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Article

Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes

1
Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20133 Milano, Italy
2
Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via G. Celoria 16, I-20133 Milano, Italy
*
Author to whom correspondence should be addressed.
Universe 2024, 10(12), 467; https://doi.org/10.3390/universe10120467
Submission received: 4 August 2024 / Revised: 4 November 2024 / Accepted: 5 November 2024 / Published: 23 December 2024
(This article belongs to the Special Issue Universe: Feature Papers 2024—'Cosmology')

Abstract

:
In the initial part of this paper, we survey (in arbitrary spacetime dimension) the general FLRW cosmologies with non-interacting perfect fluids and with a canonical or phantom scalar field, minimally coupled to gravity and possibly self-interacting; after integrating the evolution equations for the fluids, any model of this kind can be described as a Lagrangian system with two degrees of freedom, where the Lagrange equations determine the evolution of the scale factor and the scalar field as functions of the cosmic time. We analyze specific solvable models, paying special attention to cases with a phantom scalar; the latter favors the emergence of nonsingular cosmologies in which the Big Bang is replaced, e.g., with a Big Bounce or a periodic behavior. As a first example, we consider the case with dust (i.e., pressureless matter), radiation, and a scalar field with a constant self-interaction potential (this is equivalent to a model with dust, radiation, a free scalar field and a cosmological constant in the Einstein equations). In the phantom subcase (say, with nonpositive spatial curvature), this yields a Big Bounce cosmology, which is a non-absurd alternative to the standard ( Λ CDM) Big Bang cosmology; this Big Bounce model is analyzed in detail, even from a quantitative viewpoint. We subsequently consider a class of cosmological models with dust and a phantom scalar, whose self-potential has a special trigonometric form. The Lagrange equations for these models are decoupled passing to suitable coordinates ( x , y ) , which can be interpreted geometrically as Cartesian coordinates in a Euclidean plane: in this description, the scale factor is a power of the radius r = x 2 + y 2 . Each one of the coordinates x , y evolves like a harmonic repulsor, a harmonic oscillator, or a free particle (depending on the signs of certain constants in the self-interaction potential of the phantom scalar). In particular, in the case of two harmonic oscillators, the curves in the plane described by the point ( x , y ) as a function of time are the Lissajous curves, well known in other settings but not so popular in cosmology. A general comparison is performed between the contents of the present work and the previous literature on FLRW cosmological models with scalar fields, to the best of our knowledge.
PACS:
98.80.-k; 98.80.Cq; 95.36.+x; 04.40.Nr
MSC:
83-XX; 83F05; 83C15

Contents
1.Introduction4
2.Generalities on Einstein’s Gravity, Perfect Fluids and Scalar Fields11
2.1. Dimensional Aspects11
2.2. Conventions About Spacetime, the Einstein Equations, and the Gravitational Constant11
2.3. Einstein’s Gravity with Perfect Fluids and a Scalar Field12
3FLRW Cosmologies with Perfect Fluids and a Scalar Field14
3.1. Spacetime Structure14
3.2. Conditions for Timelike or Lightlike Geodesic Completeness: Nonsingular FLRW Spacetimes 15
3.3. Hubble Parameter16
3.4. The Perfect Fluids16
3.5. The Scalar Field17
3.6. The Total Stress–Energy Tensor17
3.7. Einstein Equations17
3.8. Klein–Gordon Equation18
3.9. Conservation Law for the n-th Fluid18
3.10. Summary of the Evolution Equations18
3.11. Curvature Density; Normalized Densities18
3.12. Stationary Points of the Scale Factor19
3.13. Energy Conditions19
3.14. The Scale Factor at a Special Time20
3.15. Dimensionless Formalism20
4.Analysis of the Evolution Equations24
4.1. Determining the Fluids’ Densities24
4.2. The Fluids’ Densities in the Linear Case26
4.3. The Final Form of the Einstein and Klein–Gordon Equations26
4.4. Lagrangian Formalism; Zero-Energy Constraint27
4.5. Again on the Scale Factor at a Special Time28
4.6. Maximal Solutions29
5.The Case with a Constant Potential for the Scalar Field: General Results on the Evolution of  a ( t )  and  ϕ ( t ) 29
5.1. Basic Setting29
5.2. The Constant of Motion Π 30
5.3. Reduced Lagrangian30
5.4. Zero-Energy Solutions of the Reduced System31
5.5. Time Evolution of the Scalar Field32
5.6. On Maximal Solutions32
5.7. Hubble Parameter; Densities and Pressures33
5.8. The Scale Factor at a Special Time33
5.9. Comparison with the Literature34
6.Again on the Case of  V ( ϕ )  = Constant: Big Bounce from a Phantom Scalar34
6.1. Introducing a More Specific Setting34
6.2. The Case of Π = 0  : Recovering the Standard Model of Cosmology36
6.3. Introducing the Analysis of Cases with Π 0 40
6.4. Preparing the Analysis of the Phantom Case with Π 0 40
6.5. The Phantom Case with Π 0 (Small) and Ω k 0 : Analysis of the Potential V ( a ) and the Associated Function H ( a ) : = V ( a ) / a 2 41
6.6. The Phantom Case with Π 0 (Small) and Ω k 0  : Analysis of the Zero-Energy Solutions and Classicality Condition45
6.7. The Phantom Case with Π 0   (Small), Ω k = 0 and Specific Values for All the Parameters54
6.8. The Phantom Case with Π 0 (Small) and Ω k < 0 : Sketching the Basic Results57
7.Polar and Cartesian Coordinates for Phantom Cosmologies with a Periodic Field Potential62
7.1. Polar Coordinates (Under Periodicity Assumptions for the Field Potential)62
7.2. Cartesian Coordinates63
8.An Explicitly Solvable Phantom Model with Dust and a Trigonometric Field63
8.1. Some Introductory Considerations63
8.2. Introducing the Solvable Model64
8.3. The Case W i = λ i 2 > 0 ( i = 1 , 2 )66
8.4. Cosmologies with λ 1 = λ 2 66
8.5. Cosmologies with λ 1 and λ 2 Arbitrary69
8.6. The Case W i = ω i 2 < 0 ( i = 1 , 2 ): Lissajous Cosmologies72
8.7. Lissajous Cosmologies with ω 1 = ω 2 72
8.8. Lissajous Cosmologies with ω 2 = 2 ω 1 75
8.9. On General Lissajous Cosmologies83
8.10. The Case W 1 > 0 , W 2 < 0 86
8.11. Again on the Case W 1 > 0 , W 2 < 0 88
9.Concluding Remarks93
Appendix A.Riemannian Manifolds, Spacetimes, and Dimensional Analysis93
Appendix B.FLRW Spacetimes and Generalizations94
Appendix C.Time Orientation of a Spacetime: The Case of a Generalized FLRW Spacetime95
Appendix D.A Review on Geodesics and Several Associated Notions of Completeness, Mainly in Riemannian Manifolds and in Spacetimes. Nonsingular Spacetimes96
Appendix E.Geodesics and Geodesic Completeness of Generalized FLRW Spacetimes98
Appendix F.A Review the Energy Conditions106
Appendix G.On the Determination of the Fluids’ Densities106
Appendix H.Comparing the Zero Energy Motions of Two One-Dimensional Lagrangians107
Appendix I.The Descartes’ Rule of Signs107
Appendix J.Two Statements on a Class of Polynomials108
Appendix K.Proof of Statements (185) and (186) on the Potential  V  of the Standard Cosmological Model110
Appendix L.Proof of Statements (a)–(d) in Section 6.5 about the Function  V  of Equation (201)111
Appendix M.Proof of Statements (f)–(i) in Section 6.5 about the Function  H  of Equation (220)114
Appendix N.Proof of Equation (250) (Including Convergence of an Integral Therein)116
Appendix O.Proof of Some Statements about  a ϕ r  in Equation (271)117
Appendix P.A Usefulf Inequality118
Appendix Q.Estimates on the Time  t i  in Equation (244)119
Appendix R.Further Estimates on the Time  t i  in Equation (244)122
Appendix S.On the Quantity  Δ ϕ  in Equation (292)125
Appendix T.A More Detailed Description of the Model in Section 6.7128
Appendix U.Proof of Statements ( α - ζ ) in Section 6.8 about the Function  V  of Equation (201)128
Appendix V.Derivation of Equations (363)–(366)130
Appendix W.Derivation of Equations (401) and (402)132
Notes132
References136

