentials of the form Ref. [13] suggests to analyze these potentials by means of the gauge function B and of the new coordinates x, y defined by In absence of matter and curvature, the Lagrangian L(x, y,ẋ,ẏ) obtained via these prescriptions is separable. Following our general approach, let us implement the prescriptions of [13] in our framework with matter and curvature, and search for additional separable cases. Eqs.
The Lagrangian (3.5.4) is given by the sum of two functions depending separately on (x,ẋ) and (y,ẏ), plus two additional terms proportional to Ω (m) * and k, respectively, both consisting of suitable powers of x 2 − y 2 . The only cases where the latter additional terms disappear or are themselves separable, yielding again a separable Lagrangian of the form (3.0.21), are those where k = 0 and the exponent −(1 + w)/2 equals 0 or 1. We discuss these two cases in the sequel, keeping in mind that the corresponding Lagrange equations can be reduced to quadratures as indicated in Eq. (3.0.24); correspondingly, let us also repeat that admissible solutions must also fulfill the energy constraint E = 0 and the conditions x(t) > y(t) > 0 (cf. Eq. (3.5.3)).

Class 6 potentials
Let us consider potentials of the form In connection with this class, [13] suggests to consider the gauge function and to replace the Lagrangian coordinates (A, ϕ) with a conveniently defined, complex variable z.
Regarding this complex setting, it can be useful to introduce the following conventions, somehow implicit in the cited reference: be the open region in the complex plane C obtained removing the negative real semi-axis. Correspondingly, consider the determination of the argument function given by arg : This entails, in particular, that arg z = 0 for z ∈ (0, ∞) and that arg(z) = − arg z for all z ∈ C × .
The usual, natural logarithm log : (0, ∞) → R possesses the extension log : with arg as in Eq. (3.6.3). Such an extension is a holomorphic function fulfilling e log z = z and logz = log |z| − i arg z = log z for all z ∈ C × .
Keeping these premises in mind, the complex formalism of [13] can be described as follows: the coordinates (A, ϕ) ∈ R 2 are replaced by a complex coordinate z ∈ D with The correspondence z → (A, ϕ) defined above is one-to-one between D and R 2 , with inverse Let us note that the second relation in Eq. (3.6.6) implies e −2ϕ = ℑz/ℜz, which allows to express the potential (3.6.1) as In absence of matter and curvature, [13] expresses the Lagrangian function associated to a potential of the form (3.6.1) fixing the gauge and introducing a complex coordinate z as above; the Lagrangian L(z,ż) obtained in this way is of the holomorphic type (3.0.25), so the related Lagrange equations can be integrated by quadratures. Following our general philosophy, hereafter we try to generalize the results of [13] to cases with matter or curvature. To this purpose, first note that the prescriptions (3.6.2) (3.6.6) yield the following expressions for the Lagrangian (2.4.3) and for the energy (2.4.5): (here and in the sequel, ℑz 2 stands for ℑ(z 2 )). In passing, let us point out that Eqs. (3.6.9) (3.6.10) have the same structure of Eqs. (3.5.4)(3.5.5) in the previous section, a fact which becomes evident if one considers the replacement (z,z) → (x, y).
We are interested in cases in which the Lagrangian (3.6.9) maintains the holomorphic structure (3.0.25) even in presence of matter or curvature. Clearly, this structure occurs only if k = 0 and the exponent −(1 + w)/2 equals 0 or 1, which yields the cases discussed below. As usual, let us remark that admissible solutions must fulfill the energy constraint E = 0, as well as the condition z(t) ∈ D.
i) k = 0, w = −1 . The Lagrangian (3.6.9) and the energy (3.6.10) become, respectively, (3.6.12) One can apply the methods described below Eq. (3.0.25) to solve the Lagrange equations by quadratures; in the present case the holomorphic function U and the complexified energy E of Eqs.
The Lagrangian (3.6.9) and the energy (3.6.10) reduce to (3.6.14) Again, one should refer to the methods reported below Eq.