1. Introduction

Scalar fields in gravity theories and cosmology. The scalar fields considered in this work are classical objects; the term “classical” is used in opposition to “second quantized”. Consistent with this viewpoint, throughout the present Introduction, the term “scalar field” is used in the classical sense, unless the contrary is declared explicitly.
The consideration of scalar fields in generalized theories of gravitation or within standard general relativity, and the application of such theoretical settings to cosmology are all issues with a long story. In the Brans–Dicke theory [1], proposed in the early 1960s as an alternative to Einstein’s gravitational theory, Mach’s principle about the origin of inertia is implemented by referring to a scalar field, whose reciprocal plays the role of an effective gravitational constant. Such alternative gravity theories are outside the scope of the present work; so, in the sequel, we always refer to standard general relativity (Einstein’s gravity theory) and its applications in cosmology.
In this standard framework, there are important motivations for considering cosmological models with scalar fields. On the one hand, a scalar field (“inflaton”) can be used as a model for the mechanism driving inflation. This approach originates from the work of some scholars at the beginning of the 1980s: let us mention, in particular, Linde [2], Madsen, and Coles [3].
On the other hand, one can use a scalar field as a dynamical model for dark energy; among the pioneering works in this area, let us mention a paper by Ratra and Peebles [4] in the late 1980s. Caldwell, Dave, and Steinhardt [5] are credited to have introduced the term “quintessence” ten years later in connection with scalar models of dark energy. Shortly afterwards, a paper by Peebles and Vilenkin [6] presented the first attempt (to our knowledge) to use a scalar field in a unified framework, to model both inflation in the primordial universe and dark energy in later epochs.
A long time before these contributions, an anticipatory paper by Bekenstein [7] introduced a cosmological model where a scalar field was considered for a very different purpose, namely, to prevent a singular behavior (no Big Bang); we will return to [7] later.
Self-interacting scalar fields are often considered in cosmology, describing this situation via an appropriate potential. This will also be the viewpoint of the present paper: see the action functional in our subsequent Equation (6), where Φ is the scalar field and a self-interaction potential V ( Φ ) appears.
Another possibility, frequently considered in the literature but not in the present work, concerns a direct interaction between the scalar field and the gravitational field. The most typical interaction (in the framework of Einstein gravity) involves the addition of a term ( 1 / 2 ) ξ R Φ 2 under the integral in our Equation (6), where R is the scalar curvature, and ξ is a numerical parameter. It is well known [8] that the evolution equation for the scalar field has properties of conformal invariance if ξ = ( d 1 ) / ( 4 d ) , where d + 1 is the spacetime dimension, and in this case, one speaks of a scalar field conformally coupled to gravity. The setting of the present paper corresponds to the choice ξ = 0 ; as is well known, this case is described in terms of a scalar field minimally coupled to gravity.
Obviously enough, for spacetime, most cosmological applications assume a Friedmann–Lemaître–Robertson–Walker (FLRW) geometry; this corresponds to our Equations (15)–(17), involving the cosmic time τ and the (positive, dimensionless) scale factor a ( τ ) .
Needless to say, the literature on the above issues is enormous; in this paper, we indicate just a few selected references. For a very recent overview, let us mention the review by Avsajanishvili, Chitov, Kahniashvili, Mandal, and Samushia [9] on dynamical models for dark energy, and on the constraints on such models arising from observational data; this paper devotes plenty of space to scalar field models for dark energy (and inflation). Concerning connections with observational data, let us also recall the seminal paper by Saini, Raychaudhury, Sahni and Starobinsky [10], dating back to 2000; here, dark energy is modeled via a scalar field (minimally coupled to gravity), and a probabilistic approach is developed to reconstruct the self-interaction potential V ( Φ ) from experimental data of the Supernova Cosmology Project [11] on luminosity distance vs redshift.
Canonical and phantom scalar fields. The case where the kinetic part in the action functional of a scalar field has an anomalous sign has not rarely been considered in the literature: the term “phantom” is used to describe this situation. When the kinetic part of the action functional has the usual sign, the expression “canonical scalar field” is used to avoid confusion. In the framework of this paper, the kinetic part in the action functional of the scalar field corresponds to the term ( σ / 2 ) μ Φ μ Φ in Equation (6) with σ = + 1 and σ = 1 , respectively, in the canonical and in the phantom case.
The kinetic energy density of a phantom scalar field is negative; this favors violations of the energy conditions [12,13] usually prescribed for the stress–energy tensor. On the other hand, a similar situation often occurs if one considers the (renormalized) vacuum expectation value of the stress–energy tensor of a quantized, canonical scalar field (see, e.g., [14,15]); so, a phantom scalar field (in the classical sense ascribed to this expression throughout this paper) can be used as a simplified model for a quantized, canonical scalar field in a vacuum state. This idea is not new: the statement that “a phantom scalar may be the effective description for some quantum field theory” appeared two decades ago, in the literal form reported here, in a paper by Nojiri and Odintsov [16].
Phantom scalar fields have been considered both in cosmology and in other applications of general relativity. A typical non-cosmological application concerns wormholes: since the 1960s, Ellis [17] and Bronnikov [18] proposed phantom scalars as sources for the Einstein equations to obtain wormhole solutions. In the framework of cosmology, the use of phantom scalars to model dark energy was supported at the beginning of the 2000s by Caldwell [19]; Carroll, Hoffman, and Trodden [20]; and Nojiri and Odintsov (let us cite again [16]).
Some authors also considered the possibility of a mixed, canonical or phantom behavior depending on the intensity of the scalar field: with the notations of the present paper, this would involve replacing the term ( σ / 2 ) μ Φ μ Φ in the action functional (6) ( σ = ± 1 ) with one of the form ( 1 / 2 ) ω ( Φ ) μ Φ μ Φ , where Φ ω ( Φ ) is a given real-valued function, with sign depending on Φ . This idea was applied to cosmological models by Capozziello, Nojiri, and Odinstov [21] and Elizalde, Nojiri, Odintsov, Gómez, and Faraoni [22].
Finally, let us mention that, in recent times, two of us (with D. Fermi) [23] compared the behavior of canonical and phantom scalars in connection with the horizon problem in FLRW cosmologies with nonpositive spatial curvature, ordinary types of matter and a scalar field, admitting a Big Bang; our conclusion was that, while the horizon problem is always present in the case with a canonical scalar, it disappears in the case with a phantom scalar and a suitable self-interaction potential.
Exact solutions of cosmological models involving scalar fields. It is not our intention to give a historical overview of this topic; we introduce the subject by referring to the book by Faraoni [24], which provides extensive information and a rich bibliography.
In FLRW cosmological models of this type, the unknowns are typically the scale factor; the scalar field(s); and, e.g., the densities of certain matter fluids as functions of time. The exact results presented in the literature are special solutions or even the general solution of the evolution equations.
Among the references that mostly inspired our past or present investigations about solvable FLRW cosmologies with scalar fields, we will first mention some works of the Naples school, authored by Capozziello, de Ritis, Esposito, Marmo, Piedipalumbo, Platania, Rubano, Scudellaro, and Stornaiolo [25,26,27,28]; these rely on the “Nöther symmetries method” developed by the same school [26,29], which allows to determine the general solution of the model for suitable choices of the self-interaction potential (all these papers consider canonical scalar fields, except for [25], which deals with a phantom).
Another source of inspiration for our studies in this area has been a paper by Fré, Sagnotti, and Sorin [30]. This work considers spatially flat FLRW universes containing only a self-interacting canonical scalar field and gives a list of self-potentials that enable to determine the general solution of the evolution equations; in the Ph.D. thesis of one of us [31] and in a paper by D. Fermi and two of us [32], several results of [30] were generalized to FLRW universes with nonzero spatial curvature and/or containing, besides the canonical scalar, a perfect fluid with a suitable equation of state. Most papers mentioned before will be reconsidered at the end of Section 8.2.
To continue, let us point out a peculiar “inverse viewpoint” proposed by several scholars working on exact solutions of FLRW cosmologies with scalar fields and, possibly, matter fluids. The idea of these authors is that, instead of solving the evolution equations of the model with a given self-interaction potential for the scalar field, one could give some prescription on the evolution of the model (e.g., specify the time dependence of the scale factor) and infer a posteriori all the other features, including the field self-potential. This viewpoint was proposed by Ellis and Madsen [33] and Easther [34] in the 1990s and developed in recent times by Dimakis, Karagiorgos, Zampeli, Paliathanasis, Christodoulakis, and Terzis [35], as well as Barrow and Paliathanasis [36].
The “generating function methods” and the “superpotential method” are realizations of the inverse viewpoint, where the self-potential of the scalar field is again reconstructed a posteriori; these approaches and their history are described in detail in the recent book by Chervon, Fomin, Yurov, and Yurov [37]. In the simplest version of these methods, one considers a spatially flat FLRW cosmology with a scalar field (and no matter content), whose evolution equations are rephrased in a form proposed independently by Ivanov in 1981 [38] and by Salopek and Bond in 1990 [39]; the dependence of the Hubble parameter on the scalar field is prescribed via a generating function, from which one also derives the field self-potential and an exact solution of the model. The superpotential method mentioned before applies to both canonical and phantom scalar fields; in the phantom case (with no matter content), several exact solutions of FLRW cosmologies were obtained using this method in a paper by Chervon and Panina [40].
In spite of its interesting features, the “inverse” viewpoint is not employed in the present work: our attitude is that the self-potential of the scalar field is given, and one should try to infer the time dependence of the scale factor, the scalar field, and so on. We briefly return to [38,40] at the end of Section 8.2.
Nonsingular cosmologies. The historical evolution of cosmology after Friedmann and Lemaître has led people to regard spacetime singularities as a paradigmatic fact. In FLRW spacetimes, such a singular behavior is typically associated with the initial or final vanishing of the scale factor, i.e., to a Big Bang or Big Crunch; the Big Bang is an obvious feature of the standard ( Λ CDM) cosmological model. It is known after the studies of Penrose, Hawking and Geroch in the 1960s–1970s [12,41,42,43,44] (see also [45]) that singularities are present in any cosmological model based on Einstein gravity fulfilling (i) reasonable causality conditions; (ii) some standard energy condition for the stress–energy tensor (such as the weak, or, more typically, the strong energy condition)1; (iii) suitable technical requirements, indicating the expansion (or contraction) of the universe.
In the cited references, the term “singularity” has a very precise technical meaning and indicates that some timelike or lightlike geodesics are incomplete in the past or future; the same viewpoint is adopted systematically in the present work (see Section 3.2 and Appendix D and Appendix E mentioned therein).
The facts recalled above support a view of spacetime singularities as an almost natural feature of cosmological models; yet, this feature is very disturbing. Quoting directly from Novello and Perez Bergliaffa [46], “a singularity can be naturally considered as a source of lawlessness⋯ because the spacetime description breaks down ‘there’, and physical laws presuppose spacetime”. Let us also cite the paper by Boisseau, Giacomini, Polarski and Starobinsky [47], where it is stated that a singularity-free cosmological model “can cure many of the problems occurring in Big Bang cosmology”.
Two typical classes of FLRW cosmologies with no singularities are formed by models with a Big Bounce and periodic models.
We use the term “Big Bounce” to describe a situation in which the scale factor τ a ( τ ) , well defined for all τ ( , + ) , attains a nonzero minimum value at a unique time and is always decreasing and increasing, respectively, before and after this time (the more generic expression “bounce” is employed in the literature in connection with the nonzero local minima of the scale factor).
Obviously enough, the term “periodic” refers to the case where the scale factor oscillates periodically between a nonzero minimum value and a maximum value.
We do not aim to give here a thorough overview of nonsingular cosmologies; the already-cited paper by Novello and Perez Bergliaffa [46] reviews this subject, starting from pioneering contributions in the 1930s.
Of course, to produce a cosmological model free of singularities, one must violate some of the assumptions in the above-mentioned theorems of Penrose, Hawking, and Geroch. One way to evade such theorems is to employ a modified, non-Einsteinian gravity theory (e.g., Brans–Dicke theory, f ( R ) theory or Weyl’s unified theory of gravity and electromagnetism); let us repeat that such theories fall outside the scope of our paper.
Another way to evade singularity theorems is to consider a stress–energy tensor not fulfilling the standard energy conditions as a source for the Einstein equations. While repeating the recommendation to consult [46], here, we just mention a few models of this kind (all of them with an FLRW spacetime geometry).
An early model based on Einstein’s gravity, which is nonsingular due to the violation of the energy conditions, was proposed by Bekenstein [7,48] in the 1970s: here, the universe is filled with dust (i.e., pressureless matter), radiation, and a free, massless, canonical scalar field; the scalar field is conformally coupled to gravity, and also coupled to dust in a natural way. The model can be solved exactly, and nonsingular solutions are obtained under suitable conditions on the basic parameters and integration constants; certain solutions exhibit a Big Bounce, while in other cases, the scale factor oscillates.
In 1996, Dabrowski [49] presented a nonsingular cosmological model, deserving appreciation due to its simplicity. In this model, an FLRW universe (of the usual dimension 3 + 1 ) is filled only by non-interacting perfect fluids, namely dust, radiation, and the so-called string-like and wall-like matter, with positive (mass-energy) densities and negative pressures related, respectively, with the equations p s = ( 1 / 3 ) ρ s and p w = ( 2 / 3 ) ρ w (the last two kinds of matter account for the presence of cosmic strings and domain walls. Both of them individually fulfill the weak energy condition; wall-like matter violates the strong energy condition). The model of [49] is exactly solvable; for suitable, negative values of the cosmological constant, it gives rise to nonsingular solutions in which the scale factor oscillates periodically (with a nonzero minimum).
In the subsequent decade, Dabrowski, Stachowiak, and Szydłowski [50,51] modified the previous model with the addition of “phantom” perfect fluids, with positive densities and negative pressures related through equations of the form p p h = w p h ρ p h , involving constants w p h < 1 (any fluid of this type violates both the weak and strong energy conditions). The exact solutions for this modified model obtained in [50,51] include a variety of different behaviors, e.g., nonsingular, periodic universes or universes with a single bounce, which, however, have singularities related to the divergence of the scale factor at finite times before and after the bounce.
The already-cited paper [47] presents a nonsingular model in the framework of Einstein’s gravity, again with the violation of the energy conditions. In this case, the unique content of the universe is a canonical scalar field, conformally coupled to gravity and self-interacting. The self-interaction potential is the sum of a positive constant and a quartic term in the scalar field, with a negative coefficient; the additive constant can be reinterpreted as a positive cosmological constant, and the negative quartic term makes this potential unbounded from below. The general solution of the model is computed explicitly in the case of zero spatial curvature and describes a Big Bounce for a suitable set of values of the integration constants, with a nonzero measure.
Besides scalar fields, other classical fields have been found to produce nonsingular cosmologies in the framework of Einstein’s gravity; let us mention, in particular, the models based on nonlinear generalizations of electromagnetism (e.g., the Born–Infeld theory), for which we again refer to [46].
It has been known for a long time that nonsingular FLRW cosmologies arise in the framework of semiclassical Einstein gravity, where the source for the Einstein equations is (or includes) the renormalized expectation value of the stress–energy tensor of a quantized field with respect to a suitable quantum state (e.g., a quantized, canonical scalar field; the field is typically free, massless, conformally coupled to gravity and in a vacuum state). Seminal contributions in this area were given between 1973 and 1984 by Parker and Fulling [52], Starobinsky [53,54], Gurovich and Starobinsky [55], and Anderson [56,57]. Among the subsequent investigations on FLRW cosmologies and semiclassical Einstein’s gravity, let us mention in particular the paper by Dappiaggi, Fredenhagen and Pinamonti [58] (dealing with a quantized, massive scalar field in a general Hadamard state, and discussing the stability issue for the solutions of the model in detail).
Finally, let us just mention that theories attempting to quantize gravity also yield cosmologies that are nonsingular, in a suitable sense; in particular, a “quantum Big Bounce” has been predicted by Ashtekar, Pawlowski, and Singh [59] in the framework of loop quantum gravity. Like classical modifications of Einstein’s gravity, quantum gravity theories fall outside the scope of the present paper.
Aims and contents of the present work. The present work has different aims, indicated in the sequel as (i)(ii’)(ii”). Hereafter, for each one of these aims, we establish a schematic formulation; this is followed by an illustration of the related sections in the paper, describing their contents, and giving some indications on the collocation of such contents with respect to the literature. The first aim of our work is as follows:
(i)
To review the general setting for FLRW cosmologies with non-interacting perfect fluids and with a (canonical or phantom) scalar field, possibly self-interacting and minimally coupled to gravity, with special attention to the Lagrangian formalism.
The above issues are treated in Section 2, Section 3 and Section 4 (and in Appendix A, Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G and Appendix H, cited therein).
In Section 2, we introduce a general setting for Einstein’s gravity with the above actors. Here, spacetime is an arbitrary Lorentzian manifold of dimension d + 1 , with d { 2 , 3 , 4 , } . The n-th fluid, not interacting directly with the other actors, has an arbitrary equation of state p n = P n ( ρ n ) , relating the pressure to the mass–energy density; the scalar field is coupled only to gravity, in a minimal way, and its self-potential is arbitrary. The action functional for this system is in the already-cited Equation (6); the Einstein equations and the evolution equations for the fluids and the scalar field (Klein–Gordon equation) are presented.
In Section 3, we direct our attention to FLRW cosmologies, which are the subject of the rest of the paper. In this case, spacetime geometry has the form (15)–(17), involving cosmic time τ and the scale factor a = a ( τ ) , with an arbitrary value for the constant sectional curvature in the spatial part of the metric; the fluids’ densities depend only on τ , like the scalar field. A suitable dimensionless formulation is introduced, rescaling all relevant physical quantities via powers of the gravitational constant and a second constant H ; the latter is the reciprocal of a time and in principle can be fixed arbitrarily, but is often identified with the present-time value of the Hubble parameter. This setting allows us to introduce, e.g., the dimensionless time variable t = H τ (see Equation (61)), which is employed systematically in the rest of the paper. Section 3 also reviews the general notions of nonsingular (or singular) spacetime in terms of timelike or lightlike geodesic completeness and presents the necessary and sufficient conditions derived by O’Neill [60], Romero, and Sanchez [61,62] for an FLRW spacetime to be nonsingular in any one of these senses; the (weak and strong) energy conditions, with their specializations to the FLRW case, are recalled as well.
Section 4 is devoted to a systematic study of the evolution law for the FLRW cosmologies of Section 3. The density of any fluid is shown to be a known function of the scale factor (determined by the equation of state of the fluid), and we are left with a system of ODEs corresponding to the Einstein and Klein–Gordon equations, where the unknowns are the scale factor a and the scalar field ϕ (in dimensionless form) as functions of (dimensionless) time t. This system of ODEs is recognized to be equivalent to the Lagrange equations for a suitable Lagrangian L ( a , ϕ , a ˙ , ϕ ˙ ) (with ˙ ≡ d/dt, see Equations (121) and (122)), supplemented with the condition that the energy of the Lagrangian system vanishes (zero-energy constraint). This Lagrangian setting is well known; perhaps, our treatment is more general than usual for what concerns the equations of state of the fluids (see the comments and references at the end of Section 4.4). The other aims of this paper can be described cumulatively in the following way:
(ii’-ii”)
To explore some exactly solvable cases of the previous setting for FLRW cosmologies, which have received (to our knowledge) little or no attention in the literature; these special cases often involve phantom scalar fields and yield nonsingular cosmological models.
Hereafter, we illustrate the above two items separately; the first one in this pair can be formulated as follows:
(ii’)
To discuss the case where the self-potential of the scalar field is constant, paying special attention to a subcase with pressureless matter, radiation, and a phantom scalar, yielding a Big Bounce cosmology.
Of course, the case with a constant self-potential is mathematically simple; however, its implications are nontrivial, especially in the presence of a phantom scalar. We think this case has received insufficient attention in the literature, perhaps just due to its simplicity (on this point, see Section 5.9 and the final paragraph in Section 6.6). In the present work, we give a detailed consideration to (ii’) in Section 5 and Section 6 (and in Appendix I, Appendix J, Appendix K, Appendix L, Appendix M, Appendix N, Appendix O, Appendix P, Appendix Q, Appendix R, Appendix S, Appendix T and Appendix U cited therein).
In Section 5, we present a general analysis of the constant self-potential case for the scalar field; this is equivalent to a model with a cosmological constant and a free, massless scalar field. The Lagrangian L ( a , ϕ , a ˙ , ϕ ˙ ) (see again Equations (121) and (122)) is now independent of ϕ ; so, the canonically conjugate momentum Π   : = L / ϕ ˙ is a constant of motion. This fact reduces the study of the model to the analysis of a one-dimensional system with a Lagrangian depending only on a and a ˙ ; the latter can be ultimately put into the form L ( a , a ˙ ) = a ˙ 2 V ( a ) , where V is an effective potential determined by the equations of state of the fluids and the values of the constant spatial curvature and of the momentum Π (see Equations (141) and (142); the zero-energy constraint is now a condition on this reduced system). Formally, L is the Lagrangian of a conservative mechanical system with one degree of freedom; so, the standard techniques for such systems can be applied to infer the qualitative behavior of the scale factor a = a ( t ) from the graph of V , and to compute the function a ( t ) via quadratures. (The time dependence of the scalar field can also be determined by quadratures).
In the subsequent Section 6, the previous setting is specialized assuming a positive value for the constant self-potential (i.e., a positive cosmological constant), choosing for spacetime the usual dimension 3 + 1 and considering two fluids, namely pressureless matter (dust) and a radiation gas. If the momentum Π vanishes, the (canonical or phantom) scalar field is constant and the scale factor a = a ( t ) evolves as in the standard ( Λ CDM) model of cosmology, as illustrated in Section 6.2. For Π 0 , we have a modification of the standard model, which is especially interesting in the presence of a phantom scalar field.
Our analysis concerns mainly the phantom case with Π nonzero and small, and with nonpositive spatial curvature. In this case, discussed in Section 6.5, Section 6.6 and Section 6.7, we find a Big Bounce model in which the scale factor a ( t ) attains a minimum value a 0 > 0 at t = 0 , decreases for t 0 , increases for t 0 and diverges exponentially for t (indeed the model is time-symmetric: a ( t ) = a ( t ) ); the energy conditions are analyzed, and their violations are indicated. A reasonable choice for the value of Π is one giving a very small minimum a 0 for the scale factor but ensuring that the total mass–energy density is always much smaller than the Planck density, so that the classical treatment of gravity is reasonable (“classicality condition”). In Section 6.7, we set the spatial curvature to zero and we propose numerical values for all the other parameters of the model implementing the previous ideas, and ensuring that the (normalized) mass–energy densities of dust, radiation, and the phantom scalar at the present time have the values usually ascribed to the (normalized) densities of matter, radiation, and dark energy in the standard model. The result is a model that differs significantly from the standard one in an interval of duration ≃ 10 56  s after the Big Bounce, when the energy density of the phantom scalar is negative and dominates the radiation density. After this very short time interval, the model is practically indistinguishable from the standard one; so, we have a radiation-dominated era followed by a matter-dominated era, and then the present epoch dominated by dark energy (the latter is represented by the phantom scalar, which has now a positive energy density if one includes the constant self-potential corresponding to the cosmological constant).
Finally, Section 6.8 discusses the phantom case for Π nonzero and small as well as positive values of the spatial curvature, finding nonsingular cosmologies with a Big Bounce or a periodic behavior of the scale factor.
We previously mentioned two related aims (ii’-ii”) of the present work, and we have just concluded the presentation of (ii’). The other aim can be formulated as follows:
(ii”)
To propose a solvable FLRW cosmological model with a phantom scalar, a specific self-interaction potential of trigonometric type and pressureless matter. This model is treated passing from the ( a , ϕ ) coordinates to suitably defined “Cartesian” coordinates ( x , y ) ; when the coefficients of the self-interaction potential are negative, the trajectories of the model in the ( x , y ) space are Lissajous curves.
The above issues are discussed in Section 7 and Section 8 (and in the related Appendix V and Appendix W).
Section 7 has a preliminary role and considers a phantom scalar field with an arbitrary self-interaction potential, in the presence of fluids with very general equations of state; the spacetime dimension d + 1 is also arbitrary. After a coordinate change, with a = const .   ×   r 2 / d and ϕ = const .   ×   θ , the Lagrangian L ( a , ϕ , a ˙ , ϕ ˙ ) of the system (see once more Equations (121) and (122)) becomes a function L ( r , θ , r ˙ , θ ˙ ) (see Equations (336) and (337)); the latter can be interpreted mechanically as the Lagrangian of a particle in a Euclidean plane equipped with polar coordinates ( r , θ ) , in the presence of forces with a potential depending on r and θ . This fact (which is specific to the phantom case) allows for a nice visualization of the cosmological model in which the radius r (i.e., the distance of the particle from the origin) determines the scale factor, and the angle θ represents the scalar field. Of course, an equivalent description can be given in terms of the Cartesian coordinates x = r cos θ , y = r sin θ (see Equation (345)).
In Section 8, we show that an explicitly solvable model can be obtained specializing the setting of Section 7 to the case where the spatial curvature vanishes, there is only one fluid of the dust type (i.e., pressureless matter) and the self-interaction potential of the phantom scalar, expressed in terms of the angle θ , is a function of the form θ ( 1 / 2 ) W 1 cos 2 θ + ( 1 / 2 ) W 2 sin 2 θ with W 1 , W 2 two arbitrary constants (see Equations (350) and (351)). In this case, the evolution equations of the cosmological model in Cartesian coordinates take the form x ¨ = W 1 x , y ¨ = W 2 y (see Equation (355)), thus describing two decoupled systems; each one of the two systems is interpretable as a harmonic repulsor, a harmonic oscillator, or a free particle according to the sign of W 1 or W 2 . The connections of this framework with the previous literature are discussed in the final part of Section 8.2; to our knowledge, the previous work closer to this setting is the already-cited paper [25] by Capozziello, Piedipalumbo, Rubano, and Scudellaro, who considered only the special case W 1 = 0 .
For arbitrary W 1 and W 2 , there is a large variety of behaviors that are explored throughout Section 8; in particular, if W 1 and W 2 are both negative, we have two uncoupled harmonic oscillators, and the curves t ( x ( t ) , y ( t ) ) are the already-mentioned Lissajous curves [63,64], better known for very different reasons. Our analysis also considers the cases where W 1 and W 2 are both positive or have opposite signs; in all cases, we frequently meet nonsingular cosmologies with a Big Bounce or a periodic behavior, two features that are strongly connected with the presence of a phantom scalar. All these results must be understood as describing a variety of possible universes, whose physical plausibility should be discussed separately in each subcase.

2. Generalities on Einstein’s Gravity, Perfect Fluids and Scalar Fields

The general setting introduced here, and reconsidered in Section 3 and Section 4 in the framework of FLRW cosmologies, has intersections with the Ph.D. thesis of one of us [31] and with two papers by D. Fermi and two of us [23,32]. However, there are some technical differences (e.g., in the treatment of dimensional aspects); moreover, differently from the cited works, the equations of state considered here for the perfect fluids are arbitrary, apart from the regularity conditions stipulated in Section 4.1 onward. The cosmological models discussed in Section 5, Section 6, Section 7 and Section 8 of the present work are different from those considered in [23,31,32].

2.1. Dimensional Aspects

In this paper, we always need to distinguish dimensionless quantities from those having a physical dimension, i.e., lengths, times, masses, and all the derived quantities. As usual, dimensionless quantities are viewed as elements of the space R of real numbers. Lengths, times, and masses are described as elements of appropriate real, one-dimensional, oriented vector spaces L , T , M , from which one can build by appropriate (tensorial) constructions many other real, one-dimensional, oriented vector spaces2; an example often considered in the sequel is the space of mass densities in d spatial dimensions, which is D = M L d . If X is any one of the spaces R , L , T , M , D , and so on, we will write X + (respectively, X + ) for the subset of positive (respectively, non-negative) elements of X ( X + = X + { 0 } ). Throughout the paper, we identify a time duration τ with the length c τ , where c is the speed of light. Thus, T L , and we can ultimately confuse c with the real number 1; in the sequel, we will write c = 1 to recall these identifications.

2.2. Conventions About Spacetime, the Einstein Equations, and the Gravitational Constant

All the manifolds considered in this paper are assumed to be real, smooth, Hausdorff, connected and paracompact. Functions involved in calculus considerations are assumed to be smooth, whenever this notion makes sense; the expression “smooth” always means “of class C ”. We frequently refer to Riemannian manifolds or to spacetimes (i.e., manifolds with a Lorentzian metric, of signature ( , + , , + ) ); see Appendix A for some caveats on our use of these notions, including connections with the setting of Section 2.1.
Throughout the paper, we work on a ( d + 1 ) -dimensional spacetime M d + 1 of spatial dimension d 2 . The spacetime metric (of signature as above) is denoted with g; the corresponding squared line element is written d s 2 . A coordinate system on M d + 1 is generically written as ( x μ ) (with Greek indices); of course, d s 2 = g μ ν d x μ d x ν . The metric g is employed in the usual way to raise and lower indices of tensors. The covariant derivative, the Ricci tensor and the scalar curvature corresponding to g are indicated, respectively, with ∇ , R μ ν , and R. The Einstein equations are written as
R μ ν 1 2 R g μ ν = d ( d 1 ) γ d G d T μ ν ,
where T μ ν is the stress–energy tensor; here,
G d ( L d 2 M 1 ) + = ( L 2 D 1 ) +
is the gravitational constant, and γ d is a positive numerical coefficient, giving the volume of the unit sphere in d-dimensional Euclidean space divided by d 2 for any d 3 , and chosen arbitrarily if d = 2 , so that
γ d : = π d / 2 ( d 2 ) Γ ( d / 2 + 1 ) for d 3 , γ 2 : = any   positive   real   number .
As reviewed in [31] on the grounds of [69], with the above normalization, Einstein’s gravity predicts in the Newtonian limit that the gravitational force between two particles of masses m 1 , m 2 at a Euclidean distance r equals G d m 1 m 2 / r d 1 , for any space dimension d 3 ; let us also recall that Einstein’s gravity does not possess a Newtonian limit if d = 2  [70], which explains why γ 2 is not specified. Finally, let us remark that
γ 3 = 4 3 π ; d ( d 1 ) γ d G d | d = 3 = 8 π G , G G 3 the   usual   gravitational   constant .