Class 7 potentials
Let us consider a potential of the form (3.7.1) here and in the sequel we assume To treat a potential of the above form, [13] suggests to use the gauge function B and the real coordinates x, y, determined as follows: The domain D γ,V 2 is not indicated explicitly in [13], but it is evident that we can put In fact, it can be readily checked that the map (x, y) → (A, ϕ) described in Eq. (3.7.5) is a smooth diffeomorphism between the open sets D γ,V 2 and R × I γ,V 2 , with inverse function given by In the case of no matter and zero curvature, the Lagrangian L(x, y,ẋ,ẏ) obtained with the above prescriptions is separable. As usual, let us try to use the same prescriptions adding matter and curvature. Eqs. (3.7.10) (3.7.11) Let us mention that in the subcases w = −1, w = −1/3 and w = 0 (dust), the exponents of x in L 1 (x,ẋ) and of y in L 2 (y,ẏ) become, respectively, 0, 1 and 2, so that the Lagrange equations are elementary.
vii) γ = − n−1 2n , w = 1 n . The Lagrangian (3.7.8) and the energy (3.7.9) take the forms Note the strong similarities with subcase (iv): L 1 and L 2 are as in Eq.
Like the class of potentials addressed in subsection 3.6, the present class can be treated using a complex formalism. To this purpose, let C × , arg and log be defined as in Eqs. (3.6.3) (3.6.4) of subsection 3.6 (see also the related comments); in addition, let us put For any z, λ as above, the map z → z λ is holomorphic on C × and we have z λ = |z| λ e iλ argz , z λ =z λ . Potentials of the form (3.8.1) were treated in [13] fixing the gauge function B and replacing the Lagrangian coordinates (A, ϕ) with a complex variable z, defined as follows: (here and in the sequel ℜz 2 , ℑz 2 stand for ℜ(z 2 ), ℑ(z 2 )). In the above, z is tacitly assumed to belong to a suitable domain D ⊂ C, which is not described explicitly in [13]; in any case, the coordinate transformation (3.8.4) is one-to-one between D and the set {(A, ϕ) | A ∈ R, γ ϕ ∈ (0, ∞)} if and only if D := z ∈ C ℜz > ℑz > 0 = z ∈ C × 0 < arg z < π/4 . To proceed, we claim that Eqs. (3.8.1) (3.8.4) entail the following identities ( 10 ): ℑz ; writing "cosh" and "sinh" in terms of exponentials, from the latter identity we infer The main result of [13] The subcases w = −3, −5/3, −1 are elementary, since z appears in L(z,ż) with exponents 0, 1 or 2.
In view of the above relations, starting from Eq. (3.8.1) and setting C := V e −iθ we infer the following chain of identities: iii) Ω (m) * = 0, γ = n−1 n . The Lagrangian (3.8.8) and the energy (3.8.9) become The subcases n = 2 and n = 3 are elementary, since z appears in L(z,ż) with exponents 2 and 1.

Class 9 potentials
Let us consider potentials of the form (3.9.1) [13] treats this class of potentials fixing the gauge function B and introducing new Lagrangian coordinates (x, y) related to (A, ϕ) by a "Lorentz transformation", as follows: In absence of matter and curvature, the Lagrangian L(x, y,ẋ,ẏ) obtained in this way in [13] is separable. We now add matter and curvature, and use again the above prescriptions trying to find new separable cases. Eqs.
The only situation with matter or curvature where the Lagrangian (3.9.4) is separable, is the one described hereafter.

Explicit form and detailed analysis of some solutions
In the previous section, we provided a list of integrable cases with matter or space curvature associated to the nine potential classes of Frè-Sagnotti-Sorin; let us recall that each one of these cases is solvable for one of the reasons (a-e) indicated at the beginning of section 3 (linearity of the Lagrange equations, triangular or special triangular Lagrangian, separable Lagrangian, one-dimensional holomorphic and conservative system). Of course, after indicating a reason for the solvability of the Lagrange equations one should give the explicit form of the (general) solution and analyze it both qualitatively and quantitatively. In particular, one should investigate the occurrence of an initial Big Bang singularity and the related presence of a particle horizon (see Eqs. (2.2.19) (2.2.20) and the associated comments), as well as the possible development of a Big Crunch or, in absence of it, the evolution of the system for long times. In connection with these issues, it is essential to determine the asymptotic behavior of the main elements of the model: the scale factor a, the scalar field ϕ and the related equation of state parameter w (φ) , together with the density parameters Ω (m) , Ω (φ) , Ω (k) . If the model turns out to be physically plausible, at least for some epoch in the evolution of the universe, one should also choose sensible values for the parameters in the potential V(ϕ) and for the constants of integration of the solution, so as to make contact with the available observational data. In the forthcoming subsections 4.1-4.2 we discuss the above issues (or some of them) for some specific cases, taken as examples.
All the cases to be discussed in the sequel have vanishing scalar curvature, i.e.,  Making reference to the above relation, we will say that matter dominates at the Big Bang if Ω (m) (t) → 1 (or equivalently, Ω (φ) (t) → 0) for t → 0 + ; conversely, we will say that the scalar field (or the dark energy) dominates at the Big Bang if Ω (φ) (t) → 0 (or equivalently, Ω (m) (t) → 1) in the same limit. (One can define similarly the cases where matter or the scalar field dominate at the Big Crunch, if this exists.) In addition, since a negative matter density cannot be related to any realistic physical model, we further require the parameter Ω (m) * to be positive or null, i.e., Finally let us point out that, in agreement with the general results of [12], in all cosmological models with a scalar field and a matter fluid to be discussed in the sequel there is a particle horizon at any time after the Big Bang whenever the matter fluid fulfills as a strict inequality the strong (whence, the weak) energy condition, which in the present context means (cf. Eq. (2.1.13)) w > 2 n − 1 .  Moreover, we assume that the self-interaction potential for the field is given by for notational convenience, in the sequel we shall put The model depicted above was previously studied by Rubano and Scudellaro [29], and by Piedipalumbo, Scudellaro, Esposito and Rubano [25], in the physically most relevant case with spatial dimension n = 3. Hereafter, we review within our framework the results of [25,29], generalizing them to the case of arbitrary n 2; in addition, we enrich the analysis discussing the asymptotic behavior of the density parameters Ω (m) , Ω (φ) near the Big Bang.
To begin with, let us notice that the potential (4.1.2) is clearly of the form (3.1.1) (with V 0 = 0 and the conditions stated above on V 1 , V 2 ); to be more precise, as a consequence of Eqs.
With the previously stated assumptions, the Lagrange equations (3.1.11) for u, v reduce tö 1.6) and the corresponding solutions are readily found to be where A, B, C, D ∈ R are suitable integration constants. From Eqs. (3.1.10)(4.1.7), by elementary computations we infer the following expression for the energy E ≡ E(u, v,u,v) of the system: Taking the above relation into account, to fulfill the energy constraint E = 0 we set (4.1.12) 38