2.3. Einstein’s Gravity with Perfect Fluids and a Scalar Field

We consider a general model living in a ( d + 1 ) -dimensional spacetime M d + 1 (with d 2 ), keeping all conventions of Section 2.2. We assume the model to include the following:
  • N species of non-interacting perfect fluids, all of them with the same ( d + 1 ) - velocity. For any n { 1 , , N } , we suppose that the n-th fluid has a positive mass–energy density ρ n and a pressure p n . We have ρ n : M d + 1 D + and p n : M d + 1 D where D is the space of mass densities mentioned in Section 2.1, which coincides with the space of pressures in our setting with c = 1 . We assume a barotropic equation of state of the following form:
    p n = P n ( ρ n ) , P n : D + D .
    The above-mentioned perfect fluids can include, for example, dust ( p n = 0 ) and a radiation gas ( p n = ( 1 / d ) ρ n ).3
  • A classical scalar field Φ , minimally coupled to gravity and self-interacting with potential V ( Φ ) . We have Φ : M d + 1 F , where F is an appropriate real, one-dimensional, oriented vector space. For dimensional reasons that will soon be clarified, we require F 2 = M L 2 d = D L 2 and V : F D . No direct interaction is assumed between the scalar field and the above-mentioned perfect fluids.
(Let us recall our general assumptions of smoothness, also applying to the functions mentioned above). The action functional governing the dynamics of this system is
S : = M d + 1 d v R 2 d ( d 1 ) γ d G d n = 1 N ρ n σ 2 μ Φ μ Φ V ( Φ ) .
In the above, d v is the pseudo-Riemannian volume element corresponding to the spacetime metric g ( d v = det ( g μ ν ) μ d x μ in any coordinate system ( x μ )); moreover, G d and γ d are the gravitational constant and the numerical coefficient in Equations (2) and (3). We already indicated that R and ∇ always stand for the scalar curvature and the covariant derivative induced by g; note that, Φ being a scalar function, μ Φ coincides with the usual derivative μ Φ in any coordinate system. Finally, σ { ± 1 } is a sign parameter: in compliance with the standard nomenclature, we shall call Φ a canonical scalar field (or an ordinary scalar field) if σ = + 1 , and a phantom scalar field if σ = 1 (in applications to cosmology, the terms quintessence and phantom quintessence are also employed for σ = + 1 and σ = 1 , respectively).
Note that all the summands between square brackets on the right-hand side of Equation (6) take values in the space D ; thus, S takes values in the space D L d + 1 = M L (which is the usual space of actions, due to the identifications between masses and energies, and between lengths and times, in our setting with c = 1 ).
In order to derive the dynamics of this model, the action functional S of Equation (6) must be viewed as depending on the spacetime metric g, on the fluids’ histories and on the scalar field Φ ; the fluids’ histories are defined via the world lines of the fluids’ particles and S depends on such histories through the densities ρ n , see, e.g.,  [12]4. The evolution equations for the model can be derived requiring the action S to be stationary under variations in the previously mentioned variables; the related calculations are standard (see again [12]), and here, we just report the final results.
Firstly, the stationarity condition under variations in the metric g yields the Einstein equations
R μ ν 1 2 R g μ ν = d ( d 1 ) γ d G d n = 1 N T μ ν n + T μ ν Φ ;
these involve the stress–energy tensors of the fluids and the scalar field, defined, respectively, as follows:
T μ ν n : = ( p n + ρ n ) U μ U ν + p n g μ ν for n { 1 , , N } ,
T μ ν Φ : = σ μ Φ ν Φ 1 2 g μ ν λ Φ λ Φ V ( Φ ) g μ ν ,
with U μ indicating the common ( d + 1 ) -velocity vector field of all the fluids ( U μ U μ = 1 )5.
Secondly, the stationarity of S under variations in the fluids’ histories leads to the separate conservation laws
μ T μ ν n = 0 ( n = 1 , , N )
(these manifest our assumption, codified in S , that no direct interaction exists between any two fluids or between any fluid and the scalar field).
Finally, the stationarity condition for S under variations in the field Φ yields the Klein–Gordon equation
σ μ μ Φ + V ( Φ ) = 0 ,
where V is the derivative of the function V . Indeed, Equations (7), (10), and (11) are not independent. For general information on this issue, see [23,31] or [32]; we return to this topic in Section 3 and Section 4, in a way that is exhaustive for the purposes of the present work.
In applications to the cosmology of the above general setting, the perfect fluids represent different types of matter or radiation; the motivations for considering a scalar field were recalled in the Introduction.
The case with a constant field potential. Let us specialize the previous framework into the following case:
V ( Φ ) = const . V .
Then, the summand d ( d 1 ) γ d G d T μ ν Φ in the right-hand side of the Einstein Equation (7) contains (according to (9)) a term of the form d ( d 1 ) γ d G d V g μ ν ; moving this term from the right-hand side to the left-hand side of Equation (7), we obtain
R μ ν 1 2 R g μ ν + Λ g μ ν = d ( d 1 ) γ d G d n = 1 N T μ ν n + T μ ν Φ ,
Λ : = d ( d 1 ) γ d G d V , T μ ν Φ : = σ μ Φ ν Φ 1 2 g μ ν λ Φ λ Φ .
Equation (13) includes the Einstein equations with a cosmological constant Λ , and the term T μ ν Φ on their right-hand side is the stress–energy tensor of a free, massless scalar field; note that Λ has the same sign as V .6

3. FLRW Cosmologies with Perfect Fluids and a Scalar Field

In the present Section 3, we apply the general setting of Section 2 to the framework of cosmology, assuming that the universe is spatially homogeneous and isotropic at any fixed time.

3.1. Spacetime Structure

To implement the assumptions of homogeneity and isotropy, we consider a ( d + 1 ) -dimensional Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, with d 2 ; this is a product
M d + 1 = T × S d ,
where T T ( = L ) is an open time interval with its natural coordinate τ (the “cosmic time”) and S d (the “space”) is a d-dimensional, complete Riemannian manifold of constant sectional curvature k L 2 , often referred to as the “spatial curvature” in the sequel. With the additional assumption of simple connectedness, S d is isometric to flat Euclidean space if k = 0 , a hyperbolic space if k < 0 , and a spherical hypersurface if k > 0 . Without the assumption of simple connectedness, S d is isometric to the quotient of one of the previously mentioned spaces with a suitable, discrete group of isometries; see [71]. We write h for the Riemannian metric of S d , and d 2 for the corresponding, squared line element. A coordinate system of S d is generically indicated with ( x i ) i = 1 , , d (using Latin indexes like i , j , ); of course, d 2 = h i j d x i d x j . It is well known (see again [71]) that S d can be covered by (local, L -valued) coordinates ( x i ) in which
d 2 = i = 1 d ( d x i ) 2 1 + k 4 i = 1 d ( x i ) 2 2 .
The squared line element of the spacetime (15) is supposed to have the following form:
d s 2 = d τ 2 + a 2 ( τ ) d 2 ,
where a : T R + , τ a ( τ ) is the (smooth, dimensionless) scale factor. Of course, any coordinate system ( x i ) i = 1 , , d on S d induces a spacetime coordinate system
( τ , ( x i ) i = 1 , , d ) ( x μ ) μ = 0 , , d
in which g 00 = 1 , g i j = a 2 h i j and g 0 i = g i 0 = 0 for i , j = 1 , , d . We refer to Appendix B for a slightly more formal description of this spacetime7.
The spacetime we are considering carries a natural time orientation, allowing us to distinguish between future and past; a timelike or lightlike tangent vector is future-directed (resp., past-directed) if and only if it vanishes, or its components ( X μ ) in any coordinate system of the form (18) are such that X 0 > 0 (resp., X 0 < 0 ); see Appendix C for a review of the general notion of time orientation and for some more details on the FLRW case.
If a particle is co-moving with the FLRW frame, and we use coordinates as above, along the world line of the particle, we have x i = constant for i = 1 , , d , while x 0 = τ gives the proper time; so, the ( d + 1 ) -velocity of the particle has the following components:
U μ = δ 0 μ ( μ = 0 , , d ) .
From now on, we intend
X : = d d τ
and X : = d 2 / d τ 2 ; the more standard dotted notation X ˙ will be used later for the derivative with respect to the dimensionless time variable t, defined by the forthcoming Equation (61).

3.2. Conditions for Timelike or Lightlike Geodesic Completeness: Nonsingular FLRW Spacetimes

In any spacetime, we can of course define (parametrized) geodesics and, in particular, geodesics of the timelike or lightlike type; a geodesic is said to be maximal if it cannot be extended, allowing the parameter to range in a wider interval. Given a time-oriented spacetime, a maximal, future-directed geodesic of the timelike or lightlike type is said to be past complete (resp., future complete) if it is defined on an interval with initial endpoint (resp., with final endpoint + ).
A time-oriented spacetime is said to be:
  • Past timelike complete, or past lightlike complete, or past complete if each maximal, future-directed geodesic of the timelike, lightlike , or of both types is past complete;
  • Future timelike complete, or future lightlike complete, or future complete if each maximal, future-directed geodesic of the timelike, lightlike , or of both types is future complete;
  • Timelike complete, lightlike complete or complete if it possesses the above-defined properties both in the past and future (meaning that the maximal geodesics of the types mentioned above are defined on the full interval ( , + ) ).
The adjective nonsingular will often be used as an equivalent for the adjective “complete”, in any one of the above senses; so, we will say that a spacetime is, e.g., future timelike nonsingular, or past nonsingular, or nonsingular. We refer to Appendix D for a more detailed description of all the above notions. Obviously enough, the terms incomplete or singular will be used in the sequel as opposites to “complete” or “nonsingular”.8
The case of an FLRW spacetime has been discussed by O’Neill [60] in relation to lightlike completeness, and by Romero and Sanchez [61,62] in relation to both timelike and lightlike completeness; hereafter we report (with minimal adaptations to our language) the results of these authors, which are justified in Appendix E just to make the present considerations self-contained9. Given an FLRW spacetime as in Section 3.1, let us represent the time interval T T as
T = ( τ , τ + ) , τ { } T , τ + T { + } , τ < τ + ,
and let us choose any instant τ 0 T . Then, the given, time-oriented spacetime is:
  • Past timelike complete, if and only if
    τ = , τ 0 d τ a ( τ ) 1 + a 2 ( τ ) = + ;
  • Past lightlike complete, if and only if
    τ τ 0 d τ a ( τ ) = +
    (without requiring τ = );
  • Future timelike complete, if and only if
    τ + = + , τ 0 + d τ a ( τ ) 1 + a 2 ( τ ) = + ;
  • Future lightlike complete, if and only if
    τ 0 τ + d τ a ( τ ) = +
    (without requiring τ + = + ).
In Appendix E it is also shown that the violation of any one of conditions (22)–(25) implies that all maximal, future-directed geodesics of the corresponding type (timelike or lightlike ) are past or future incomplete (apart from some exceptional cases, where the projection of the geodesic on S d or the geodesic itself are constant).
As an example, an FLRW spacetime with
T = ( , + ) T , inf τ T a ( τ ) = a 0 > 0
fulfills all conditions (22)–(25), and is therefore complete or nonsingular10. In the sequel of the present work, we will present several bouncing cosmological models that fulfill (26).
Other FLRW spacetimes may appear to be nonsingular from a naive viewpoint but are in fact singular in the rigorous sense described before; for example, the case
T = ( , + ) , a ( τ ) = e H τ for   τ T ( H T , H 0 )
violates conditions (22) and (23) if H > 0 and conditions (24) and (25) if H < 0 11; note that the above exponential law for the scale factor appears in the de Sitter universe, which has spatial curvature k = 0 (see, e.g., [12] (page 125)).

3.3. Hubble Parameter

From here to the end of the present Section 3, we again consider a (singular or nonsingular) FLRW spacetime. The Hubble parameter is defined by the usual prescription
H : = a a ;
at each time τ T , H ( τ ) T 1 = L 1 gives the ratio between the instantaneous separation velocity and the distance of any two particles co-moving with the FLRW frame (as measured by the line element a ( τ ) d ).

3.4. The Perfect Fluids

In the scenario of an FLRW spacetime, we wish to consider N perfect fluids and a scalar field as in Section 2.3, with some additional features prescribed in the sequel; let us start from the fluids, leaving the discussion of the scalar field for Section 3.5.
The perfect fluids are assumed to be co-moving with the FLRW frame; so, for n = 1 , , N , the ( d + 1 ) -velocity appearing in the stress–energy tensors (8) has the form (19), in any coordinate system ( x μ ) μ = 0 , , d as in (18). In agreement with the homogeneity hypothesis, we further suppose that the mass–energy density of each fluid depends only on cosmic time:
ρ n ρ n ( τ ) .
Consequently, the same happens for the pressure: p n = p n ( τ ) = P n ( ρ n ( τ ) ) (recall Equation (5)).

3.5. The Scalar Field

Again to comply with the homogeneity hypothesis, we assume that the scalar field Φ depends only on cosmic time:
Φ Φ ( τ ) .
In this case, the stress–energy tensor (9) also has the perfect fluid form with a suitable density ρ Φ and pressure p Φ , namely
T μ ν Φ = ( p Φ + ρ Φ ) U μ U ν + p Φ g μ ν ,
ρ Φ : = σ 2 Φ 2 + V ( Φ ) , p Φ : = σ 2 Φ 2 V ( Φ )
(note that ρ Φ , p Φ actually take values in the space of densities D , due to the dimensional assumptions about Φ and V in Section 2.3).

3.6. The Total Stress–Energy Tensor

Due to Equations (8) and (31), this has the following form:
T μ ν = ( p + ρ ) U μ U ν + p g μ ν ,
ρ : = n = 1 N ρ n + ρ Φ , p : = n = 1 N p n + p Φ .

3.7. Einstein Equations

The components of the Ricci tensor R μ ν and the scalar curvature R for the spacetime metric (17) are readily computed in any coordinate system ( x μ ) as in (18); it is found that
R 00 = d a a , R i j = a a + ( d 1 ) a 2 + ( d 1 ) k h i j , R 0 i = R i 0 = 0 ( i , j = 1 , , d ) ,
R = 2 d a a + d ( d 1 ) a 2 a 2 + d ( d 1 ) k a 2
(just for completeness, this calculation is reviewed in Appendix B). From these facts, one infers that the Einstein tensor also has a “perfect fluid” form; in fact,
R μ ν 1 2 R g μ ν = d ( d 1 ) γ d G d ( P + R ) U μ U ν + P g μ ν ,
where the “density” R and the “pressure” P are given by
R : = 1 2 γ d G d a 2 a 2 + k a 2 , P : = 1 d γ d G d a a + d 2 2 a 2 a 2 + d 2 2 k a 2 .
Due to (33), (37) and (38), the Einstein Equations (7) read as follows:
1 2 γ d G d a 2 a 2 + k a 2 = ρ ,
1 d γ d G d a a + d 2 2 a 2 a 2 + d 2 2 k a 2 = p ,
with ρ = n = 1 N ρ n + ρ Φ , p = n = 1 N p n + p Φ as in (34), p n = P n ( ρ n ) , as in (5), and ρ Φ , p Φ as in (32).

3.8. Klein–Gordon Equation

In the present framework, μ μ Φ = Φ d a a Φ 12; thus, the Klein–Gordon Equation (11) for the scalar field reads as follows:
σ ( Φ + d a a Φ ) + V ( Φ ) = 0
(note that a / a is the Hubble parameter H of Equation (28)).

3.9. Conservation Law for the n-th Fluid

Let n { 1 , , N } . In the present setting, one has μ T μ ν n = [ ρ n + d a a ( p n + ρ n ) ] U ν ; thus, the conservation law (10) reads as follows:
ρ n + d a a ( p n + ρ n ) = 0 .

3.10. Summary of the Evolution Equations

The cosmological model evolves according to Equations (39)–(42). We have a system of ODEs for a family of smooth functions
T τ a ( τ ) R + , Φ ( τ ) F , ρ n ( τ ) D + ( n = 1 , , N ) ,
T T an   open   interval .
A solution of the cosmological model is a family of functions as above, fulfilling all the above equations. We shall see later that such equations are not independent.

3.11. Curvature Density; Normalized Densities

Let a , Φ , ρ n ( n = 1 , , N ) be as in (43). We define ρ Φ as usually, following Equation (32); for a reason that will be clear hereafter, we also define the curvature density
ρ k : = k 2 γ d G d 1 a 2
(taking values in D ). That said, it is clear that the first Einstein Equation (39) means 1 2 γ d G d a 2 a 2 = ρ + ρ k , i.e.,
H 2 2 γ d G d = ρ ˜ ,
where H (we repeat it) is the Hubble parameter, and we have introduced the total density, including the curvature contribution; the latter is defined by13
ρ ˜ : = ρ + ρ k = n = 1 N ρ n + ρ k + ρ Φ .
For comparing the different densities considered here, it is customary to introduce the normalized densities
Ω n : = ρ n ρ ˜ ( n = 1 , , N ) , Ω k : = ρ k ρ ˜ , Ω Φ : = ρ Φ ρ ˜ ,
which are well defined and dimensionless at any time τ such that ρ ˜ ( τ ) 0 ; of course, these fulfill by constructing the following condition:
n = 1 N Ω n + Ω k + Ω Φ = 1 .
If the first Einstein equation (i.e., Equation (39) or (45)) holds, we have ρ ˜ ( τ ) 0 if and only if H ( τ ) 0 , and we can also write
Ω n = 2 γ d G d H 2 ρ n ( n = 1 , , N ) , Ω k = 2 γ d G d H 2 ρ k = k a 2 ,
Ω Φ = 2 γ d G d H 2 ρ Φ = 2 γ d G d H 2 σ 2 Φ 2 + V ( Φ ) .

3.12. Stationary Points of the Scale Factor

Let a , Φ , ρ n ( n = 1 , , N ) be as in (43), and assume the first Einstein Equation (39) or (45) holds; let τ 0 T . It is readily seen from (45) that τ 0  is a stationary point for the scale factor if and only if the total density, including the curvature contribution, vanishes at this time:
a ( τ 0 ) = 0 0 = ρ ˜ ( τ 0 ) = n = 1 N ρ n + ρ k + ρ Φ ( τ 0 ) .
We are assuming ρ n > 0 at all times; moreover, the definition (44) indicates that ρ k 0 at all times if k 0 . Thus,
k 0 and a ( τ 0 ) = 0 ρ Φ ( τ 0 ) < 0 .
Of course, the density ρ Φ = ( σ / 2 ) Φ 2 + V ( Φ ) is more likely to become negative in the phantom case σ = 1 ; thus, for k 0 , phantom scalars are interesting candidates to produce solutions of the present model with stationary points for the scale factor. These remarks should be taken into account when searching for solutions with such features (e.g., bouncing or oscillating).

3.13. Energy Conditions

The weak and strong energy conditions (WEC and SEC) for an arbitrary stress–energy tensor on the ( d + 1 ) -dimensional spacetime are reviewed in Appendix F, following [12,13]; in the case of a perfect fluid with density ρ and pressure p,
WEC   holds ρ 0 , ρ + p 0 ;
SEC   holds ( d 2 ) ρ + d p 0 , ρ + p 0 .
In the present framework, whenever the Einstein equations (39) and (40) hold, we have14:
ρ = 1 2 γ d G d a 2 a 2 + k a 2 , ρ + p = 1 d γ d G d a a + a 2 a 2 + k a 2 ,
( d 2 ) ρ + d p = 1 γ d G d a a .
From here, one easily infers that certain assumptions on the instantaneous values of a , a , and on k are sufficient to fulfill or violate the above energy conditions. For example, at any time τ 0 , we have the following implications:
k < 0 , a ( τ 0 ) = 0 ρ ( τ 0 ) < 0 WEC   fails   at   τ 0 ;
k 0 , a ( τ 0 ) = 0 , a ( τ 0 ) > 0 ( ρ + p ) ( τ 0 ) < 0 WEC   and   SEC   both   fail   at   τ 0 ;
a ( τ 0 ) > 0 ( d 2 ) ρ + d p ( τ 0 ) < 0 SEC   fails   at   τ 0 .
Note that the above three violations refer, respectively, to a generic stationary point for a ( ) , a local minimum point for a ( ) (bouncing point) detectable via the first two derivatives, and an arbitrary time at which the scale factor has a positive acceleration.

3.14. The Scale Factor at a Special Time

In the sequel, we often consider the following condition about the scale factor T τ a ( τ ) , depending on an assigned constant H ( T 1 ) + = ( L 1 ) + and also involving the Hubble parameter H of Equation (28):
There   is   time   τ   such   that   a   ( τ ) = 1   and   a ( τ ) = H ( i . e . , H ( τ ) = H ) .
In cosmological models which are presumed to be anyhow realistic, the condition a ( τ ) = 1 (and a ( τ ) > 0 ) typically defines the present time; if so, the equality H ( τ ) = H indicates that H is the present value of the Hubble parameter.
For future use, let us recall that the observational data yield for the present-time Hubble parameter the estimate (see [72,73], “s” means “second”):
H 2.18 × 10 18 s 1 .

3.15. Dimensionless Formalism

In the present Section 3.15, we fix a constant
H ( T 1 ) + = ( L 1 ) + ;
this is not necessarily related to condition (58) on the scale factor, which, for the moment, we are not prescribing; in other words, the interpretation of H as the present-time Hubble parameter is not required for the moment (even though being adopted in many subsequent applications).
Hereafter, we use H to build a fully dimensionless reformulation of the FLRW cosmological model introduced previously.
Several points in this reformulation are obvious, but we are forced to write explicitly the corresponding equations since in the sequel we often need to cite them.
Time variable. In place of cosmic time τ , we will employ the dimensionless time
t : = H τ ;
if τ ranges in an open interval T T , t ranges in the open interval
J : = { H τ | τ T } R .
Given any function τ u ( τ ) of cosmic time, we can associate to it a function t u ( t ) of dimensionless time, so that u ( τ ) = u ( t ) t = H τ (note the slightly abusive use of the same symbol u with two different meanings). In this sense, we can speak of, e.g., the scale factor t a ( t ) , the scalar field t Φ ( t ) , the density t ρ n ( t ) , and so on. We will intend
X ˙ : = d d t ;
with the previous notation X for the derivative with respect to τ , and for each smooth function u of time, we have of course u = H u ˙ , u = H 2 u ¨ , and so on.
Completeness conditions. We now want to rephrase the completeness conditions in Section 3.2; this is one of the cases in which the dimensionless reformulation is obvious, but it is convenient to write it down for subsequent citation.
Let us represent the real interval (62) as
J = ( t , t + ) , t < t + + ,
and let us choose any point t 0 J . The FLRW spacetime under consideration is:
  • Past timelike complete, if and only if
    t = , t 0 d t a ( t ) 1 + a 2 ( t ) = + ;
  • Past lightlike complete, if and only if
    t t 0 d t a ( t ) = +
    (without requiring t = );
  • Future timelike complete, if and only if
    t + = + , t 0 + d t a ( t ) 1 + a 2 ( t ) = + ;
  • Future lightlike complete, if and only if
    t 0 t + d t a ( t ) = +
    (without requiring t + = + ).
All the above completeness conditions are fulfilled, e.g., if15
J = ( , + ) , inf t J a ( t ) = a 0 > 0 .
Let us repeat that, throughout the present work, the terms “nonsingular” and “singular” are often used as equivalents for “complete” and “incomplete”.
Hubble parameter. Equation (28) implies
H = H h , h : = a ˙ a ;
we refer to h as the dimensionless Hubble parameter.
Spatial curvature. The constant sectional curvature k L 2 of the spatial metric will be represented as
k = H 2 Ω k , Ω k R .
Note that sign k = sign Ω k ; this is a standard convention, whose convenience will be clear in the sequel.
Curvature scalar. Equations (36) and (71) imply
R = H 2 2 d a ¨ a + d ( d 1 ) a ˙ 2 a 2 d ( d 1 ) Ω k a 2 .
Scalar field. In Section 2.3, we indicated that the scalar field Φ takes values in a space F such that F 2 = M L 2 d = D L 2 , and its potential is a function V : F D = M L d . We now represent each Φ F as
Φ = 1 2 γ d G d ϕ , ϕ R ,
and introduce a dimensionless field potential V : R R such that
V ( Φ ) = H 2 2 γ d G d V ( ϕ ) for   all Φ F ,   with   ϕ R as   in ( 73 ) .
Densities and pressures. For n { 1 , , N } , let us consider a pair ρ n D + , p n D , representing possible values for the density and pressure of the n-th fluid. The corresponding, dimensionless density and pressure r n R + , p n R are defined by
ρ n = H 2 2 γ d G d r n , p n = H 2 2 γ d G d p n .
Let us recall the equation of state (5) p n = P n ( ρ n ) , with P n : D + D smooth. It is readily seen that there is a unique smooth function P n : R + R such that
p n = P n ( ρ n ) p n = P n ( r n )
for all ρ n D + , r n R + and p n D , p n R related as in (75).
To continue, let us consider a possible time dependence for the scalar field Φ and its dimensionless equivalent ϕ . Due to (32), (73), and (74) (and also due to the remark after Equation (63)), we have
ρ Φ = H 2 2 γ d G d r ϕ , p Φ = H 2 2 γ d G d p ϕ ,
where we have introduced the dimensionless density and pressure
r ϕ : = σ 2 ϕ ˙ 2 + V ( ϕ ) , p ϕ : = σ 2 ϕ ˙ 2 V ( ϕ ) .
Of course, the total density and pressure ρ , p of Equations (33) and (34) have dimensionless analogs r , p ; indeed,
ρ = H 2 2 γ d G d r , p = H 2 2 γ d G d p ,
r : = n = 1 N r n + r ϕ , p : = n = 1 N p n + p ϕ .
On the use of ϕ and Φ as labels. In the sequel, for convenience, we will use the symbol ϕ as a label equivalent to Φ for quantities related to the scalar field. In particular, the densities ρ Φ , Ω Φ and the pressure p Φ of Equations (32) and (47) will also be indicated with ρ ϕ , Ω ϕ and p ϕ . This convention yields some simplification in our notations: for example, Equation (75) about the n-th fluid and Equation (77) about the scalar field can be written in the unified form ρ A = ( H 2 / 2 γ d G d ) r A , p A = ( H 2 / 2 γ d G d ) p A , holding true both for A = n { 1 , , N } and for A = ϕ .
Einstein and Klein–Gordon equations; conservation laws for fluids. Using Equations (61), (63) and (71)–(77), we readily see that the Einstein equations (39) and (40), the Klein–Gordon equation (41), and the conservation laws (42) for fluids can be converted, respectively, to the following forms:
a ˙ 2 a 2 Ω k a 2 = r ,
2 d a ¨ a + d 2 2 a ˙ 2 a 2 d 2 2 Ω k a 2 = p ,
σ ϕ ¨ + d a ˙ a ϕ ˙ + V ( ϕ ) = 0 ,
r ˙ n + d a ˙ a ( p n + r n ) = 0 ,
with r = n = 1 N r n + r ϕ , p = n = 1 N p n + p ϕ as in (80), p n = P n ( r n ) (see (76)), and r ϕ , p ϕ as in (78). In the dimensionless setting, we can regard this as a system of ODEs for the unknown smooth functions J t a ( t ) R + , ϕ ( t ) R , r n ( t ) R + ( n = 1 , , N ), with J R an open interval.
Total density, including the curvature contribution; normalized densities. Due to (44) and (71), the curvature density can be represented as
ρ k = H 2 2 γ d G d r k ,
where we have introduced the dimensionless curvature density as
r k : = Ω k a 2 .
The first Einstein Equation (81) means
h 2 = r ˜ ,
where h : = a ˙ / a is the dimensionless Hubble parameter (see Equation (70)), and we have introduced the total dimensionless density including the curvature contribution, which is by definition
r ˜ : = r + r k = n = 1 N r n + r k + r ϕ .
To continue, we note that the definition (47) of the normalized densities is equivalent to
Ω n = r n r ˜ ( n = 1 , , N ) , Ω k = r k r ˜ , Ω ϕ = r ϕ r ˜
(intending Ω ϕ as an equivalent notation for Ω Φ ; see the paragraph after Equation (80)).
If Equation (87) (i.e., Equation (81)) is satisfied, we can also write
Ω n = r n h 2 ( n = 1 , , N ) , Ω k = r k h 2 = Ω k a ˙ 2 , Ω ϕ = r ϕ h 2 = 1 h 2 σ 2 ϕ ˙ 2 + V ( ϕ ) .
Again on stationary points of the scale factor. The dimensionless analogs of statements (50) and (51) are as follows:
a ˙ ( t 0 ) = 0 0 = r ˜ ( t 0 ) = n = 1 N r n + r k + r ϕ ( t 0 ) ;
Ω k 0 ( i . e . , k 0 ) and a ˙ ( t 0 ) = 0 r ϕ ( t 0 ) < 0
(with r ϕ = ( σ / 2 ) ϕ ˙ 2 + V ( ϕ ) , which is more likely to become negative in the phantom case σ = 1 ).
Again on the energy conditions. The dimensionless reformulation of the contents of Section 3.13 is straightforward. First of all,
WEC   holds r 0 , r + p 0 ;
SEC   holds ( d 2 ) r + d p 0 , r + p 0 .
Whenever the Einstein equations (81) and (82) hold, we have
r = a ˙ 2 a 2 Ω k a 2 , r + p = 2 d a ¨ a + a ˙ 2 a 2 Ω k a 2 , ( d 2 ) r + d p = 2 a ¨ a .
In particular, we have violations of the energy conditions in certain situations that involve, respectively, a stationary point for the scale factor, a bouncing point for the scale factor detectable via the first two derivatives, and an arbitrary time of positive acceleration; more precisely, for any time t 0 ,
Ω k > 0 , a ˙ ( t 0 ) = 0 r ( t 0 ) < 0 WEC   fails   a t   t 0 ;
Ω k 0 , a ˙ ( t 0 ) = 0 , a ¨ ( t 0 ) > 0 ( r + p ) ( t 0 ) < 0 WEC   and   SEC   both   fail   at   t 0 ;
a ¨ ( t 0 ) > 0 ( d 2 ) r + d p ( t 0 ) < 0 SEC   fails   at   t 0 .
Again on the scale factor at a special time. Given H as in (60), let us reconsider condition (58) on the scale factor. Clearly, the dimensionless analog of (58) is the following:
There   is   a   time   t   such   that   a   ( t ) = 1   and   a ˙ ( t ) = 1 ( i . e . , h ( t ) = 1 ) .
In agreement with the considerations after Equation (58), in realistic models, one often regards t as representing the present time.

4. Analysis of the Evolution Equations

Let us stick to the setting of Section 3 and, in particular, to the dimensionless formalism of Section 3.15 (which assumes the specification of a constant H as in (60)). The present Section 4 presents some basic results about Equations (81)–(84).

4.1. Determining the Fluids’ Densities

Let us choose n { 1 , , N } and consider Equation (84) r ˙ n + d a ˙ a ( p n + r n ) = 0 , with p n = P n ( r n ) . We make the ansatz r n ( t ) = F n ( a d ( t ) ) , with an unknown smooth function F n of a positive variable α . It is readily seen that r n fulfills Equation (84) if F n ( α ) = [ P n ( F n ( α ) ) + F n ( α ) ] / α .
To formalize the above considerations, we introduce the function space
F n : = { F n C ( R + , R + ) | F n ( α ) = P n ( F n ( α ) ) + F n ( α ) α for   all   α R + } .
Moreover, we consider the following condition, discussed in the next paragraph and often assumed to hold in the sequel:
For   each   α 0 R +   and   C n 0 R + , there   exists   F n F n such   that   F n ( α 0 ) = C n 0
(the function F n in (101) is clearly unique, by the standard theory of the Cauchy problem).
That said, let us choose arbitrarily a smooth function a : J R + , t a ( t ) (with J R an open interval). In Appendix G, we prove rigorously that, if (101) holds, for each smooth function r n : J R + , t r n ( t ) , we have the following equivalence:
r n   fulfills ( 84 ) ( with   p n = P n ( r n ) ) r n = F n ( a d ) for   some   F n F n .
To continue, assume the first Einstein Equation (81) (or (87)) to be fulfilled by certain functions J t a ( t ) , ϕ ( t ) , r n ( t ) ( n = 1 , , N ), and Equations (84) and (101) to hold at least for some n; then, from the relation r n = F n ( a d ) ( F n F n ) and from (90), we infer that the n-th normalized density is
Ω n = F n ( a d ) h 2 ,
with h the dimensionless Hubble parameter; see (70).
On the condition (101). Let us again choose n { 1 , , N } . By the standard theory of the Cauchy problem, for any α 0 , C n 0 R + , there is certainly a function F n C ( A , R + ) (with A R + an open interval) such that F n ( α ) = ( P n ( F n ( α ) ) + F n ( α ) ) / α for all α A and α 0 A , F n ( α 0 ) = C n 0 . Equation (101) requires this to happen with A = R + (the domain of the functions in the space F n ) and this is, in fact, a regularity condition about the function  P n .
Equation (101) is certainly fulfilled in the following cases (i) and (ii):
(i)
It is
P n ( r ) + r 0 for   all   r R + ,
0 C d r | P n ( r ) + r | = + and C + d r | P n ( r ) + r | = + for   all   C R + .
In this case, for each α 0 , C n 0 R + , the function F n of (101) is defined implicitly by the following quadrature formula:
C n 0 F n ( α ) d r P n ( r ) + r = ln α α 0 for   all   α R + .
(This formula actually individuates a unique smooth function F n : R + R + ; see again Appendix G).
(ii)
It is
P n ( r ) + r = 0 for   all   r R + .
In this trivial case, Equation (100) gives
F n = { F n : R + R + | F n = constant }
and the function F n in (101) is given by
F n = constant = C n 0 .
Again on the fluids’ densities. Let us assume (101) for some n. By the uniqueness theorem for the Cauchy problem, for any fixed α 0 R + , the mapping F n C n 0 : = F n ( α 0 ) is a bijection between the function space F n of Equation (100) and R + . In particular, let us choose α 0 = 1 and consider the bijection
F n R + , F n Ω n : = F n ( 1 )
(here we write Ω n instead of C n 0 , for reasons that will become clear shortly afterward). In case (i) of the previous paragraph, the inverse of the map (109) sends Ω n R + to the function F n F n described by Equation (105) with α 0 = 1 and C n 0 = Ω n , i.e.,
Ω n F n ( α ) d r P n ( r ) + r = ln α for   all   α R + .
To continue, assume the first Einstein equation (81) (or (87)) to be fulfilled by certain functions J t a ( t ) , ϕ ( t ) , r n ( t ) ( n = 1 , , N ), and Equations (84) and (101) to hold at least for some n. Then, considering the n-th normalized density, we infer from Equation (103) that
Ω n ( t ) = F n ( 1 ) Ω n if   a   ( t ) = 1 ,   h ( t ) = 1   for   some   t
(here, we consider condition (99) about t , which has been already commented).

4.2. The Fluids’ Densities in the Linear Case

The equation of state (5) of the n-th fluid is often assumed to have the following linear form:
p n = w n ρ n , i . e . , P n ( ρ ) = w n ρ for   all   ρ D + ,
with w n R a constant, dimensionless coefficient; we already mentioned the subcases of dust and radiation gas, in which w n = 0 and w n = 1 / d , respectively. The dimensionless equivalent of (112) has exactly the same structure:
p n = w n r n , i . e . , P n ( r ) = w n r for   all   r R + .
It should be noted that the linear case (112) and (113) fits items (i) and (ii) of the penultimate paragraph for w n 1 and w n = 1 , respectively. For any w n R , the space of Equation (100) is given by
F n = { F n : R + R + | F n ( α ) = Ω n α 1 + w n for   all α R + ,   with Ω n R + } ;
note that the above representation is consistent with the relation (109) F n ( 1 ) = Ω n . Due to (114) and (102), we can state the following: the conservation law (84) implies for the n-th dimensionless density and pressure the expressions
r n = Ω n a ( 1 + w n ) d , p n = w n Ω n a ( 1 + w n ) d ( Ω n R + ) .
If the Einstein equation (81) is fulfilled, Equations (90) and (115) give for the n-th normalized density the following expression:
Ω n = 1 h 2 Ω n a ( 1 + w n ) d .
Of course, statement (111) applies to the linear case under consideration.

4.3. The Final Form of the Einstein and Klein–Gordon Equations

From here to the end of the paper, the condition (101) is supposed to hold for n = 1 , , N ; this allows us to treat the conservation law (84) for each fluid via Equation (102). Let us return to the (dimensionless) Einstein Equations (81) and (82). We substitute therein the expressions for r n and p n = P n ( r n ) coming from (102), which involve a function F n in the space (100); moreover, we use the explicit expressions (78) for r ϕ , p ϕ . In this way, Equations (81) and (82) are converted to
E = 0 , E : = a ˙ 2 a 2 Ω k a 2 n = 1 N F n ( a d ) + σ 2 ϕ ˙ 2 + V ( ϕ ) ,
A = 0 ,
A : = 2 d a ¨ a + d 2 2 a ˙ 2 a 2 d 2 2 Ω k a 2 n = 1 N ( P n F n ) ( a d ) + σ 2 ϕ ˙ 2 V ( ϕ ) .
For future reference, we also report the Klein–Gordon Equation (83) that we rephrase as
F = 0 , F : = σ ϕ ¨ + d a ˙ a ϕ ˙ + V ( ϕ ) .
Equations (117)–(119) form a system in two unknown smooth functions J t a ( t ) R + , ϕ ( t ) R ; here and in the sequel, J will always indicate an unspecified, open real interval.
The equations in this system are not fully independent. In fact, if A = 0 and F = 0 at all times, for E = 0 to be fulfilled at all times, it is sufficient that E = 0 holds at a particular time. This statement can be checked directly16, but it is more instructive to infer the same result with the methods of the forthcoming Section 4.4.

4.4. Lagrangian Formalism; Zero-Energy Constraint

Let us return to the action functional S of Equation (6). In the present framework, based on the spacetime (15) with the metric (17), one has d v = a d d τ d ω , where d ω is the volume element of the Riemannian metric h; in terms of the dimensionless time (61), ranging in a real interval J , we have d τ = d t / H . We also insert in Equation (6) the expression  (72) for R, the expression for ρ n arising from Equations (75) and (102) and the expressions (73) and (74) for Φ and V ( Φ ) . In this way, we obtain the following:
S = H 2 γ d G d S d d ω ×
× J d t 2 ( d 1 ) a d 1 a ¨ + a d 2 a ˙ 2 Ω k a d 2 n = 1 N a d F n ( a d ) + σ 2 a d ϕ ˙ 2 a d V ( ϕ ) =
= H 2 γ d G d S d d ω × J d t L ( a , ϕ , a ˙ , ϕ ˙ ) 2 ( d 1 ) d d t a d 1 a ˙ ,
where
L : R + × R 3 R , ( a , ϕ , a ˙ , ϕ ˙ ) L ( a , ϕ , a ˙ , ϕ ˙ ) : = a d 2 a ˙ 2 σ 2 a d ϕ ˙ 2 + U ( a ) + a d V ( ϕ ) ,
U : R + R , a U ( a ) : = n = 1 N a d F n ( a d ) + Ω k a d 2 .
The above manipulations are to some extent formal, since the volume S d d ω can be infinite (this certainly happens if S d is simply connected and it has curvature k 0 ). Forgetting this problem (and recalling that total derivatives like ( d / d t ) ( a d 1 a ˙ ) are irrelevant from the Lagrangian viewpoint), we expect L to be a Lagrangian function describing the evolution of our cosmological model. This can be checked a posteriori, independently of the manipulations that led us to Equation (121).
In fact, the Lagrange equations induced by L read as follows:
δ L δ a = 0 , δ L δ ϕ = 0 , δ L δ q : = d d t L q ˙ + L q for   q = a , ϕ .
It is readily found that
δ L δ ϕ = a d F ,
with F as in Equation (119). The calculation of δ L / δ a is a bit more engaging and involves amongst else the derivative ( / a ) [ F n ( a d ) ] , to be computed recalling the expression for F n in (100); one ultimately obtains
δ L δ a = d a d 1 A ,
with A as in Equation (118). The energy function E : R + × R 3 R associated with the Lagrangian (121) is
E ( a , ϕ , a ˙ , ϕ ˙ ) : = L a ˙ a ˙ + L ϕ ˙ ϕ ˙ L ( a , ϕ , a ˙ , ϕ ˙ ) = a d 2 a ˙ 2 σ 2 a d ϕ ˙ 2 U ( a ) a d V ( ϕ ) .
Of course, along any solution t ( a ( t ) , ϕ ( t ) ) of the Lagrange equations, we have E ( t ) = constant; in particular, E ( t ) = 0 at all times if and only if E ( t 0 ) = 0 at some particular time t 0 . On the other hand, it is evident that
E = a d E ,
with E an in Equation (117). To summarize, we should highlight the following:
(i)
The Einstein equation A = 0 and the Klein–Gordon equation F = 0 are equivalent to the Lagrange equations induced by L.
(ii)
The Einstein equation E = 0 is equivalent to the zero-energy constraint E = 0 ; along solutions of the Lagrange equations, this constraint is fulfilled at all times if and only if it is fulfilled at a particular time t 0 .
This also justifies the statements at the end of Section 4.3 from a Lagrangian viewpoint. From now on, we will discuss the time evolution of our cosmological model using the Lagrangian formalism. This approach is well known in the literature on FLRW cosmologies: let us mention, e.g., the contributions of the Naples school [25,26,27,28,29] (already indicated in the Introduction, in connection with the Nöther symmetry method). Our Lagrangian setting is perhaps more general than usual for what concerns the equations of the state of perfect fluids: in fact, our equations of state p n = P n ( r n ) just assume some regularity properties for the function P n , while most authors limit their attention to special cases like the linear one.
The linear case. If the equation of state of the n-th fluid has the linear form (112) or (113) for some w n R , functions F n F n have the form (114) depending on a constant Ω n R + ; so, in Equation (122), for U, one must put
F n ( a d ) = Ω n a ( 1 + w n ) d .

4.5. Again on the Scale Factor at a Special Time

We have already considered for the scale factor t a ( t ) the condition a ( t ) = 1 , a ˙ ( t ) = 1 (i.e., h ( t ) = 1 ), to be fulfilled at some special time t : see Equation (99), recalling that this condition is the dimensionless equivalent of (58).
Making reference to the Lagrangian L of Equations (121) and (122) and the energy function (126), we now claim the equivalence of the forthcoming statements (i)(ii):
(i)
There is a zero-energy solution J t ( a ( t ) , ϕ ( t ) ) R + × R of the Lagrange equations induced by L, such that a ( t ) = 1 , a ˙ ( t ) = 1 at some time t J .
(ii)
There are ϕ , ϕ ˙ R such that
n = 1 N Ω n + Ω k + σ 2 ϕ ˙ 2 + V ( ϕ ) = 1 , where Ω n : = F n ( 1 ) .
(Concerning the position Ω n : = F n ( 1 ) , recall Equation (109) and the subsequent discussion).
Let us first prove that (i) implies (ii). In fact, assume there is a zero-energy solution J t ( a ( t ) , ϕ ( t ) ) of the Lagrange equations as in (i), and put ϕ : = ϕ ( t ) , ϕ ˙ : = ϕ ˙ ( t ) . Then, recalling Equations (126) and (122), we can write 0 = E ( a ( t ) , ϕ ( t ) , a ˙ ( t ) , ϕ ˙ ( t ) ) = E ( 1 , ϕ , 1 , ϕ ˙ )   = 1 ( σ / 2 ) ϕ ˙ 2 U ( 1 ) V ( ϕ )   = 1 ( σ / 2 ) ϕ ˙ 2 n = 1 N F n ( 1 ) Ω k V ( ϕ ) , whence Equation (129) of (ii).
Conversely, let us assume (ii) and infer (i). For this purpose, we arbitrarily choose t R ; let J t ( a ( t ) , ϕ ( t ) ) be the solution17 of the Lagrange equations with initial data a ( t ) = 1 , ϕ ( t ) = ϕ , a ˙ ( t ) = 1 , ϕ ˙ ( t ) = ϕ ˙ . Clearly, (i) is true if we prove this to be a zero-energy solution. Indeed, from the assumption (129), we readily infer E ( a ( t ) , ϕ ( t ) , a ˙ ( t ) , ϕ ˙ ( t ) ) = 0 , whence E ( a ( t ) , ϕ ( t ) , a ˙ ( t ) , ϕ ˙ ( t ) ) = 0 at each time t J .