Big Bang analysis
Let us wonder under which conditions the solution (4.1.7) produces a Big Bang at some instant, that we conventionally choose as the time origin t = 0 (cf. Eq. (4.0.2)). Such conditions entail that a(t) → 0 (i.e., A(t) → −∞) for t → 0 + and, even prior to this, that a(t) (hence, A(t)) is well defined in a right neighborhood of t = 0; in terms of the functions u(t), v(t), this amounts to require u 2 (t) − v 2 (t) → 0 and u(t) > |v(t)| for t → 0 + (4.1.13) (here and in the sequel, an expression of the form "f (t) > 0 for t → 0 + " means that there exits some ǫ > 0 such that f (t) > 0 for all t ∈ (0, ǫ)).
From the explicit expressions reported in Eq. (4.1.7), it follows straightforwardly that 14) The above relations show that the first condition in Eq. (4.1.13) is fulfilled if and only if A 2 − C 2 = 0, while it is necessary to assume that A 0 in order to satisfy the second condition in the same equation; thus, we require A = |C| 0 . In the sequel we will distinguish three subcases of the latter constraint.  To go on, we note that the previously mentioned expressions for A, ϕ, w (φ) and Ω (m) (see Eqs. (4.1.5) (4.1.9) (4.1.11) (4.1.12)) imply the following, for t → 0 + : for τ /θ → 0 + .  Then, for t → 0 + we have for τ /θ → 0 + , (4. 1.27) which allows us to infer that there is a particle horizon. Moreover, Eq. (4.1.26) shows that the scalar field dominates at the Big Bang.
iii) A = C = 0. In this setting the second condition in Eq.  To investigate the asymptotic behavior of the system near the Big Bang, note that for t → 0 + we have

Behavior of the model in the far future
First of all, let us remark that the bare solutions written in Eq. (4.1.7) make sense for any t ∈ R. However, one should not forget that the second condition in Eq. (4.1.10) puts severe restrictions on the maximal admissible domain I ⊂ R for such solutions. In presence of a Big Bang at t = 0, the most enticing scenarios are those corresponding to an endless evolution of the universe, namely, I = (0, +∞) .   Thus, for large times the scale factor diverges, the field equation of state resembles the cosmological constant case (2.2.29) and the scalar field is dominant (Ω (φ) (t) → 1 since Ω (m) (t) → 0). All these features are attained with exponential speed.

Quantitative analysis of one of the previous cases
Hereafter we reconsider the general model described at the beginning of the present subsection 4.1 and show how to fix all the (so far unspecified) associated parameters n, θ, Ω (m) * , V 1 , V 2 , A, B, C, D so as to provide a physically plausible scenario. In all the cases mentioned above, to infer that u(t) > |v(t)| for all t ∈ (0, ∞) it suffices to notice that cosh(z) > | cos(z)| and sinh(z) > | sin(z)| for any z > 0.

Spatially flat solutions for class 2 potentials with γ = w = ±1
In this subsection we analyze (n + 1)-dimensional, spatially flat (k = 0; see Eq. (4.0.1)) cosmological models whose matter content consists of a perfect fluid with a arbitrary equation of state parameter w ∈ R\{±1} , (4.2.1) and a scalar field with self-interaction potential For notational convenience, in the sequel we put To proceed, recall that with the above positions the Lagrangian function has the special triangular structure (3.0.15), and the related Lagrange equations (3.2.8) becomë The corresponding solutions can be determined explicitly, treating the cases ε = −1, 0, +1 separately. Before providing a detailed analysis of these cases, let us point out that Eqs. (2.2.11) (2.2.12) (2.2.32) 14 To make a quantitative comparison with the results of [31], it should be noted that the dimensionless scalar field ϕ and the potential V(ϕ) of the present work (with n = 3 and c = 1) are related as follows to the analogues φ [31] , V [31] (φ [31] ) employed in [31] (see [15, App. G] for more details): After possibly a time translation t → t + const. and a time reflection t → −t, any admissible solution of Eqs. (4.2.10)(4.2.11) can be written in one of the following forms, for the values of w indicated contextually (see Appendix B for the derivation of the following expressions): for A > 0 and w ∈ R\{±1}, w = − 3 + 2h 1 + 2h for all h ∈ {0, 1, 2, ...} ; (4.2.12) x(t) = A cosh(ω t) , for A > 0 and w ∈ R\{±1} ; (4.2.13) y(t) = C cosh(ω t) + D sinh(ωt) 1+w ω t for A > 0 and w ∈ R\{±1}, w = 0, −1/2 .  I ⊂ R such that y(t) > 0 for all t ∈ I .
and of elementary trigonometric identities, the expression for y(t) in Eq. (4.2.14) reduces to Taking the above relations into account and recalling that in the present case V 1 = 0 (see Eq. (4.2.9)), we infer that to fulfill the zero-energy constraint E = 0 we must put in Eqs. (4.2.12) (4.2.13) (4.2.14), respectively, After a time translation t → t + const. and possibly a time reflection t → −t, any solution of Eqs. (4.2.27) (4.2.28) can be written in one of the following forms (see Appendix B for more details): for A > 0 and w ∈ R\{±1} ; (4.2.29) for A > 0 and w ∈ R\{±1}, w = −3. I ⊂ R such that x(t), y(t) > 0 for all t ∈ I .

Big Bang analysis
In the present subsection we proceed to investigate the presence of an initial Big Bang singularity and the asymptotic behavior close to it for one of the previously described solutions, taken as an example. More precisely, let us assume that  Hereafter we proceed to analyze these two cases separately.
which can be locally inverted to give   Similarly to the case with C > 0 discussed in the previous paragraph, the above relations show that a(t) → 0 for t → 0 + and imply the integrability of e B(t) in a right neighborhood of t = 0 (for −1 < w < 1); so, we have a Big Bang at t = 0. To say more, Eqs. (4.2.57)(4.2.60) give which, together with Eq. (2.2.20), indicates that the particle horizon is finite whenever 2 n(1+w) < 1. In our case with n ≥ 2 and −1 < w < 1, this is equivalent to w > (2/n) − 1 (cf. Eq. (4.0.7) and the related comments); especially, let us point out that the latter condition is fulfilled in the case of radiation where w = 1/n. Concerning the coefficient in the field equation of state and the matter density parameter Ω (m) , from Eqs. ( which entails, by inversion,

Behavior of the model in the far future
The qualitative behavior on large time scales of the model under analysis depends sensibly on the choice of the parameters which characterize the solutions x(t), y(t) of the Lagrange equations (4.2.5)(4.2.6). In the sequel we account (at least partially) for this rather predictable fact, referring once more to the exemplary case whose Big Bang phenomenology was examined in the previous paragraph. Correspondingly, we assume again that V 1 > 0 and −1 < w < 1 (see Eq. (4.2.43)), and consider the solution given in Eqs. (4.2.12)(4.2.23); in addition we suppose that either C > 0, or C = 0 and Ω (m) * > 0 (see Eq. (4.2.46)), which grants the occurrence of a Big Bang t = 0. Next, let us recall that the maximal admissible domain of definition I ⊂ (0, +∞) for the said solutions x(t), y(t) must be such that x(t) > 0 and y(t) > 0 for all t ∈ I (see Eq. (4.2.4)). Of course, the expression x(t) = A sinh(ω t) with A > 0 (see Eq. (4.2.12)) ensures x(t) > 0 on the whole positive half-line (0, +∞); therefore, any restriction on the domain I comes from the requirement y(t) > 0 for t ∈ I. In general, the maximal interval where the expression for y(t) in Eq. (4.2.12) gives y(t) > 0 cannot be determined by purely analytical means and one must perform a numerical evaluation. On the other hand, let us notice that for t → +∞ we have ( 18 ) Recalling the previous assumptions on the parameters, the above asymptotics show that in order to fulfill the inequality y(t) > 0 in the limit t → ∞, it is necessary to demand ( 19 ) Incidentally, let us point out that the above condition on V 2 for 0 < w < 1 is certainly fulfilled if Whenever the conditions in Eq. (4.2.68) are violated, y(t) eventually becomes negative; since y(t) is positive close to the Big Bang (for t → 0 + ), it follows that y(t) must vanish at some finite time, namely, ∃ t * ∈ (0, +∞) such that y(t * ) = 0 . In this case the maximal admissible interval I is a (finite) subset of (0, t * ). Besides, given that x(t) = A sinh(ω t) is strictly positive and finite for all t ∈ (0, t * ), from Eqs. (2.2.4)(3.2.3) we see that The above relation suggests that a Big Crunch could occur at t = t * ; in this regard, it should be recalled that the very definition of Big Crunch also requires that e B(t) is integrable in a left neighborhood of t * . Noting that Eqs.
we see that e B(t) is certainly integrable for t → t − * if 0 w < 1, while a finer analysis is needed when −1 < w < 0. On the other side, let us stress that the fulfillment of the conditions in Eq. (4.2.68) is certainly not sufficient to ensure y(t) > 0 for all t ∈ (0, +∞). Notwithstanding, in the upcoming subsection 4.2.3 we are going to show that this condition is actually attained at least for a specific choice of the parameters. By continuity arguments, this fact indicates that there also exist other values of the parameters for which the maximal admissible domain is in fact I = (0, +∞) . (4.2.73) Assuming the maximal domain I to be as above and restricting the attention to the case