4.6. Maximal Solutions

The notion of maximality has been already considered in the present work, in connection with geodesics in a spacetime (see the first lines in Section 3.2). In general, a solution of a system of ODEs (on a manifold, e.g., on R m , with domain an open interval) is said to be maximal if it cannot be extended to a solution of the same ODEs, defined on a larger interval.
Maximality will reappear in many subsequent considerations; in particular, we will frequently refer to the maximal, zero-energy solutions J t ( a ( t ) , ϕ ( t ) ) R + × R of the Lagrange equations in Section 4.4. Another typical application will concern the maximal, zero-energy solutions J t a ( t ) R + of the Lagrange equations induced by a certain “reduced” Lagrangian L ( a , a ˙ ) , which is introduced in Section 5.3.

5. The Case with a Constant Potential for the Scalar Field: General Results on the Evolution of a ( t ) and ϕ ( t )

5.1. Basic Setting

Throughout the present Section 5, we assume the following for the dimensionless field potential:
V ( ϕ ) = constant Υ R .
Due to Equation (74), the corresponding dimensioned potential is
V ( Φ ) = constant = H 2 Υ 2 γ d G d .
According to Equations (12)–(14), this setting is equivalent to considering the Einstein equations with a cosmological constant as follows:
Λ = d ( d 1 ) H 2 Υ 2
(having the sign of Υ ), and a free, massless scalar field.
Due to Equation (130), the Lagrangian (121) and its energy function (126) take the following form:
L ( a , ϕ , a ˙ , ϕ ˙ ) : = a d 2 a ˙ 2 σ 2 a d ϕ ˙ 2 + U ( a ) + Υ a d ,
E ( a , ϕ , a ˙ , ϕ ˙ ) : = a d 2 a ˙ 2 σ 2 a d ϕ ˙ 2 U ( a ) Υ a d ,
with U ( a ) : = n = 1 N a d F n ( a d ) + Ω k a d 2 , as in (122).
Let us recall again that a linear equation of state p n = w n ρ n for the n-th fluid ( w n R ) implies F n ( α ) = Ω n α 1 + w n with Ω n R + , see (114). However, the subsequent analysis is not confined to linear cases and refers to the general setting of Section 4.1.
In the sequel of the present Section 5, we will show that any model of this kind possesses a second constant of motion besides the energy; this allows us to solve by quadratures the evolution equations, as well as to describe qualitatively the behavior of the scale factor and of the scalar field.

5.2. The Constant of Motion Π

Clearly, the coordinate ϕ is cyclic for the Lagrangian (133): L / ϕ = 0 . Due to this, the canonically conjugate momentum
Π : = L ϕ ˙ = σ a d ϕ ˙
is constant along any solution of the Lagrange equations. The relation between ϕ ˙ and Π in Equation (135) can be inverted, thus obtaining
ϕ ˙ = σ Π a d .

5.3. Reduced Lagrangian

Let us consider a solution J t ( a ( t ) , ϕ ( t ) ) of the Lagrange equations with an assigned value of Π (recalling that J indicates any open real interval). By the standard theory of Lagrangian systems with cyclic coordinates, the function J t a ( t ) can be characterized as a solution of the Lagrange equation induced by the reduced Lagrangian
L : R + × R R , ( a , a ˙ ) L ( a , a ˙ ) : = L ( a , a ˙ , ϕ ˙ ) Π ϕ ˙ ϕ ˙ = σ Π a d = a d 2 a ˙ 2 V ( a ) ,
V ( a ) : = σ Π 2 2 a d U ( a ) Υ a d = σ Π 2 2 a d n = 1 N a d F n ( a d ) Ω k a d 2 Υ a d
(in the last passage, we have expressed U via Equation (122)). The energy function associated with L is
E ( a , a ˙ ) : = L a ˙ a ˙ L ( a , a ˙ ) = a d 2 a ˙ 2 + V ( a ) .
By comparison with the energy function E of Equation (134), we see that
E ( a , a ˙ ) = E ( a , ϕ , a ˙ , ϕ ˙ ) ϕ ˙ = σ Π a d ;
so, the energy constraint E = 0 of Section 4.4 is equivalent to an energy constraint E = 0 on the motion t a ( t ) .
To continue, let us note that we can write
L ( a , a ˙ ) = a d 2 L ( a , a ˙ ) , E ( a , a ˙ ) = a d 2 E ( a , a ˙ )
where we have put
L ( a , a ˙ ) : = a ˙ 2 V ( a ) , E ( a , a ˙ ) : = a ˙ 2 + V ( a ) ,
V ( a ) : = a 2 d V ( a ) = σ Π 2 2 a 2 2 d n = 1 N a 2 F n ( a d ) Ω k Υ a 2 .
Of course, L is a Lagrangian with energy function E ; the corresponding Lagrange equation reads as follows:
2 a ¨ ( t ) = V ( a ( t ) ) .
It turns out that the zero-energy solutions of the Lagrange equations for L  and  L  coincide (this reflects a general result; see Appendix H); so, in the sequel, we will refer to the simplest Lagrangian L .

5.4. Zero-Energy Solutions of the Reduced System

Clearly, L and E in Equation (141) are the Lagrangian and the energy function of a fictitious one-dimensional, conservative mechanical system with kinetic energy a ˙ 2 and potential energy V ( a ) . The corresponding motions can be analyzed by the usual, qualitative and quantitative methods for one-dimensional conservative systems. We must direct our attention to the solutions J t a ( t ) of the Lagrange equations fulfilling at all times the constraint
0 = E ( t ) = a ˙ 2 ( t ) + V ( a ( t ) ) ,
which leads us to the following statements:
(i)
For all t in the (open, real) interval J , one has V ( a ( t ) ) = a ˙ 2 ( t ) 0 . So, the image of the function t a ( t ) is an interval contained in the following set:
V 0 : = { a R + | V ( a ) 0 } .
Of course, this is the union of the subsets
V = 0 : = { a R + | V ( a ) = 0 } , V < 0 : = { a R + | V ( a ) < 0 } ,
which must be distinguished for a qualitative analysis of the solution.
(ii)
The equations V ( a ( t ) ) = a ˙ 2 ( t ) and 2 a ¨ ( t ) = V ( a ( t ) ) have well known implications. In particular, if t 1 J is an instant, one has a ˙ ( t 1 ) = 0 if and only if V ( a ( t 1 ) ) = 0 ; if, in addition, V ( a ( t 1 ) ) 0 , then t 1 is an inversion time, i.e., a ˙ ( t ) is nonzero with different signs when t ranges in two suitable intervals ( t 1 δ , t 1 ) and ( t 1 , t 1 + δ ) (this is readily inferred from the relations a ˙ ( t 1 ) = 0 and a ¨ ( t 1 ) = ( 1 / 2 ) V ( a ( t 1 ) ) 0 ).
(iii)
If ( t , t ) J is an interval such that sign a ˙ ( t ) = constant s a { ± 1 } for all t ( t , t ) , everywhere in this interval, we have
a ˙ ( t ) = s a V ( a ( t ) )
so that, by separation of variables,
a ( t ) a ( t ) d a V ( a ) = s a ( t t ) .
(If t or t is an endpoint of J , in the above formula, a ( t ) or a ( t ) should be intended as the limit of a ( t ) for t t or t t ).
(iv)
The considerations in (iii) bring to our attention integrals of the following form:
a a d a V ( a ) ;
let us assume, e.g., 0 < a < a < + and V ( a ) < 0 for all a ( a , a ) . If V ( a ) < 0 and V ( a ) < 0 , the integral (149) is certainly convergent. Convergence is also ensured if V vanishes but V is nonvanishing, at one or both endpoints a , a . For example, let V ( a ) = 0 and V ( a ) < 0 ; then, by Taylor’s formula, for a ( a ) + , we have 1 / V ( a ) 1 / V ( a ) × 1 / a a , so that the integrand in (149) diverges in an integrable way near a .
(v)
If a 1 R + is such that V ( a 1 ) = 0 , V ( a 1 ) = 0 , the constant function a ( t ) : = a 1 for all t R is a zero-energy solution (indicated in the sequel as an equilibrium solution).
(vi)
A nonconstant, zero-energy solution requires an infinite time to reach a point a 1 R + such that V ( a 1 ) = 0 , V ( a 1 ) = 0 . To explain this statement, let us consider, e.g., the case of a zero-energy solution t a ( t ) such that a ˙ ( t 0 ) > 0 at some time t 0 and assume, with a 0 : = a ( t 0 ) , that a 0 < a 1 and V ( a ) < 0 for a [ a 0 , a 1 ) . If the solution is maximal (i.e., if its time domain cannot be furtherly extended; see Section 4.6), for t > t 0 , it will exist, with a ˙ ( t ) > 0 , until reaching point a 1 . According to item (iii), a 1 will be reached at the time t 1 such that
t 1 t 0 = a 0 a 1 d a V ( a ) ;
on the other hand, the above integral diverges, i.e.,
t 1 = + .
In fact, the assumptions V ( a 1 ) = 0 , V ( a 1 ) = 0 and Taylor’s formula grant the existence of δ , C R + such that V ( a ) C ( a 1 a ) 2 for a ( a 1 δ , a 1 ] ; this implies 1 / V ( a ) ( 1 / C ) × 1 / ( a 1 a ) , which ensures divergence of the integral in  (150).
The above argument has obvious variants. For example, again with V ( a 1 ) = 0 and V ( a 1 ) = 0 , let us consider a maximal zero-energy solution such that a ˙ ( t 0 ) > 0 at some time t 0 and assume, with a 0 : = a ( t 0 ) , that a 1 < a 0 and V ( a ) < 0 for a ( a 1 , a 0 ] . Then, a ( t ) exists for t < t 0 until reaching a 1 in the past of t 0 ; this occurs at time .
(vii)
Under specific assumptions, one can also discuss the time required for a maximal solution to diverge to + . In this discussion, one essentially uses Equation (148), sending to + one of the extremes of integration; an example of these considerations will appear in Section 6.6, in the lines before Equation (241).
(viii)
Due to Equation (143) 2 a ¨ ( t ) = V ( a ( t ) ) , we have
s i g n a ¨ ( t ) =   sign V ( a ( t ) ) ;
so, the concavity and the inflexion points in the graph of a solution t a ( t ) can be determined by studying the sign of V along the solution.

5.5. Time Evolution of the Scalar Field

After analyzing the zero-energy solutions t a ( t ) of the reduced system, let us proceed to discuss the zero-energy solutions J t ( a ( t ) , ϕ ( t ) ) of the complete system. The following statements hold:
(i)
Due to (136), for all t J , we have
sign ϕ ˙ ( t ) = constant = σ sign Π ;
the function t ϕ ( t ) is constant if Π = 0 , and strictly monotonic if Π 0 .
(ii)
Consider an interval ( t , t ) J on which a ˙ has constants sign s a { ± 1 } , as in item (iii) of Section 5.4. Then, the function t a ( t ) is a diffeomorphism between ( t , t ) and the interval ( a m i n , a m a x ) where a m i n : = min ( a ( t ) , a ( t ) ) and a m a x : = max ( a ( t ) , a ( t ) ) . We now consider the inverse function ( a m i n , a m a x ) a t ( a ) , and the composition ( a m i n , a m a x ) a ϕ ( a ) : = ϕ ( t ( a ) ) . Recalling Equations (136) and (147), we find
d ϕ d a ( a ) = ϕ ˙ ( t ( a ) ) a ˙ ( t ( a ) ) = σ Π a d s a V ( a ) = σ s a Π a 2 d V ( a )
so that, integrating between a ( t ) and a ( t ) ,
ϕ ( t ) ϕ ( t ) = σ s a Π a ( t ) a ( t ) d a a 2 d V ( a ) .
(as in the comment after Equation (148), here, a ( t ) or a ( t ) must be intended as a limit if t or t is an endpoint of J ; the same can be said of ϕ ( t ) or ϕ ( t ) ).

5.6. On Maximal Solutions

We again refer to the notion of maximality described in Section 4.6, for the solutions of any system of ODEs. Let us consider a solution J t ( a ( t ) , ϕ ( t ) ) of the Lagrange equations induced by the Lagrangian (133), with zero energy and with an assigned value of Π . The function t ϕ ( t ) is essentially determined by the function t a ( t ) , as indicated in Section 5.5; from here, one infers that J t ( a ( t ) , ϕ ( t ) ) is maximal if and only if J t a ( t ) is a maximal zero-energy solution for the reduced system with the Lagrangian in (141).

5.7. Hubble Parameter; Densities and Pressures

Let us consider again a solution J t ( a ( t ) , ϕ ( t ) ) of the Lagrange equations with zero energy and an assigned value of Π . For the dimensionless Hubble parameter h : = a ˙ / a , we infer from (144) and (147) that
h 2 = V ( a ) a 2 , h = s a V ( a ) a ;
the second equality holds on any time interval ( t , t ) J on which a ˙ has a constant sign s a { ± 1 } .
For the densities and pressures of the fluids and for the curvature density in dimensionless form, we have the usual expressions r n = F n ( a d ) , p n = ( P n F n ) ( a d ) ( n = 1 , , N ), r k = Ω k / a 2 (see (102) and (86)).
Equation (78) on the dimensionless density and pressure of the scalar field and Equations (130) and (136) on V, ϕ ˙ give
r ϕ = σ Π 2 2 a 2 d + Υ , p ϕ = σ Π 2 2 a 2 d Υ .
Let us also recall that, according to Equations (87) and (88), h 2 = V ( a ) / a 2 also equals the total dimensionless density r ˜ : = n = 1 N r n + r k + r ϕ .
Finally, according to (90), the normalized densities have the following expressions:
Ω A = r A h 2 for   A = n , k , ϕ , h 2   as   in ( 156 ) .

5.8. The Scale Factor at a Special Time

The condition (99) on the scale factor, which is the dimensionless equivalent of (58), was rediscussed in Section 4.5. The results obtained therein are easily adapted to the present framework, making reference to the Lagrangian L for the complete system (see Equation (133)), or to the Lagrangian L for the reduced system (see Equations (141) and (142)). The conclusion is the equivalence of the forthcoming statements (i)(i’)(ii), for any Π R :
(i)
There is a zero-energy solution J t ( a ( t ) , ϕ ( t ) ) R + × R of the Lagrange equations induced by L such that the canonically conjugate momentum of the scalar field has the assigned value Π , and a ( t ) = 1 , a ˙ ( t ) = 1 at some time t J .
(i’)
There is a zero-energy solution J t a ( t ) R + of the Lagrange equations induced by L , with the assigned value Π , such that a ( t ) = 1 , a ˙ ( t ) = 1 at some time t J .
(ii)
It is
n = 1 N Ω n + Ω k + σ Π 2 2 + Υ = 1 , where Ω n : = F n ( 1 ) .
Let us remark that Equation (159) is just the relation (129), where we have expressed V as in (130) and we have written ϕ ˙ 2 = Π 2 , as prescribed by (136) when a = a ( t ) = 1 . By comparison with the definition (142) of V , we readily see that Equation (159) is equivalent to
V ( 1 ) = 1 .
Assuming the above conditions to hold, let us consider a zero-energy solution t a ( t ) of the Lagrange equations induced by L ; if t is such that a ( t ) = 1 we have 0 = E ( a ( t ) , a ˙ ( t ) ) = a ˙ 2 ( t ) + V ( 1 ) = a ˙ 2 ( t ) 1 , whence a ˙ ( t ) = ± 1 . To summarize, given a zero-energy solution J t a ( t ) of the reduced system and any time t J , Equation (159) ensures the following equivalence:
a ( t ) = 1 and a ˙ ( t ) = 1 a ( t ) = 1 and a ˙ ( t ) > 0 .
Let us recall the comment after Equation (99) on the possible interpretation of t as the present time.

5.9. Comparison with the Literature

We already mentioned in the Introduction that the case of a constant self-potential for the scalar field has (to our knowledge) received insufficient attention in the literature, perhaps due to its simplicity. The only reference on this issue of which we are aware is a paper by Dabrowski, Kiefer, and Sandhöfer [74]. Section II.B of that work introduces a toy model in which spacetime has the usual dimension 3 + 1 , and the only content of the universe is a canonical or phantom scalar field with zero self-potential (meaning that there is a free scalar field and the cosmological constant vanishes); this corresponds to put d = 3 , Υ = 0 and N = 0 (no perfect fluids) in all equations of the present Section 5 (and especially, in Equations (133), (134), (141) and (142) on the Lagrangian formalism). We briefly return to [74] at the end of Section 6.6.

6. Again on the Case of V ( ϕ ) = Constant: Big Bounce from a Phantom Scalar

6.1. Introducing a More Specific Setting

In the present Section 6, we again stick to the framework of Section 5; let us recall that the dimensionless formalism employed therein requires the specification of a constant H as in (60), on which we will return later.
Our present aims are to specialize the setting of Section 5 making assumptions with a minimum of realism, and to analyze the resulting cosmologies; as we will show, these are especially interesting in the case of a phantom scalar field. Our specialization of the framework of Section 5 is as follows:
(a)
For the spatial dimension, we assume the realistic value
d = 3 ;
let us recall Equation (4), involving the usual gravitational constant G.
(b)
There are N = 2 fluids, referred to as radiation and matter and indicated with the labels r and m, respectively; consequently, all formulas of the previous sections will be rephrased assuming the values r , m for the fluids’ index n. Radiation is described in the standard way and matter is supposed to behave like dust, so the corresponding equations of state are as follows:
p n = w n ρ n ( n = r , m ) , w r : = 1 3 , w m : = 0 .
Due to (b), all equations of the previous sections involving the evolution laws for the densities must be applied with
F r ( α ) = Ω r α 4 / 3 , F m ( α ) = Ω m α for   all   α R + ,
Ω r , Ω m R +
(recall Equation (114)). In particular, the Lagrangian L and the energy E of Equations (133) and (134) take the following forms:
L ( a , ϕ , a ˙ , ϕ ˙ ) : = a a ˙ 2 σ 2 a 3 ϕ ˙ 2 + Ω r a + Ω m + Ω k a + Υ a 3 ,
E ( a , ϕ , a ˙ , ϕ ˙ ) : = a a ˙ 2 σ 2 a 3 ϕ ˙ 2 Ω r a Ω m Ω k a Υ a 3 .
Let us recall that we have the constant of motion Π ; see Equations (135) and (136). As in Equation (141), we have a reduced Lagrangian and an energy function
L ( a , a ˙ ) : = a ˙ 2 V ( a ) , E ( a , a ˙ ) : = a ˙ 2 + V ( a )
where, according to (142),
V : R + R , a V ( a ) = σ Π 2 2 a 4 Ω r a 2 Ω m a Ω k Υ a 2 .
It is easy to include in or adapt to the present framework the formulas of the previous sections for the relevant observables of the cosmological model. In a few words, we have the following:
  • The spatial curvature has the general representation (71).
  • The Hubble parameter H and its dimensionless analog h are described by Equations (70) and (156).
  • The densities and pressures of radiation, matter, and the scalar field, as well as the curvature density, in their dimensioned or dimensionless versions, are described by Equations (75), (77), (85), (86), (115), and (157), which, in the present case, give
    ρ A = 3 H 2 8 π G r A ( A = r , m , k , ϕ ) , p A = 3 H 2 8 π G p A ( A = r , m , ϕ ) ;
    r r = Ω r a 4 , p r = Ω r 3 a 4 ; r m = Ω m a 3 , p m = 0 ; r k : = Ω k a 2 ;
    r ϕ = σ Π 2 2 a 6 + Υ , p ϕ = σ Π 2 2 a 6 Υ .
  • Equations (80) and (88) for the total dimensionless pressure and the total dimensionless densities, excluding or including the curvature contribution, read, in this case,
    r : = r r + r m + r ϕ , p : = p r + p ϕ ;
    r ˜ : = r r + r m + r k + r ϕ ;
    let us recall Equation (87).
  • Equation (158) for the normalized energy densities reads, in this case,
    Ω A = r A h 2 ( A = r , m , k , ϕ ) ,
    with h = a ˙ / a the dimensionless Hubble parameter.
  • According to Equation (132), the present setting with a field potential V ( ϕ ) = const. = Υ can be rephrased in terms of the Einstein equations with a cosmological constant
    Λ = 3 H 2 Υ .
To continue, we add the following assumptions to items (a) and (b):
(c)
It is
Υ > 0
(positive field potential or positive cosmological constant).
(d)
The values considered for the constants Ω A ( A = r , m , k ), Π and Υ are in any case related as follows:
Ω r + Ω m + Ω k + σ Π 2 2 + Υ = 1 .
As a comment on Equation (176), let us recall our assumptions on the signs of the other parameters in this model: we have asked from the very beginning that Ω r , Ω m > 0 , while we have left the s i g n Ω k = s i g n k unspecified.
Equation (177) is just the specific form of condition (159) for the case under examination. According to the discussion in Section 5.8, Equation (177) (or (159)) is the necessary and sufficient condition for the existence of solutions of the (complete or reduced) Lagrange equations with zero energy and the given value for Π , fulfilling the condition (99)
There   is   a   time   t such   that   a   ( t ) = 1   and   a ˙ ( t ) = 1 ( i . e . ,   h ( t ) = 1 ) ,
or its dimensioned analog (58)
There   is   a   time   τ such   that   a ( τ ) = 1   and   a ( τ ) = H ( i . e . ,   H ( τ ) = H ) .
Since the cosmological models of the present Section 6 aim for minimal realism, we will interpret t , τ as describing present time, and thus H will be the present value of the Hubble parameter.
Following again Section 5.8, let us emphasize that condition (177) is equivalent to Equation (160) V ( 1 ) = 1 , where V is now the potential (168); let us also repeat that, for each zero-energy solution t a ( t ) of the reduced system and for any t in its time domain, Equation (177) ensures the equivalence (161)
a ( t ) = 1 and a ˙ ( t ) = 1 a ( t ) = 1 and a ˙ ( t ) > 0 .
To continue, let us reconsider the normalized densities Ω A . We already know (and, in any case, we can recover from Equations (170) and (174)) that
Ω A ( t ) = Ω A for   A = r , m , k ;
moreover, we readily obtain from Equations (171) and (174) that
Ω ϕ ( t ) = σ Π 2 2 + Υ .

6.2. The Case of Π = 0  : Recovering the Standard Model of Cosmology

Recalling that we are assuming Ω r , Ω m , Υ > 0 , we momentarily consider the solutions t ( a ( t ) , ϕ ( t ) ) of our model with
Π = 0 .
Equation (177) becomes
Ω r + Ω m + Ω k + Υ = 1 ;
according to item (i) in Section 5.5, we have
ϕ ( t ) = constant ,
and Equation (179) allows us to interpret Υ as the present time value Ω ϕ ( t ) . Equation (168) becomes
V ( a ) = Ω r a 2 Ω m a Ω k Υ a 2 ;
of course,
V ( a ) Ω r a 2 for a 0 + , V ( a ) Υ a 2 for a + .
It can be shown that there is a unique a 2 such that
a 2 R + , V ( a 2 ) = 0 ;
moreover,
V ( a ) > 0 for   a ( 0 , a 2 ) , V ( a ) < 0 for   a ( a 2 , + )
(see Appendix K and Appendix I cited therein; the notation a 2 is employed to favor a comparison with the potentials of the subsequent Section 6.5, Section 6.6, Section 6.7 and Section 6.8). The above facts indicate that a 2 is the absolute maximum point of V .
The zero-energy solutions t a ( t ) of the reduced system with Lagrangian L ( a , a ˙ ) = a ˙ 2 V ( a ) are easily analyzed with the methods of Section 5.4; in particular, we have the quadrature formula (148), with V as in (183). This is just the evolution law of the standard (or benchmark, or  Λ CDM) model of cosmology with radiation, matter, curvature and a positive cosmological constant, representing dark energy (see, e.g., [75] or [76]), provided that we identify Υ with the present time value of the normalized dark energy density.
To fix ideas, let us consider the subcase
Ω k 0
(i.e., k 0 ). Then, we see from (183) that
V ( a ) < 0 for   all   a R + ;
Figure 1 gives a qualitative plot of the function V under the assumption (187).18
Let t a ( t ) be a maximal zero-energy solution; this can be analyzed using the general method of Section 5.4. Due to (188) (and to (144)), the range of the solution is the whole R + and a ˙ ( t ) never vanishes, thus having a constant nonzero sign. It turns out that the domain of the solution is a time interval unbounded from above if a ˙ > 0 , and unbounded from below if a ˙ < 0 . After a time translation t t + constant , this interval can be reduced to the form ( 0 , + ) if a ˙ > 0 , and to the form ( , 0 ) if a ˙ < 0 ; as usual for the standard model, hereafter, we consider the case a ˙ > 0 .
In this case, for t ranging in ( 0 , + ) , a ( t ) increases monotonically from 0 to + according to the following law (from (148)):
0 a ( t ) d a V ( a ) = t
(with 1 / V ( a ) = 1 / Ω r / a 2 + Ω m / a + Ω k + Υ a 2 = a / Ω r + Ω m a + Ω k a 2 + Υ a 4 ; this function vanishes for a 0 + , so the above integral is nonsingular). Equation (189) and the first relation (184) imply
a ( t ) 2 Ω r t for   t 0 + ;
of course, the vanishing of the scale factor in this limit indicates a Big Bang. For t + , an exponential behavior can be predicted for a ( t ) since (by (147) and the second relation (184)) a ˙ ( t ) = V ( a ( t ) ) Υ a ( t ) .
The concavity features of the function t a ( t ) are inferred from Equation (152) sign a ¨ ( t ) = sign V ( a ( t ) ) , from Equation (186) about V , and from the previous description of this solution. Clearly, there is a unique
t 2 ( 0 , + ) s . t . a ( t 2 ) = a 2 ,
and a ( t ) ( 0 , a 2 ) for t ( 0 , t 2 ) , a ( t ) ( a 2 , + ) for t ( t 2 , + ) . These facts imply
a ¨ ( t ) < 0 for   t ( 0 , t 2 ) , a ¨ ( t 2 ) = 0 , a ¨ ( t ) > 0 for   t ( t 2 , + ) ;
so, t 2 is an inflexion point. Needless to say, the present time t ( 0 , + ) is uniquely determined by the condition a ( t ) = 1 (recall the discussion in the final part of Section 6.1). According to Equation (189),
t 2 = 0 a 2 d a V ( a ) , t = 0 1 d a V ( a ) .
Figure 2 describes qualitatively the behavior of the scale factor t a ( t ) . Making reference to the formal notions of singular or nonsingular behavior in Section 3.2 and Section 3.15 (here, applied to the time domain ( t , t + ) = ( 0 , + ) ), we can say that the spacetime under consideration is past timelike and lightlike singular since it violates conditions (65) and (66). By contrast, this spacetime is future timelike and lightlike nonsingular since it fulfills (67) and (68).
Let us recall that the comparison between observational data [72,73] and the standard model suggests specific values for the normalized energy densities at the present time, which, in our language, correspond to the following prescriptions:
Ω r 9.2 · 10 5 , Ω m 0.31 , Ω k 0 , Υ 0.69 ;
these reference values will be taken into account in the sequel.
To fix the ideas, let us assume exactly the above numerical values for Ω r , Υ and set Ω m = 0.31 Ω r , Ω k = 0 (so that (181) holds exactly). Using these values and solving Equation (185) for a 2 (i.e., numerically), we obtain
a 2 0.608 .
From these values and from Equation (193), we obtain (computing numerically the corresponding integrals)
t 2 0.528 , t 0.955 .
Using Equation (59) involving the present-time Hubble parameter, we infer that the cosmic times τ i = t i / H ( i = 2 , ) have the values
τ 2 7.68 × 10 9 yr , τ 13.9 × 10 9 yr ;
of course, τ represents the present age of the universe.
We could continue our analysis and point out other facts, corresponding to known features of the standard model. Here, we will just remark that Equation (188) still holds for Ω k < 0 and | Ω k | not too large, while this equation fails for Ω k < 0 and | Ω k | sufficiently large (i.e., for k > 0 large enough); in the latter case, the behavior of the scale factor is qualitatively different from that described here, as pointed out, e.g., in [75].

6.3. Introducing the Analysis of Cases with Π 0

Maintaining all the assumptions of Section 6.1, we now pass from the case Π = 0 (the standard model of cosmology) to cases with Π 0 ; these will be treated assuming frequently that Π is small. The deviation of these cases from the standard model is controlled by the term σ Π 2 / ( 2 a 4 ) in Equation (168) for V .
In the sequel, most of our attention is devoted to the case
(i)
σ = 1 (phantom scalar field), Π 0 (typically small), Ω k 0 (i.e., k 0 );
this will be analyzed in detail in Section 6.5, Section 6.6 and Section 6.7. In comparison with the standard model of cosmology with Ω k 0 , case (i) exhibits an important qualitative difference: instead of the Big Bang of the standard model, there is a Big Bounce, with a minimum value a 0 (possibly very small but nonvanishing) for the scale factor.
For t + , in case (i), the scale factor presents an exponential growth similar to that of the standard model with Ω k 0 ; the same exponential growth appears for t , i.e., very far in the past before the Big Bounce. As we shall see, specifying values close to those in (194) for Ω r , Ω m , Ω k and Υ and an extremely small value for Π , we obtain a Big Bounce model with a minimum of physical plausibility; this could perhaps be considered a non-absurd alternative to the standard Big Bang cosmology.
We will subsequently address the case
(ii)
σ = 1 (phantom scalar field), Π 0 (typically small), Ω k < 0 (i.e., k > 0 );
this will be sketched in Section 6.8. If Ω k (i.e., k) is below a threshold value, the behavior of the model is similar to that of case (i), with a Big Bounce and exponential growth of the scale factor for t ± . If Ω k is above the threshold, the scale factor oscillates periodically (with a positive minimum) or, alternatively, it exhibits a Big Bounce with a (typically large) minimum value and exponential growth for t ± (different behaviors occur if Ω k has exactly the threshold value; see again Section 6.8). The physical plausibility of the model when Ω k is above (or equal to) the threshold is dubious; to say the least, further investigations on this issue would be necessary.
Finally, in this paper, we will not discuss the case
(iii)
σ = 1 (canonical scalar field), Π 0 ;
this presents a Big Bang19 and can be considered, at least for small Π , as a perturbation of the standard model of cosmology.

6.4. Preparing the Analysis of the Phantom Case with Π 0

From here to the end of the present Section 6, we assume the scalar field is a phantom:
σ = 1 .
We consider solutions of the model with Π nonvanishing; for future convenience, we represent this quantity as
Π = s 2 Ω r μ , s { ± 1 } , μ R + .
We confirm the general conditions (165), (176), and (177), thus requiring that
Ω r , Ω m , Υ > 0 , Ω r + Ω m + Ω k Ω r μ 2 + Υ = 1
(note that Ω r μ 2 is just the expression for Π / 2 following from (199); the sign of Ω k is unspecified for the moment; of course, all comments after Equation (177) apply here). The potential of Equation (168) is given in the present case by
V : R + R , a V ( a ) = Ω r μ 2 a 4 Ω r a 2 Ω m a Ω k Υ a 2 .
Needless to say,
V ( a ) Ω r μ 2 a 4 + for   a 0 + ; V ( a ) Υ a 2 for   a + .
Before going on, let us record for future citation that Equation (171) for the dimensionless density and pressure of the scalar field and Equations (198) and (199) give
r ϕ = Ω r μ 2 a 6 + Υ , p ϕ = Ω r μ 2 a 6 Υ .

6.5. The Phantom Case with Π 0 (Small) and Ω k 0 : Analysis of the Potential V ( a ) and the Associated Function H ( a ) : = V ( a ) / a 2

Throughout the present Section 6.5, we make the assumptions (198)–(200) and also prescribe
Ω k 0
(i.e., k 0 ). In the sequel, we will often put conditions of smallness on μ (i.e., on Π ); the most important of such conditions will be the forthcoming inequality (211).
The potential V . In Appendix L (which makes reference to the preceding Appendix I and Appendix J), we prove the following statements (a)–(d) about V :
(a)
There is a unique point a 0 such that
a 0 R + , V ( a 0 ) = 0 ;
in addition,
V ( a ) > 0   for   a ( 0 , a 0 ) , V ( a ) < 0   for   a ( a 0 , + )
and
a 0 < 1 .
(b)
Let
N : = Ω m + Ω k μ + Υ μ 3 Ω r , F : = Ω m + Ω k μ + Υ μ 3 2 Ω r + 3 Ω m μ + 4 Ω k μ 2 + 6 Υ μ 4 ;
then, N > 0 , F > 0 and F μ < 1 . Moreover,
μ 1 + N μ < a 0 < μ ( 1 F μ ) .
(Note that the upper bound on a 0 in (209) implies the rougher bound a 0 < μ ).
(c)
Let us consider the derivative
V : R + R , a V ( a ) = 4 Ω r μ 2 a 5 + 2 Ω r a 3 + Ω m a 2 2 Υ a ,
and assume
4 2 Υ Ω m 1 / 3 μ < 1 .
Then, there are just two points a 1 , a 2 such that
a 1 , a 2 R + , a 1 < a 2 , V ( a 1 ) = V ( a 2 ) = 0 ;
moreover
V ( a ) < 0 for   a ( 0 , a 1 ) ( a 2 , + ) , V ( a ) > 0 for a ( a 1 , a 2 ) ,
so that a 1 is a local minimum point, and a 2 is a local maximum point for V . Also, we have the following (somehow rough) inequalities:
a 0 < a 1 < 2 μ < a 2 .
(d)
Maintaining the assumption (211), let
M : = Ω m 2 Ω r , G : = Ω m 4 2 Υ μ 3 2 2 Ω r + 3 Ω m μ ;
then, M > 0 , G > 0 by (211) and G μ < 1 . Moreover, we have the following bounds (refining (214)) for a 1 :
2 μ 1 + M μ < a 1 < 2 μ 1 G μ .
Remarks. (i) Figure 3 presents a qualitative plot of the potential V under the conditions (198)–(200), (204) and (211). To help visualization, in the figure, a 0 and a 1 are not so small in comparison with a 2 while, for physically plausible choices of the parameters, we have a 0 , a 1 a 2 : see the forthcoming Section 6.7.
This plot should be compared with that of the potential in the standard cosmological model (see Section 6.2 and Figure 1).
(ii) Let us consider a μ -dependent family of models in which
Π = s 2 Ω r μ , s { ± 1 } , Υ = Υ + Ω r μ 2 ( μ R + )
Ω r > 0 , Ω m > 0 , Ω k 0 , Υ > 0   constants   such   that   Ω r + Ω m + Ω k + Υ = 1 .
The above requirement of sum one for Ω r , , Υ is the equivalent of the condition of sum one in (200); note that Equation (179) for the present-time normalized energy density of the scalar field gives Ω ϕ ( t ) = Υ .
For such a family of models, the quantities a 0 , a 1 are functions of μ , and it makes sense to discuss their behavior in the limit of vanishing μ . For μ 0 + , we have
a 0 = μ 1 Ω m 2 Ω r μ + O ( μ 2 ) ,
a 1 = 2 μ 1 Ω m 2 2 Ω r μ + O ( μ 2 ) .
In fact, both the lower and the upper bound for a 0 in (209) behave like the right-hand side of  (218); similarly, the lower and the upper bound for a 1 in (216) behave like the right-hand side of  (219).
The function H . Maintaining the assumptions (198)–(200) and (204), we now define
H : R + R , a H ( a ) : = V ( a ) a 2 = Ω r μ 2 a 6 + Ω r a 4 + Ω m a 3 + Ω k a 2 + Υ .
This function is especially interesting since, according to Equations  (156) and (87), along any solution of the cosmological model, we have
H ( a ( t ) ) = h 2 ( t ) = r ˜ ( t ) ,
where h is the dimensionless Hubble parameter (see Equation (70)) and r ˜ is the total dimensionless density, including the curvature contribution (see Equation (173)). Of course,
H ( a ) Ω r μ 2 a 6 for a 0 + , H ( a ) Υ for a + .
Moreover, H has the following features (e) (an obvious fact) and (f)–(i) (proved in Appendix M):
(e)
For all a R + , we have
sign H ( a ) = sign V ( a ) .
Thus, if a 0 R + is the point mentioned in Equations (205) and (206), we have
H ( a ) < 0   for   a ( 0 , a 0 ) , H ( a 0 ) = 0 , H ( a ) > 0   for   a ( a 0 , + ) .
(f)
Let us consider the derivative
H : R + R , a H ( a ) = 6 Ω r μ 2 a 7 4 Ω r a 5 3 Ω m a 4 2 Ω k a 3 .
Then, there is a unique point a 1 / 2 such that
a 1 / 2 R + , H ( a 1 / 2 ) = 0 ;
moreover,
H ( a ) > 0   for   a ( 0 , a 1 / 2 ) , H ( a ) < 0   for   a ( a 1 / 2 , + ) .
Due to these facts and to the asymptotics (222), a 1 / 2 is the absolute maximum point of H and
H ( a 1 / 2 ) > Υ .
The last inequality obviously implies H ( a 1 / 2 ) > 0 so that, by comparison with (224), we also infer
a 0 < a 1 / 2 .
(g)
Let
S : = 3 ( 3 / 2 Ω m + Ω k μ ) 4 Ω r , T : = 3 ( 3 / 2 Ω m + Ω k μ ) 8 Ω r + 9 3 / 2 Ω m μ + 12 Ω k μ 2 ;
then, S > 0 , T > 0 and T μ < 1 . Moreover,
3 / 2 μ 1 + S μ < a 1 / 2 < 3 / 2 μ 1 T μ .
(Note that the upper bound on a 1 / 2 in (231) implies the rougher bound a 1 / 2 < 3 / 2 μ .)
(h)
The bounds (231) imply
4 Ω r 27 μ 4 · 1 + X μ ( 1 T μ ) 4 < H ( a 1 / 2 ) < 4 Ω r 27 μ 4 · ( 1 + S μ ) 2 ( 1 + Y μ ) ,
with S , T as in (230), and
X : = 2 S + 3 3 / 2 Ω m Ω r · 1 1 + S μ + 9 Ω k 2 Ω r · μ 1 + S μ + 27 Υ 4 Ω r · μ 3 ( 1 + S μ ) 2 ,
Y : = 4 T ( 1 T μ ) 2 + 3 3 / 2 Ω m Ω r ( 1 T μ ) + 2 T 2 μ ( 1 T μ ) 2
+ 9 Ω k 2 Ω r μ ( 1 T μ ) 2 + 27 Υ 4 Ω r μ 3 ( 1 T μ ) 4 .
(i)
Assume (211) holds so that we can speak of the point a 1 defined by (212); also, assume
M μ < 1 3 , with   M   as   in ( 215 ) .
Then,
a 1 / 2 < a 1 .
Remarks. (i) Figure 4 presents a qualitative plot of the function H under the conditions (198)–(200), (204), (211), and (234).
(ii) Let us consider a family of models as in (217). For μ 0 + , Equation (231) implies
a 1 / 2 = 3 / 2 μ 1 3 3 / 2 Ω m 8 Ω r μ + O ( μ 2 ) ;
in fact, both the lower and the upper bound on a 1 / 2 in Equation (231) behave like the right-hand side of (236). Again for μ 0 + , we have
H ( a 1 / 2 ) = 4 Ω r 27 μ 4 1 + 3 3 / 2 Ω m Ω r μ + O ( μ 2 ) .
The last result can be obtained by directly inserting the expansion (236) of a 1 / 2 into Equation (220) for H , or using the estimates (232) for H ( a 1 / 2 ) ; in fact, both the lower and upper bound on H ( a 1 / 2 ) in Equation (232) behave like the left-hand side of Equation (237).