Qualitative analysis of one of the previous cases. A model for inflation
We now proceed to examine in more detail a particular case of the cosmological model analyzed before in the present subsection 4.2, selecting specific values for the associated free parameters. Let us anticipate that the rationale behind the said choice of parameters is to realize an inflationary scenario, where an early stage inflation occurs during the radiation dominated era. This scenario would allow, among else, to resolve the flatness, horizon and monopole problems. The above considerations and the arguments to be presented in the sequel are largely inspired by the model portrayed in [30,Sec. 11.4], where inflation is triggered by a true cosmological constant; on the contrary, here we plan to mimic this cosmological constant contribution in terms of the scalar field ϕ with self-interaction potential as in Eq. (4.2.2).
To begin with, let us fix the space dimension and the spatial curvature as (see Eq. (4.0.1)) We further suppose that the ordinary matter content of the universe can be described by means of a perfect fluid of radiation type, i.e., we posit w = 1/3 . ; these indicate that the field ϕ can effectively reproduce a cosmological constant contribution whenever the self-interaction potential V(ϕ) possesses a stationary point (see Eq. (2.2.26)). Taking this fact and the gauge invariance ϕ → ϕ + const. into account, we require the potential in Eq. (4.2.2) to attain a maximum at ϕ = 0; accordingly, we set (see Fig. 7 for the plot of the map ϕ ∈ R → V(ϕ)/V ). Since the condition in Eq. (4.2.9) is certainly fulfilled in the case under analysis, we can refer to the Lagrange equations (4.2.10)(4.2.11) and assume that the corresponding solutions x(t), y(t) are as in Eq. (4.2.12); taking also into account the associated zero-energy constraint (4.2.23) and using some known relations for the hypergemetric functions 2 F 1 appearing in Eq. (4.2.12) ( 20 ), we obtain Quite understandably, the analysis of the model at issue becomes significantly simpler if we arbitrarily fix the parameter A (left unspecified until now) so as to get rid of the hypergeometric function 2 F 1 appearing in the expression for y(t) in Eq. (4.2.85); to this purpose, from now on we set Next, let us require a Big Bang to occur at t = 0 and recall that, for this to happen, it is necessary to assume that C 0 (see Eq. (4.2.46)); accordingly, for later convenience we put This identity can be derived with some elementary computations starting from the Gauss series representation of 2F1 (see, e.g., [22,Eq. 15.2.1]) and using some known relations for the Euler gamma functions Γ appearing therein. Let us mention that the same identity could also be derived (with some more effort) from the relations for contiguous hypergeometric functions (see, e.g., [ In the sequel we first discuss the exceptional case ζ = 0, and then proceed to analyze the more generic configuration with ζ > 0.
The case ζ = 0. This case deserves a special mention because it corresponds to a scenario where the field ϕ behaves exactly as a cosmological constant. Eqs.   The case ζ > 0. Let us return to Eqs. (4.2.91-4.2.95), that we now use with ζ > 0. We first derive the asymptotic expansions of τ (t)/θ, a(t), w (φ) (t), Ω (m) (t) in the limit of small and large t. The behaviour of τ (t)/θ in these limits can be derived from the general asymptotic expansions (4.2.54) (4.2.79), which in the present setting reduce to ( 22 )  For our purposes it is also important to consider the behavior of the above observables when t ranges in a compact interval, and ζ is sent to zero. Indeed, let us fix any two times 0 < t 1 < t 2 ; then, from Eqs. ( (4.2.104) 22 More precisely, the asymptotics in Eq.
(4.2.105) Eq. (4.2.103) implies the following, for any pair 0 < τ 1 < τ 2 of cosmic time instants: Let us briefly comment the above results. According to Eq. (4.2.107), on each compact interval [τ 1 , τ 2 ], with τ 1 > 0 so as to ensure a strict separation from the Big Bang, for ζ sufficiently small the scale factor a(τ ) grows exponentially and w (φ) (τ ) is close to −1, indicating that the field behaves approximately like a cosmological constant. As a consequence, the behavior of the system for τ ∈ [τ 1 , τ 2 ] and small ζ is similar to that described in the previous paragraph for ζ = 0 and all τ ∈ (0, +∞). The situation is completely different if we approach the Big Bang, or we consider the very far future; for example, Eq. (4.2.104) shows that a(τ ) has a power law dependence on τ /θ with exponents 1/3 and 3, respectively, for τ /θ → 0 + and τ /θ → +∞. The presence of an epoch of exponential growth for a(τ ), preceded and followed by periods with slower growth, is typical of inflationary models. In the sequel we will show that one can adjust the parameters of the system so as to obtain a rather realistic model for inflation, even from a quantitative viewpoint.
An interlude on the quantitative determination of cosmic time. Let us recall that τ (t)/θ is expressed via Eq. (4.2.91) as a nontrivial integral over the interval (0, t]. The numerical computation of this integral (for specified values of all parameters) is problematic, especially in the situation of greatest interest for us. In fact, for t ′ → 0 + the integrand function in Eq. (4.2.91) behaves like √ ζ (ωt ′ ) −1/4 , the product of the divergent factor (ωt ′ ) −1/4 by the parameter √ ζ. To make things worse, in the sequel we are mostly interested in a case where ζ is very small. Fortunately, the problem that we have just outlined can be overcome. In fact, starting from the integral representation (4.2.91) it is possible to determine analytically two elementary functions T ± ζ such that T − ζ (t) τ (t)/θ T + ζ (t) for arbitrary ζ, t > 0; we refer to Appendix C for a detailed description of such functions. The same Appendix shows that, in the application with small ζ considered in next paragraph, T + ζ (t) and T − ζ (t) are very close for all values of t taken in account, so that the mean (1/2)(T − ζ + T + ζ )(t) is a very accurate approximant for τ (t)/θ. In the calculations mentioned in the next paragraph, τ (t)/θ has always been approximated with the previous mean.
A plausible scenario with inflation. Let us now present a reasonable choice of the parameters left unspecified for the model under analysis, which can in fact lead to a physically plausible inflationary scenario. The key idea that we are going to pursue in the sequel is that the scale factor grows exponentially in a compact interval of cosmic time, at least for very small values of ζ (a fact made evident by the asymptotic expansion written in Eq.  and V be independent of N and comparable to unity, so that the same holds true for ω (due to Eq. (4.2.86)); as an example, let us fix  Figure 14. Ω (m) (τ ) as a function of τ /θ.  Figure 15. Ω (m) (τ ) as a function of τ /θ.

An integrable case with a class 7 potential. The nonlinear repulsor/oscillator model
In this section, we refer to the integrable subcase (i) in the analysis of class 7 potentials (see page 31), with an additional prescription: the exponent 2/γ − 2 in the potential V of Eq. To proceed, let us recall that the subcase (i) of page 31 requires k = 0, γ = (1 − w)/2; on top of that, we assume V 1 and V 2 to be positive. Thus, the complete list of our choices is the following: Note that, for A as above one has A → −∞ when x 2 − y 2 → 0 + . Correspondingly, noting that the scale factor for the cosmology under analysis is given by (see Eq. (2.2.4)) a = e A/n = x 2 − y 2 ℓ+1 2n , (4.3.6) we have a → 0 + in the limit x 2 −y 2 → 0 + ; this fact is especially relevant for the presence of a Big Bang.
To say more, assuming that a Big Bang does actually occur at t = 0 in agreement with Eq. (4.0.2) ( 23 ) and adding the conventional prescription τ (t) → 0 + for t → 0 + , from Eqs. (4.0.3)(4.3.3) we derive the following expression for the cosmic time coordinate: Next let us recall that, in the subcase (i) for the potential class 7, the Lagrangian (3.7.8) and the corresponding energy (3.7.9) take the separable forms (3.7.10)(3.7.11). With the prescriptions stated in Eq. (4.3.1), the cited Eqs. (3.7.10)(3.7.11) give: In the sequel we discuss the solutions of the Lagrange equations corresponding to the above 1dimensional Lagrangians L 1 , L 2 , providing explicit quadrature formulas for them and discussing their asymptotic behaviors in different regimes of interest for the applications to be discussed in the sequel.
Before proceeding with this analysis, let us express in terms of the coordinates x, y two other relevant observables: namely, the coefficient w (φ) in the equation of state for the field and the dimensionless density parameter for matter Ω (m (4.3.11)