6.6. The Phantom Case with Π 0 (Small) and Ω k 0  : Analysis of the Zero-Energy Solutions and Classicality Condition

Throughout the present Section 6.6, we make the assumptions (198)–(200), (204), (211), and (234).
Let us consider a zero-energy solution J t a ( t ) of the reduced system with Lagrangian (141) ( J meaning, as usual, an open real interval); we assume this to be maximal, in the sense of Section 4.6, and analyze it according to the general scheme of Section 5.4. Bearing in mind the results of V in Section 6.5, we are led to the description reported in the following paragraphs:
Basic features of the solution; Big Bounce. The set V 0 : = { a R + | V ( a ) 0 } coincides in this case with [ a 0 , + ) , with a 0 as in (205). The motion J t a ( t ) has range [ a 0 , + ) and possesses a unique inversion time t 0 , characterized by the condition a ( t 0 ) = a 0 ; up to a time shift t t + constant, we can assume t 0 = 0 . Thus,
a ( 0 ) = a 0 ;
moreover
sign a ˙ ( t ) =   sign t
(even at t = 0 , when a ˙ vanishes). From the quadrature formula (148) with t , t replaced with 0 , t (or with t , 0 ), and from Equations (238) and (239), we obtain
a 0 a ( t ) d a V ( a ) = ( sign t ) t = | t | .
Since V ( a 0 ) = 0 and V ( a 0 ) < 0 , the function 1 / V ( a ) diverges like 1 / a a 0 for a ( a 0 ) + , so the above integral converges; this remark is related to item (iv) in Section 5.4.
The motion continues until a ( t ) diverges in the far future and in the far past; divergence of a ( t ) occurs for t t ± , where t ± : = ± a 0 + d a / V ( a ) . On the other hand, the last integral diverges since 1 / V ( a ) 1 / ( Υ a ) for a + (recall Equation (184)); thus, t ± = ± , and we can say that
J = ( , + ) , a ( t ) + for t ±
(a posteriori, this also ensures that (239) and (240) hold for all t R ). By construction, the function R R + , t a ( t ) solves the Cauchy problem 2 a ¨ ( t ) = V ( a ( t ) ) (recall (143)), a ( 0 ) = a 0 , a ˙ ( 0 ) = 0 . It is evident that the function R R + , t a ( t ) solves the same Cauchy problem, so by the standard uniqueness theorem, we have
a ( t ) = a ( t ) for   all   t R .
To summarize, the solution under consideration describes an eternal universe with a Big Bounce at time t = 0 , when the scale factor attains the nonzero, minimum value a 0 ; the history of the scale factor is time-symmetric with respect to the instant of the Big Bounce.
Figure 5 shows a qualitative plot of the scale factor (also taking into account the concavity considerations in the second subsequent paragraph).
Nonsingular nature of the model. Since the time domain of the model is the whole real axis ( , + ) and the function t a ( t ) has a strictly positive minimum, the present spacetime is nonsingular in all senses formalized in Section 3.2 and Section 3.15 (concerning this statement, let us recall Equation (69) and the related comments).
A result frequently used in the sequel. Let a i ( a 0 , + ) . From the features of the function t a ( t ) described previously, it appears that there is a unique time
t i R + such   that a ( t i ) = a i ;
according to Equation (240), this is given by
t i = a 0 a i d a V ( a ) .
In the sequel, we will often refer to the above statements.
Concavity features of the solution. Let us reconsider Equation (152) sign a ¨ ( t ) = sign V ( a ( t ) ) . The sign of V is described by Equations (212) and (213) (and by the related Equation (214)), involving two distinguished values a 1 , a 2 such that a 0 < a 1 < a 2 . For i = 1 , 2 , there is a unique time t i fulfilling Equation (243), and this is given by Equation (244). Moreover,
t 1 < t 2 ; a ( t ) [ a 0 , a 1 ) for   t [ 0 , t 1 ) ,
a ( t ) ( a 1 , a 2 ) for   t ( t 1 , t 2 ) , a ( t ) ( a 2 , + ) for   t ( t 2 , + ) .
Equations (212) and (213) on the sign of V and Equations (152), (243) and (245) imply
a ¨ ( t ) > 0 for   t [ 0 , t 1 ) ( t 2 , + ) , a ¨ ( t 1 ) = a ¨ ( t 2 ) = 0 , a ¨ ( t ) < 0 for   t ( t 1 , t 2 ) ;
thus, t 1 and t 2 are inflection times for the function a ( ) . One can make similar statements on the behavior of a ( ) at negative times, noting that (242) implies a ¨ ( t ) = a ¨ ( t ) .
Let us recall that, according to Equation (98), the strong energy condition (SEC) is violated whenever  a ¨ ( t ) > 0 ; in the model that we are analyzing, this happens in the time interval ( t 1 , t 1 ) containing the Big Bounce, and in the time intervals ( , t 2 ) , ( t 2 , + ) 20. A more detailed analysis of the energy conditions will be presented in the sequel.
The present time. Let us recall the inequality (207) a 0 < 1 and the considerations after Equations (58) and (99). In the model under discussion, the present time t is individuated by the conditions
t R + , a ( t ) = 1 ,
that automatically imply
a ˙ ( t ) = 1 ( i . e . , h ( t ) = 1 ) .
In fact, a ˙ ( t ) > 0 since t > 0 , and we have the equivalence (161) reviewed in the discussion after Equation (177). The present time is described by Equation (244) with i = and a = 1 , i.e.,
t = a 0 1 d a V ( a ) .
In Section 6.7, assuming physically plausible values for the constants of the model, we will find a 0 , a 1 a 2 < 1 and t 1 t 2 < t .
The large t limit: exponential growth of the scale factor. For all t R + , we have a ˙ ( t ) = V ( a ( t ) ) ; for t + , we have a ( t ) + and this fact, with Equation  (202) about V , gives a ˙ ( t ) Υ a ( t ) (note that a similar statement was made in Section 6.2). Due to this asymptotic feature, a ( t ) is expected to behave like const.× e Υ t for t + . In Appendix N, we rigorously prove that
a ( t ) K e Υ t for   t + ,
K : = exp 1 + d a 1 a Υ V ( a ) Υ t .
The integral from 1 to + in the above definition of K converges, since 1 / a Υ / V ( a ) = O ( 1 / a 3 ) for a + (see again Appendix N); let us also recall the expression (249) for t .
Needless to say, the relation (250) and the symmetry property a ( t ) = a ( t ) also imply
a ( t ) K e Υ t for   t .
For k = 0 , let us reconsider the spatial and spacetime line elements (16) and (17) and, referring to the dimensioned time τ = t / H , let us insert the expression for the scale factor corresponding to the right-hand side of Equation (250) into Equation (17); then, the spacetime line element becomes of the de Sitter form (let us recall again [12] (page 125)). In this sense, the present cosmological model with k = 0 is asymptotically de Sitter for τ + (a feature that also exists in the standard cosmological model with a positive cosmological constant, reviewed in Section 6.2). Due to (251), the present model with k = 0 is asymptotically de Sitter (in an obvious sense) even for τ .
The (dimensionless) Hubble parameter h : = a ˙ / a . Let us note that Equation (242) implies that h is an odd function of time:
h ( t ) = h ( t ) for   all   t R .
Moreover, due to (239), we have
sign h ( t ) = sign t ( in   particular ,   h ( 0 ) = 0 ) .
Let us further discuss the behavior of the function t h ( t ) ; it will suffice to see what happens for t 0 . With H as in Equation (220), we see from Equation (221) that
h ( t ) = H ( a ( t ) ) for   t [ 0 , + ) ;
let us repeat that, while t ranges in this interval, a ( t ) increases monotonically from a 0 to + . We can combine these remarks with the available information about H (see the paragraph starting from Equation (220), and Figure 4); let us recall, in particular, that H has a unique maximum point a 1 / 2 (see Equation (226) and subsequent statements), where a 1 / 2 ( a 0 , a 1 ) (see (229) and (235)), and H ( a ) approaches Υ for a + (see (222)).
In conclusion, the time behavior of h for t 0 is as follows: First of all, there is a unique time t 1 / 2 R + such that a ( t 1 / 2 ) = a 1 / 2 , and this is given by Equation (244) with i = 1 / 2 . Moreover,
t 1 / 2 < t 1 ;
h   is   strictly   increasing   on   [ 0 , t 1 / 2 ] and   strictly   decreasing   on   [ t 1 / 2 , + ) ;
max t [ 0 , + ) h ( t ) = h ( t 1 / 2 ) = H ( a 1 / 2 ) , h ( t ) Υ for   t + .
Let us also note that h ˙ = a ¨ / a h 2 ; in particular, (recalling the relation h ( 0 ) = 0 and Equation (143))
h ˙ ( 0 ) = a ¨ ( 0 ) a ( 0 ) = V ( a 0 ) 2 a 0 > 0 .
Preliminaries on the densities. Let us consider for A = r , m , k , ϕ the corresponding densities, e.g., in the dimensionless versions r A ; these are given by Equations (170) and (203), and can be regarded as functions of the scale factor or of the dimensionless time: r A = r A ( a ) or r A = r A ( t ) . In the sequel, we also refer to the total dimensionless density r ˜ , including the curvature contribution, which is the sum of all terms r A (see Equation (173)). Let us remark that the relation a ( t ) = a ( t ) implies the following for all t R :
r A ( t ) = r A ( t ) ( A = r , m , k , ϕ ) , r ˜ ( t ) = r ˜ ( t ) .
Time evolution of the total density. Let us recall again Equation (221), ensuring that
r ˜ ( t ) = h 2 ( t )
for all t R . The results of the previous paragraph about h yield the following conclusions on the behavior of the function t r ˜ ( t ) , say for t 0 :
r ˜   is   strictly   increasing   on   [ 0 , t 1 / 2 ] and   strictly   decreasing   o n   [ t 1 / 2 , + ) ;
max t [ 0 , + ) r ˜ ( t ) = r ˜ ( t 1 / 2 ) = H ( a 1 / 2 ) , r ˜ ( t ) Υ for   t +
(with t 1 / 2 as in (255)). Let us also note that
r ˜ ( 0 ) = h 2 ( 0 ) = 0 , r ˜ ˙ ( 0 ) = 2 h ( 0 ) h ˙ ( 0 ) = 0 .
Comparison with the Planck density; classicality condition. Let us recall that, in our setting with c = 1 , the Planck length, time and mass are
P = τ P = G 1.62 × 10 33 cm 5.39 × 10 44 s , m P = G 2.18 × 10 5 gr .
From these quantities, we define a Planck density
ρ P : = m P P 3 = 1 G 2 5.16 × 10 93 gr / cm 3 .
Let us compare this quantity with the total (dimensioned) density, including the curvature contribution; this is
ρ ˜ : = ρ r + ρ m + ρ k + ρ ϕ = 3 H 2 8 π G r ˜ ,
where r ˜ is given again by Equation (173), and we have used Equation (169). The comparison can be made by introducing the ratio between the maximum of ρ ˜ ( 0 ) and the constant ρ P ; due to Equations (259), (261), and (265), we have
1 ρ P max t R ρ ˜ ( t ) = 1 Q max t R r ˜ ( t ) = H ( a 1 / 2 ) Q ,
where Q : = 8 π G 3 H 2 ρ P . On the other hand, Equation (264) gives Q = 8 π 3 H 2 G , i.e., by Equation (263),
Q = 8 π 3 ( H τ P ) 2 .
We note that the parameter Q is dimensionless; using for H , τ the values in Equations (59) and (263), we obtain
Q 6.04 × 10 122 .
It is reasonable to prescribe the classicality condition for the cosmological model under analysis, that reads
1 ρ P max t R ρ ˜ ( t ) 1 , i . e . , H ( a 1 / 2 ) Q 1 ;
this ensures, a posteriori, that the classical treatment of gravity given herein is reasonable at any stage of the cosmological model under examination21,22. As we shall see, for small μ this condition is, essentially, a lower bound on μ (i.e., on the minimum a 0 of the scale factor).
Sign of the phantom energy density, and comparisons among the partial densities. Independent of the previous classicality condition, we now determine the sign of the phantom scalar energy density, and we make comparisons among this density and those associated with radiation, matter, and curvature. Obviously enough, the comparison of these densities gives exactly the same results for any of the available versions (dimensioned, dimensionless, or normalized); hereafter, we refer to the dimensionless versions r A ( A = ϕ , r , m , k ), viewing them as functions of the variable a R + on the grounds of Equations (170) and (203)23; let us also remark that for all a, we have r r ( a ) , r m ( a ) > 0 and r k ( a ) 0 (since we are assuming Ω k 0 ).
That said, for any a R + , we have
r ϕ ( a ) 0 Ω r μ 2 a 6 + Υ 0 a a ϕ , a ϕ : = Ω r μ 2 Υ 1 / 6 .
Again, for any a R + ,
r ϕ ( a ) r r ( a ) Ω r μ 2 a 6 + Υ Ω r a 4 Υ a 6 + Ω r a 2 Ω r μ 2 0 a a ϕ r ,
a ϕ r : =   the   unique   point   in   R +   such   that   Υ a ϕ r 6 + Ω r a ϕ r 2 Ω r μ 2 = 0 ;
for the proof that a ϕ r exists and is unique, and for a justification of the conditions a a ϕ r in (271), see Appendix O. In the same appendix, we prove that
a ϕ r > a 0 ;
μ 1 + Υ μ 4 / Ω r < a ϕ r < μ 1 Υ μ 4 2 Ω r + 6 Υ μ 4 .
It should be noted that, for small μ , a ϕ r is very close to a 0 : Equations (209) and (273) give the bounds
1 ( 1 F μ ) 1 + Υ μ 4 / Ω r < a ϕ r a 0 < 1 + N μ 1 Υ μ 4 2 Ω r + 6 Υ μ 4
(with N , F as in Equation (208)). For a family of models as in (217), Equation (273) implies the following for μ 0 + :
a ϕ r = μ 1 Υ μ 4 2 Ω r + O ( μ 6 )
(since both the lower and the upper bounds on a ϕ r in (273) behave like the right-hand side of (275)); similarly, we infer from Equation (274) that
a ϕ r a 0 = 1 + Ω m 2 Ω r μ + O ( μ 2 ) .
To continue, we note that, for any a R + , we have
r ϕ ( a ) r r ( a ) Ω r μ 2 a 6 + Υ Ω r a 4 Υ a 6 Ω r a 2 Ω r μ 2 0 a a ϕ r + ,
a ϕ r + : =   the   unique   point   in   R +   such   that   Υ a ϕ r + 6 Ω r a ϕ r + 2 Ω r μ 2 = 0 ;
the existence and uniqueness of a ϕ r + , as well as the above conditions a a ϕ r + , are justified with arguments similar to those employed in connection with Equation (271) and a ϕ r .
Using again similar arguments, one proves the following for any a R + 24:
r ϕ ( a ) r m ( a ) Ω r μ 2 a 6 + Υ Ω m a 3 Υ a 6 + Ω m a 3 Ω r μ 2 0 a a ϕ m ,
a ϕ m : =   the   unique   point   in   R +   such   that   Υ a ϕ m 6 + Ω m a ϕ m 3 Ω r μ 2 = 0 ,
i . e . , a ϕ m = Ω m + Ω m 2 + 4 Υ Ω r μ 2 2 Υ 1 / 3 ;
r ϕ ( a ) r m ( a ) Ω r μ 2 a 6 + Υ Ω m a 3 Υ a 6 Ω m a 3 Ω r μ 2 0 a a ϕ m + ,
a ϕ m + : =   the   unique   point   in   R +   such   that   Υ a ϕ m + 6 Ω m a ϕ m + 3 Ω r μ 2 = 0 ,
i . e . , a ϕ m + = Ω m + Ω m 2 + 4 Υ Ω r μ 2 2 Υ 1 / 3 ;
r ϕ ( a ) r k ( a ) Ω r μ 2 a 6 + Υ Ω k a 2 Υ a 6 + Ω k a 4 Ω r μ 2 0 a a ϕ k ,
a ϕ k : =   the   unique   point   in R +   such   that   Υ a ϕ k 6 + Ω k a ϕ k 4 Ω r μ 2 = 0 ;
r ϕ ( a ) r k ( a ) Ω r μ 2 a 6 + Υ Ω k a 2 Υ a 6 Ω k a 4 Ω r μ 2 0 a a ϕ k + ,
a ϕ k + : =   the   unique   point   in   R +   such   that   Υ a ϕ k + 6 Ω k a ϕ k + 4 Ω r μ 2 = 0
Finally, for any a R + , we have
r r ( a ) r m ( a ) Ω r a 4 Ω m a 3 a a r m , a r m : = Ω r Ω m ;
r r ( a ) r k ( a ) Ω r a 4 Ω k a 2 a a r k , a r k : = Ω r Ω k ;
r m ( a ) r k ( a ) Ω m a 3 Ω k a 2 a a m k , a m k : = Ω m Ω k
(intending Ω r / Ω k : = Ω m / Ω k : = + if Ω k = 0 ).
Let i denote anyone of the above subscripts ϕ , ϕ r , ϕ r + , , r k , m k . Of course, given the motion t R a ( t ) , if a i > a 0 , there is a unique time t i > 0 such that a ( t i ) = a i , and this is given by Equation (244); for any t 0 , we have that a ( t ) a i t t i .
On the computation of certain times. Let us consider a time t i admitting an integral representation of the form (244) t i = a 0 a i d a / V ( a ) , with a i > a 0 .
We have already noted that 1 / V ( a ) diverges (like 1 / a a 0 ) for a ( a 0 ) + ; moreover, we are mainly interested in cases where a 0 is an extremely small number. For these reasons, the numerical computation of the integral in (244) is not so trivial. In Appendix Q and Appendix R, the representation (244) is used as a starting point to derive rigorous lower and upper bounds on t i , which are precise in two different situations; estimation of t i by means of these bounds is a reliable alternative to the direct numerical computation of the integral in (244). We will return to these issues in Section 6.7.
The analysis in Appendix Q also allows us to consider a family of models as in (217) and discuss the limit μ 0 + for the times t 1 / 2 , t 1 , and t ϕ r , which are determined as in (244) with a i = a 1 / 2 , a 1 , and a ϕ r (see Equations (212), (226), and (271)). In this limit, it is found that
t 1 / 2 = K 1 / 2 Ω r μ 2 ( 1 + O ( μ ) ) , K 1 / 2 : = 3 4 + arcsinh 1 2 3 2 1 2 0.7623 ;
t 1 = K 1 Ω r μ 2 ( 1 + O ( μ ) ) , K 1 : = 1 2 + arcsinh 1 2 1 2 1.148 ;
t ϕ r = Ω m Ω r μ 5 / 2 ( 1 + O ( μ ) ) .
Time evolution of the scalar field. Equation (136) with σ = 1 , d = 3 and Equation (199), ensure that everywhere on the time domain ( , + ) ,
ϕ ˙ = s 2 Ω r μ a 3 , sign ϕ ˙ = constant = s .
In particular,
ϕ ˙ ( 0 ) = s 2 Ω r μ a 0 3 ;
the bounds (209) on a 0 , involving F , N as in Equation (208), imply
2 Ω r μ 2 · 1 ( 1 F μ ) 3 < | ϕ ˙ ( 0 ) | < 2 Ω r μ 2 · ( 1 + N μ ) 3 / 2 ,
showing that ϕ ˙ ( 0 ) is very large for small μ . From (288) and Equations (250) and (251) on the large t behavior of a ( t ) , we also infer that
ϕ ˙ ( t ) s 2 Ω r μ K 3 e 3 Υ t 0 for   t .
Again from (288), we see that the function t R ϕ ( t ) is strictly increasing or decreasing in the two cases s = ± 1 ; this agrees with the predictions of item (i) in Section 5.5.
Let us refer to item (ii) in the same section; using Equation (155) therein with σ = 1 , d = 3 and t = , t = 0 or t = 0 , t = + , and recalling again Equation (199), we conclude that
ϕ ( t ) ϕ ( 0 ) s Δ ϕ for   t , Δ ϕ : = 2 Ω r μ a 0 + d a a 6 V ( a ) .
According to (202), for a + , we have 1 / a 6 V ( a ) 1 / ( Υ a 4 ) (while, for a ( a 0 ) + , we have 1 / a 6 V ( a ) 1 / ( a 0 3 V ( a 0 ) a a 0 )  ); so, the integral defining Δ ϕ converges.
In Appendix S, we derive precise lower and upper bounds on Δ ϕ . For a family of models as in (217), these bounds ensure that, in the limit μ 0 + ,
Δ ϕ = π 2 ( 1 + O ( μ ϑ ) ) for   any   ϑ 0 , 1 4 .
Energy conditions. This subject has been already touched on in a few lines after Equation (246); here, we reconsider it in detail. The weak and strong energy conditions for the general cosmological models considered in this paper are presented in Equations (52) and (53) and reformulated in Equations (93) and (94) in the dimensionless form. In the case under analysis, the space dimension is d = 3 , and Equations (93) and (94) read as follows:
WEC   holds r 0 , r + p 0 ;
SEC   holds r + 3 p 0 , r + p 0 .
Again, in the case under analysis, the total dimensionless density and pressure r , p are determined by Equations (170), (203), and (172), which imply
r = Ω r μ 2 a 6 + Ω r a 4 + Ω m a 3 + Υ , r + p = 2 Ω r μ 2 a 6 + 4 Ω r 3 a 4 + Ω m a 3 ,
r + 3 p = 4 Ω r μ 2 a 6 + 2 Ω r 3 a 4 + Ω m a 3 2 Υ .
Comparing the above results with the expressions (201), (210), and (225) for the function V and for the derivatives V , H we see that25
r = V ( a ) a 2 Ω k a 2 , r + p = 1 3 a H ( a ) 2 Ω k 3 a 2 , r + 3 p = V ( a ) a .
In the sequel of this paragraph, just for simplicity, we consider the zero curvature case
Ω k = 0 .
Due to this assumption, the signs of the quantities in Equation (297) are determined by the signs of V , V , and H , which were studied in Equations (205), (206), (212), (213), (226), and (227) involving the special values a 0 < a 1 / 2 < a 1 < a 2 . In this way, we find
r = 0   for   a = a 0 , > 0   for   a ( a 0 , + ) , r + p < 0   for   a [ a 0 , a 1 / 2 ) , = 0   for   a = a 1 / 2 , > 0   for   a ( a 1 / 2 , + ) ,
r + 3 p < 0   for   a [ a 0 , a 1 ) ( a 2 , + ) , = 0   for   a = a 1 , a 2 , > 0   for   a ( a 1 , a 2 ) .
These facts and (294) and (295) imply
WEC   fails   for   a [ a 0 , a 1 / 2 ) ,   holds   for   a [ a 1 / 2 , + ) ;
SEC   fails   for   a [ a 0 , a 1 ) ( a 2 , + ) ,   holds   for   a [ a 1 , a 2 ] .
The equivalent statements in terms of the dimensionless time are obvious; for example, SEC fails for t ( , t 2 ) ( t 1 , t 1 ) ( t 2 , + ) and holds for t [ t 2 , t 1 ] [ t 1 , t 2 ] .
For extending the above results to the case Ω k > 0 , one should return to Equation (296) and study directly the signs of the quantities appearing therein as functions of a; the problem is dealt with methods very similar to those employed in Appendix L and Appendix M to study the signs of V , V , and H .
Comparison with the literature. In Section 5.9, we already mentioned the toy model introduced in [74] (Section II.B), with an FLRW universe of dimension 3 + 1 filled only by a canonical or phantom scalar field with zero self-potential. The authors of [74] consider, in particular, the case with a phantom scalar and negative spatial curvature; their analysis indicates (somehow implicitly) that the scale factor exhibits a Big Bounce, while the scalar field tends to finite values very far in the past and the future; we have met the same qualitative behavior in the more realistic model of the present Section 6.6.