Constants of motion. Qualitative analysis and quadrature formulas
From here to the end of the present subsection 4.3, we stick to the configuration described by Eq. (4.3.1) and refer to Eqs. (4.3.8)(4.3.9) for the Lagrangian L and the corresponding energy E. The Lagrangian system described by L can be analyzed in terms of the separate 1-dimensional subsystems with Lagrangians L 1 , L 2 , whose energies E 1 , E 2 are constants of motion. Of course, the Lagrangians L 1 , L 2 (as well as the energies E 1 , E 2 ) are well defined and smooth for any real x and y.
Keeping this in mind, in the sequel we shall first study separately the subsystems with Lagrangians L 1 , L 2 assuming x, y ∈ (−∞, +∞), and reserve to a second step the implementation of the condition (x, y) ∈ D (see Eq. (4.3.5)). From the expression (4.3.9) for the total energy E, we see that the constraint E = 0 is fulfilled if and only if E 1 and E 2 are expressed as follows: The expression for E 2 in Eq. (4.3.9) makes evident that F 0, and that F = 0 only along motions with y(t) = 0 andẏ(t) = 0 for all t; in the sequel we exclude such motions, fixing F ∈ (0, +∞) . Since we are assuming Ω (m) * 0 (see Eq. (4.0.6)), the above condition also grants that E ∈ (0, +∞). Let us now consider two motions t → x(t) and t → y(t) fulfilling the Lagrange equations, with energies fixed according to the above prescriptions (and t ranging within suitable intervals). Then, from Eqs. (4.3.9)(4.3.12) we infer The above equations can be interpreted as the conservation laws for the energies of two fictitious mechanical systems with kinetic energies (ℓ+1) 2 2ẋ 2 , (ℓ+1) 2 2ẏ 2 and potential energies −V 1 x 2ℓ , V 2 y 2ℓ , that we can denominate, respectively, as a nonlinear repulsor and a nonlinear oscillator. From the second equality in Eq. (4.3.14) we see that t → y(t) is an oscillatory motion such that for all t (4.3.15) (the above interval is the set {y ∈ R | V 2 y 2ℓ F} and the times t such that y(t) = ± F/V 2 1/(2ℓ) are inversion times for the motion). Since V 1 x(t) 2ℓ 0 and E > 0, from the first equality in Eq. (4.3.14) we inferẋ(t) 2 > 0, i.e.ẋ(t) = 0 for all t. Thusẋ(t) has a constant sign, and the function t → x(t) is strictly monotonic. From Eq. (4.3.14) we also infer quadrature formulas containing the hypergeometric-type function More precisely, we have the following ( 24 ): For future use, let us mention the asymptotics ( 25 ) here the dominating term is C ℓ /z 1/(2ℓ) , since 0 < 1 2ℓ 1 4 . Again for future use, let us also mention the special value ( 26 ) dx/ 1+(V1/E)x 2ℓ and then use for i = 1, 2 the following relations, based on the change of variable x = x(ti) s 1/(2ℓ) with s ∈ [0, 1]: One proceeds similarly to express via F ℓ the integral in Eq. (4.3.18). 25 This asymptotics follows from the definition (4.3.16) of F ℓ and from some known identities about hypergeometric functions, namely: a Kummer transformation (see, e.g., [22,Eq. 15.8.2]) and the elementary relations 2F1 (a, 0, c, ζ) = 1 for all ζ, 2F1 (a, b, c, ζ) = 1 + O(ζ) for ζ → 0. 26 Eq. (4.3.20) follows the definition (4.3.16) of F ℓ and from a general result about 2F1 (a, b, c, 1) (see, e.g., [22,Eq. 15.4.20]).

Choosing the initial data. More on the qualitative and quantitative analysis
We now fix the attention on the solutions t → x(t) and t → y(t)) of the Lagrange equations for L 1 , L 2 with energies as in Eqs. (4.3.12)(4.3.13) and with the following initial data, specified at time t = 0 by convention: The upper bound (F/V 2 ) 1/(2ℓ) prescribed here for Y is motivated by Eq. (4.3.15). The equality x(0) = y(0) will be employed in the sequel to infer that the scale factor a(t) vanishes for t → 0 + (see the considerations after Eqs. Each one of the solutions t → x(t) and t → y(t) is intended to be defined on the maximal admissible domain, that is on the largest interval containing t = 0 on which the solution is well defined. The discussion of subsection 4.3.1, combined with the present assumptions, ensures that t → x(t) is a strictly increasing function, while t → y(t) oscillates. The map t → x(t) has a bounded domain of the form The finite times t min , t max are characterized by the fact that (with ( ) ± indicating the limit from above or below). To determine t max , it suffices to employ the quadrature formula (4.3.17) with ξ = +1, t 1 = 0, t 2 = t max and x(t 1 ), x(t 2 ) replaced by Y, +∞, respectively; in this way we obtain (the limit x → +∞ indicated above is computed using the asymptotic expansion written in Eq. (4.3.19); the cited equation also defines the constant C ℓ ). The time t min could be determined by similar computations, but is irrelevant for the subsequent applications.
Let us now pass to the function t → y(t), oscillating in the range indicated by Eq. (4.3.15). This function is well defined for all t ∈ (−∞, +∞), and periodic: The period T is twice the time needed by y(t) to pass from the minimum to the maximum of the interval in Eq. (4.3.15), and this time can be computed using the quadrature formula (4.3.18); this gives Behavior of x(t) for positive times. The limits t → 0 + and t → t − max . For t ∈ (0, t max ) the function x(t) increases, starting from the initial value x(0) = Y > 0 and ultimately diverging. The small t behavior of x(t),ẋ(t) is determined by the smoothness of these functions and by the initial data (4.3.21), which of course imply On the other hand, the quadrature formula (4.3.17) with ξ = +1, t 1 = 0, t 2 = t ∈ (0, t max ) and To go on, let us view the observables of the model as functions of the cosmic time τ ; inserting the asymptotics (4.3.45) into Eqs.(4.3.41)-(4.3.43), we find the following for τ → 0 + : (4.3.47) (4.3.48) From the above asymptotic expansions it appears that, for τ → 0 + : the scale factor a(τ ) vanishes, which indicates the occurrence of a Big Bang; the reciprocal 1/a(τ ) diverges in a non-integrable way if n = 2, indicating the absence of a particle horizon in the case of space dimension 2, while it diverges in an integrable way if n > 2, showing that the model has finite particle horizon when the space dimension is equal to 3 or greater; Ω (m) (τ ) → 0, which on account of Eq. (4.0.5) proves that the field energy density dominates on matter density close to the Big Bang. (4.3.50) It is convenient to describe the limit t → t − max in terms of the cosmic time. To this purpose, we must first determine the asymptotics of τ (t) in this limit starting from the integral representation (4.3.7); since it is not so obvious how to proceed, we have given some detail on this computation in Appendix D. Here we only report the final result, which is Thus τ (t) → +∞ for t → t − max . Considering the inverse function t → t(τ ), it can be readily checked that Eq. (4.3.52) implies the following, for τ → +∞: (4.3.53) Insert the above asymptotic relation into Eq.s (4.3.49-4.3.51), we find the following for τ → +∞: (4.3.56) As we can see, for τ → +∞ the following phenomena occur with exponential speed: the scale factor diverges, the field behaves like a cosmological constant (w (φ) (τ ) → −1) and the field energy density dominates on matter density. 68

A Appendix. On the calculations of Section 2
Let us consider the spacetime metric (2.2.5) and the coordinate system (2.2.6) (recalling that Greek indexes range from 0 to n, while Latin indexes range from 1 to n). Moreover, we make all the assumptions stated in Section 2 about the scalar field and the matter fluid.
The metric g µν . From Eq. (2.2.5) we infer (for i, j = 1, ..., n) recall that A, B are functions of x 0 ≡ t, while the h ij 's are functions of the space coordinates (x i ) ≡ x. For future use, let us mention that g := det(g µν ) can be expressed as follows in terms of h := det(h ij ) > 0: The Ricci tensor R µν and scalar curvature R. Given the metric (A.1), these are (i, j = 1, ..., n) Since ϕ depends only on t as indicated in (2.2.10) and g µν is as in (A.1), for i, j = 1, ..., n we have Comparing this result with Eq. (2.2.8) for U µ and Eq. (A.1) for g µν , it follows that T (φ) µν can be written in the fluid-like form recall that, according to (2.2.10), ρ (m) depends only on t.

(B.2)
The related solutions of Eq. (4.2.28) can be easily derived via the following integral representation, evaluating the basic integrals which result upon substitution of the expressions (B.1)(B.2) for x(t): where C 0 , D 0 ∈ R are integration constants and t 0 ∈ (0, +∞) is fixed arbitrarily. On the one hand, from Eqs. (B.1)(B.3) we infer, for any w ∈ R\{±1} such that w = −3,

B.3 The case ε = +1
Recall that in this case Eqs. (4.2.5)(4.2.6) reduce, respectively, to Eqs. (4.2.37)(4.2.38). By direct inspection it appears that any positive solution of Eq. (4.2.37) can be written as follows, after a time translation t → t + const.: x(t) = A sin(ω t) for A > 0 and t ∈ (0, π/ω) . where C, D ∈ R are integration constants. This representation is understood to hold for all values of w ∈ R\{±1} granting the convergence of the integral over s ∈ (0, t); however, also in this case the final result can be extended to other values of w (see the remark at the end of this subsection).

79
The integrals for s ∈ (0, ℓ) and for s ∈ (ℓ, L) can be treated as described in the previous paragraph. On the other hand, the integrals for s ∈ (L, ω t) can both be recast in the following form, performing the change of the integration variable σ := η e s/2 (for η = ζ where Q is defined as in Eq. (C.3). Summing up, the above arguments allow us to infer that for L/ω < t < +∞. The obtained upper and lower bounds are just T ± ζ (t). The arguments described in the previous two paragraphs prove Eq. (C.8) for all t ∈ (0, +∞).
Asymptotics of T ± ζ (t) for small and large t. The asymptotic behavior of P(z), Q(z) for z → 0 + and z → +∞ is readily derived from the definitions (C. By comparison with Eq. (4.2.99) about τ (t)/θ, we see that T − ζ (t) has just the same asymptotics as τ (t)/θ both for small and large t; on the other hand, the asymptotics of T + ζ (t) and τ (t)/θ are very similar in both limits.  The statement in (4.3.39) about x(t) is obvious, since x(0) = Y > 0 and t → x(t) is a strictly increasing function. In the rest of this subsection we show how to derive the inequalities −x(t) < y(t) < x(t) for t ∈ (0, t max ). Our arguments also involve the inversion time t * of Eqs. (4.3.34) (4.3.37); we will treat separately the cases t max t * and t max > t * . i) t max t * . Let us consider any time t ∈ (0, t max ); then we have Y < x(t), Y < y(t) < (F/V ) 1/(2ℓ) and Eqs. (4.3.29)(4.3.36) give Clearly, the above chain of identities is verified if and only if for all t ∈ (0, t max ) .