6.7. The Phantom Case with Π 0   (Small), Ω k = 0 and Specific Values for All the Parameters

Maintaining (198) and (199), we now try to give physically plausible values to the parameters of the model by prescribing
Ω r = 9.2 × 10 5 , Ω m = 0.31 Ω r , Ω k = 0 ( i . e . , k = 0 ) , Υ = 0.69 + Ω r μ 2
and
0 < μ 1 .
The value of the parameter μ will be fixed later. Let us note that conditions (200) and (204) are satisfied; the same can be said of conditions (211) and (234), if the requirement μ 1 is intended appropriately. We also remark that Equation (179) for the present-time normalized energy density of the scalar field gives in this case
Ω ϕ ( t ) = 0.69 .
The above values of Ω r , Ω m , and Ω ϕ ( t ) are close to the ones ascribed within the standard model of cosmology to the normalized densities of radiation, matter, and dark energy at the present time; the assumption of zero curvature is standard as well (concerning these statements, let us again mention the observational data from [72,73]).
According to Section 6.5, the functions a V ( a ) , H ( a ) of Equations (201) and (220) can be described referring to some special values a 0 , a 1 / 2 , a 1 , a 2 of the variable a, such that
a 0 μ , a 1 / 2 3 / 2 μ , a 1 2 μ , a 1 < a 2 ;
in particular, a 0 is the minimum of the zero-energy solution t a ( t ) analyzed in Section 6.6, to which we systematically refer from now on. We now fix the value of μ on the grounds of the classicality condition (269); this involves the ratio
1 ρ P max t R ρ ˜ ( t ) = H ( a 1 / 2 ) Q ,
with ρ P the Planck density (see Equation (264)) and Q as in Equations (267) and (268). However, due to Equations (232) and the smallness of μ ,
H ( a 1 / 2 ) 4 Ω r 27 μ 4 ;
therefore,
1 ρ P max t R ρ ˜ ( t ) 4 Ω r 27 Q μ 4 2.26 × 10 128 μ 4 ,
where, in the last step, we used the numerical values in Equations (268) and (302) for Q and Ω r .
The classicality condition (269) requires the ratio (307) to be very small; we now choose an extremely small value for μ , which, however, is sufficiently large to fulfill (269). We will adopt the value
μ : = 10 31 ,
implying
1 ρ P max t R ρ ˜ ( t ) 2.26 × 10 4 .
From here to the end of the present Section 6.7, we stick to the choices (302) and (308) for Ω r , Ω m , Ω k , Υ , and μ , confirming that all conditions (200) (204) and (211) (234) are fulfilled; whenever necessary, we use the value (59) for H .
In the sequel, we report many distinguished values a i of the scale factor and the corresponding times t i (or τ i ), all of them computed on the grounds of the above choices for the basic parameters of this model; the penultimate paragraph of the present section gives some information about these calculations. In the rest of the section, we discuss the sign of the phantom energy density, and we make comparisons among this quantity and the densities of radiation and matter, at any time in the history of this model; moreover, we consider the present epoch.
The values of a 0 , , a 2 and the corresponding times. Hereafter, we report the values of the following quantities for i = 0 , 1 / 2 , 1 , 2 : a i , the dimensionless time t i 0 such that a ( t i ) = a i and its equivalent τ i = t i / H in terms of cosmic time. Here are the values:
a 0 10 31 , t 0 = 0 , τ 0 = 0 s ; a 1 / 2 1.22 × 10 31 , t 1 / 2 7.95 × 10 61 , τ 1 / 2 3.65 × 10 43 s ; a 1 1.41 × 10 31 , t 1 1.20 × 10 60 , τ 1 5.49 × 10 43 s ; a 2 0.608 , t 2 0.528 , τ 2 7.68 × 10 9 y r .
Let us recall that the function t a ( t ) is even and, for 0 t < + , increases from the minimum value a 0 to + . The concavity features of this function are described by Equation (246), indicating that t 1 and t 2 are inflection points. The time t 1 / 2 is the maximum point of the Hubble parameter (see Equation (257)) and (we repeat it) of the total density over the interval [ 0 , + ) .
The present time. The present time t , individuated by the conditions t > 0 and a ( t ) = 1 , is determined by Equation (249). It is found that t and its cosmic time equivalent τ = t / H have the values
t 0.955 , τ 13.9 × 10 9 y r .
τ is the cosmic time elapsed from the Big Bounce to the present epoch; it is quantitatively indistinguishable from the age of the universe (time elapsed from the Big Bang to now) in the standard model of cosmology (see Section 6.2, in particular, Equation (197))26.
Sign of the phantom energy density, and comparisons among the partial densities. These issues are related to some distinguished values a i of the scale factor ( i = ϕ , ϕ r , ϕ m , r m ), described by Equations (270)–(279) and (282)27; the values of a i are attained at certain times t i > 0 , which have cosmic time correspondents τ i = t i / H . These special values, arranged in increasing order, are as follows:
a ϕ r ( 1 + 1.68 × 10 28 ) a 0 , t ϕ r = 1.91 × 10 74 , τ ϕ r = 8.78 × 10 57 s ; a ϕ m 1.44 × 10 22 , t ϕ m 1.08 × 10 42 , τ ϕ m 4.94 × 10 25 s ; a ϕ 1.05 × 10 11 , t ϕ 5.74 × 10 21 , τ ϕ 2.63 × 10 3 s ; a r m 2.97 × 10 4 , t r m 3.59 × 10 6 , τ r m 52.2 × 10 3 y r ; a ϕ r + 0.107 , t ϕ r + 4.20 × 10 2 , τ ϕ r + 0.611 × 10 9 y r ; a ϕ m + 0.766 , t ϕ m + 0.707 , τ ϕ m + 10.3 × 10 9 y r .
Let us consider the evolution of the cosmological model for τ 0 , recalling that the behavior for τ 0 is specular due to a ( τ ) = a ( τ ) ; using the inequalities (270), (271), (277)–(279) and (282), we obtain the following description:
0.
s i g n ( r ϕ ) =   sign   ( a a ϕ ) =   sign   ( τ τ ϕ ) . We repeat that a ϕ 1.05 × 10 11 and τ ϕ 2.63 × 10 3 s .
1.
a 0 a a ϕ r , i.e., 0 τ τ ϕ r : a field dominated era, with r ϕ < 0 and | r ϕ | r r , r m ;
we repeat that a 0 10 31 , a ϕ r ( 1 + 1.68 × 10 28 ) a 0 and τ ϕ r 8.78 × 10 57 s .
2.
a ϕ r a a r m , i.e., τ ϕ r τ τ r m : a radiation dominated era, with r r | r ϕ | , r m .
We repeat that a r m 2.97 × 10 4 and τ r m 52.2 × 10 3 y r ; according to 0., r ϕ changes sign shortly after the beginning of this era.
3.
a r m a a ϕ m + , i.e., τ r m τ τ ϕ m + : a matter dominated era, with r ϕ > 0 and r m r ϕ , r r ; we repeat that a ϕ m + 0.766 and τ ϕ m + 10.3 × 10 9 yr .
4.
a ϕ m + a < + , i.e., τ ϕ m + τ < + : a field dominated era, with r ϕ > 0 and r ϕ r m , r r . This era contains the present time with a = 1 and τ = τ 13.9 × 10 9 yr .
For a more detailed description, see Appendix T.
Comparison with the standard model. As stressed after Equation (303), the present-time values for the normalized densities of radiation, matter, and energy of the scalar field in the model of the current Section 6.7 coincide with corresponding values for radiation, matter, and dark energy in the standard cosmological model with curvature k = 0 ; for the parameter μ , determining the discrepancy from the standard model, we assumed an extremely small value. The aim of these choices was to produce a model very close to the standard cosmological model at all times subsequent to the Big Bounce, except for an extremely short epoch immediately after the Big Bounce.
The results in the previous paragraphs confirm this expectation: the agreement between the two models has already been exemplified in the discussion on the age of the universe (see Equation (311) and related comments), and many other examples could be extracted from the previous analysis. To give just another example, let us consider the values of the scale factor and of the cosmic time yielding the equality of radiation and matter densities, which are a r m and τ r m with our notations; the values of these quantities in Equation (312) agree with the corresponding values in the standard model.
In general, epochs 2, 3, and 4 of the previous paragraph juxtapose (both qualitatively and quantitatively) with the radiation-dominated era, the matter-dominated era, and the dark energy-dominated era in the standard cosmological model. Concerning epoch 4, let us repeat that the present model is asymptotically de Sitter for τ + , like the standard one for k = 0 (recall the comments after Equations (250) and (251)).
On the first era after the Big Bounce, dominated by the scalar field. This is epoch 1 of the penultimate paragraph, corresponding to the extremely short time interval [ 0 , τ ϕ r ] ; of course, here, the model is significantly different from the standard one. It should be mentioned that epoch 1 is not inflationary, if one defines inflation in terms of an explosive growth of the scale factor: on the contrary, the initial value a 0 and the final value a ϕ r of the scale factor are essentially equal (see the first line in Equation (312)).
On the computation of the previous values of a i and t i . All computations related to (310)–(312) were performed with more significant digits than those reported here28.
The values of a i for i = 0 , 1 / 2 , 1 and the value of a ϕ r / a 0 were estimated referring to the bounds (209), (216), (231), and (274) and the asymptotic expressions (218), (219), (236), and (276). Due to the extremely small value of μ , in each one of these cases the numerical values of the lower bound, of the upper bound, and of the asymptotic expression neglecting the reminders O ( μ 2 ) , are practically indistinguishable.
By definition, a 2 is the second positive solution of the equation V ( a ) = 0 (or of the equivalent algebraic equation Q ( a ) = 0 , where Q is the polynomial in Equation (A123) of Appendix L); the value of a 2 was estimated by solving numerically this equation. The value of a ϕ r + was estimated by solving numerically the algebraic equation in (277).
The values of a i for i = ϕ , ϕ m , r m were obtained from the analytic expressions in (270), (278), (279), and (282).
For all choices of i in (310)–(312) we have t i = a 0 a i d a / V ( a ) , as in Equation (244) (note that the present time t corresponds to a : = 1 ). Again, for all such choices of i (except i = 0 , giving immediately t 0 = 0 ), the time t i was estimated by direct numerical computation of the integral in (244). To test the reliability of these calculations, we computed the lower bounds, upper bounds, and the asymptotic values for t i provided by Appendix Q for i = 1 / 2 , 1 , ϕ r (see, in particular, the paragraph containing Equation (A200) and the subsequent one). For the same purpose, we computed the lower and upper bounds for t i provided by Appendix R for i = 2 , , ϕ r + , ϕ m , r m (see, in particular, the paragraph containing Equation (A215) and the subsequent one). In all cases, the lower and upper bounds on t i are indistinguishable from the value provided by direct numerical computation of the previous integral. For i = 1 / 2 , 1 , ϕ r , these values are also indistinguishable from those provided by the asymptotic expressions (285)–(287) neglecting the reminders O ( μ 2 ) .
Other topics. Of course, all statements in the last two paragraphs of Section 6.6 about the field history t ϕ ( t ) and the energy conditions apply to the present framework.

6.8. The Phantom Case with Π 0 (Small) and Ω k < 0 : Sketching the Basic Results

Throughout the present Section 6.8, we keep the assumptions (198)–(200), but we prescribe
Ω k < 0
(i.e., k > 0 ). We also put three smallness conditions on μ ; the first one is Equation (211), the other two are
2 Ω k Ω r μ < 1 ,
Ω k Ω m μ < 1 .
The potential V . Again, this is given by Equation (201) and behaves as in Equation (202) for a 0 + and a + . The following facts about V are proved in Appendix U.
( α )
There are just two points a 1 , a 2 as in Equation (212), reproduced here:
a 1 , a 2 R + , a 1 < a 2 , V ( a 1 ) = V ( a 2 ) = 0 .
Moreover, we have again the inequalities (213)
V ( a ) < 0 for   a ( 0 , a 1 ) ( a 2 , + ) , V ( a ) > 0 for   a ( a 1 , a 2 ) ,
so that a 1 is a local minimum point, and a 2 is a local maximum point for V . Also, we have the (rough) inequalities
a 1 < 2 μ < a 2 .
( β )
Defining M , G as in Equation (215), we have again the inequalities M > 0 , G > 0 , G μ < 1 and the bounds (216) 2 μ / 1 + M μ < a 1 < 2 μ 1 G μ .
( γ )
We have
V ( a 1 ) < 0 .
( δ )
Let
W : = Ω r μ 2 ( a 2 ) 4 + Ω r ( a 2 ) 2 + Ω m a 2 + Υ ( a 2 ) 2 ;
then,
W > 0 .
The sign of V ( a 2 ) is determined by a comparison between W and Ω k , and it determines the sign of V on R + as in the forthcoming items ( δ 1 ) ( δ 2 ) ( δ 3 ).
( δ 1 )
We have the equivalence
W < Ω k < 0 V ( a 2 ) < 0 .
In the case (320), there is a unique point a 0 such that
a 0 R + , V ( a 0 ) = 0 ;
moreover,
a 0 < a 1 ,
V ( a ) > 0   for   a ( 0 , a 0 ) , V ( a ) < 0   for   a ( a 0 , + )
(so, we again have the situation described by Figure 3).
( δ 2 )
We have the equivalence
Ω k = W V ( a 2 ) = 0 .
In the case (324), there is a unique point a 0 such that
a 0 R + , a 0 a 2 , V ( a 0 ) = 0 ;
(i.e., V has just two zeroes a 0 , a 2 in R + ). We have again the inequality (322) a 0 < a 1 . Moreover,
V ( a ) > 0   for   a ( 0 , a 0 ) , V ( a ) < 0   for   a ( a 0 , a 2 ) ( a 2 , + )
(see Figure 6).
( δ 3 )
We have the equivalence
Ω k < W V ( a 2 ) > 0 .
In the case (327), there is a unique triplet of points a 0 , a 3 / 2 , a 5 / 2 such that
a 0 < a 3 / 2 < a 5 / 2 R + , V ( a 0 ) = V ( a 3 / 2 ) = V ( a 5 / 2 ) = 0 ;
moreover,
a 0 < a 1 < 2 μ < a 3 / 2 < a 2 < a 5 / 2 ,
V ( a ) > 0   for   a ( 0 , a 0 ) ( a 3 / 2 , a 5 / 2 ) , V ( a ) < 0   for   a ( a 0 , a 3 / 2 ) ( a 5 / 2 , + )
(see Figure 7).
( ε )
In any one of the cases ( δ 1 )–( δ 3 ), we have again the inequality (207) a 0 < 1 .
( ζ )
Let
N : = Ω m + Υ μ 3 Ω r , F : = Ω m + Ω k μ + Υ μ 3 2 Ω r + 3 Ω m μ + 6 Υ μ 4 ;
then, N > 0 , F > 0 and F μ < 1 . Moreover, in any one of the cases ( δ 1 )–( δ 3 ), we have
μ 1 + N μ < a 0 < μ ( 1 F μ ) .
Behavior of the scale factor. Let us refer to the cases ( δ 1 )–( δ 3 ) in the previous paragraph, which correspond, respectively, to the assumptions (320), (324) and (327). Recalling the definitions (146) of V = 0 , V < 0 , we have the following:
  • Under the assumption (320), we have V = 0 = { a 0 } and V < 0 = ( a 0 , + ) . The behavior of a (maximal) zero-energy solution a ( ) : t a ( t ) is similar to that described in the previous Section 6.6. So, a ( ) is defined on the whole real axis ( , + ) with image [ a 0 , + ) . There is a Big Bounce at a time that we take conventionally to be t = 0 , with a ( 0 ) = a 0 ; moreover, a ( t ) = a ( t ) and a ( t ) + (exponentially) for t .
  • Under the assumption (324), we have V = 0 = { a 0 , a 2 } and V < 0 = ( a 0 , a 2 ) ( a 2 , + ) , with V ( a 0 ) < 0 and V ( a 2 ) = 0 ; there are three types of zero-energy solutions, described hereafter.
    First type: There is a Big Bounce at a 0 , and a 2 is approached in the past and the future. We can assume the Big Bounce to occur at time t = 0 , i.e., a ( 0 ) = a 0 ; then, a ( t ) = a ( t ) for all t. The function a ( ) increases from a 0 to a 2 for t 0 and decreases from a 2 to a 0 for t 0 . The value a 2 is attained at times (see item (vi) in Section 5.4). In conclusion, a ( ) has domain ( , + ) , range [ a 0 , a 2 ) and a 2 = lim t a ( t ) . See Figure 8, where we also indicated the time t 1 > 0 such that a ( t 1 ) = a 1 ; recalling item (viii) in Section 5.4, we see that t 1 is an inflection point for the scale factor since V ( a 1 ) = 0 and V has different signs on the left and on the right of a 1 .
    Second type: a ( ) has domain ( , + ) and range ( a 2 , + ) ; a ˙ ( t ) > 0 for all real t, or a ˙ ( t ) < 0 for all real t. If a ˙ > 0 , we have a ( t ) a 2 for t and a ( t ) + for t + ; if a ˙ < 0 , the values of the limits for t are interchanged (the reason why the time required to reach a 2 is infinite is explained again by item (vi) in Section 5.4; the discussion of the time required for a ( ) to diverge is related to item (vii) in the same section). In Figure 8, we illustrate the case a ˙ > 0 .
    Third type: This corresponds to the equilibrium solution a ( t ) = a 2 for all t ( , + ) (see item (v) in Section 5.4).
  • Under the assumption (327), we have V = 0 = { a 0 , a 3 / 2 , a 5 / 2 } and V < 0 = ( a 0 , a 3 / 2 ) ( a 5 / 2 , + ) , with V ( a 0 ) < 0 , V ( a 3 / 2 ) > 0 and V ( a 5 / 2 ) < 0 ; there are two types of zero-energy solutions, described hereafter.
    First type: The function a ( ) has domain ( , + ) and range [ a 0 , a 3 / 2 ] ; it oscillates periodically with period
    T : = 2 a 0 a 3 / 2 d a V ( a ) .
    (see items (ii)–(iv) in Section 5.4; the above convergent integral gives the time required for the scale factor to pass from a 0 to a 3 / 2 , that equals the time to return to a 0 ). This type of motion is represented in Figure 9 (note that the function a ( ) has inflection points at the infinitely many times t such that a ( t ) = a 1 ; these times are not indicated in the figure).
    Second type: The function a ( ) has domain ( , + ) and range [ a 5 / 2 , + ) . There is a Big Bounce at the time when a ( ) equals a 5 / 2 , which we can assume to be t = 0 ; we have a ( t ) = a ( t ) for all real t, a ( ) is strictly increasing (resp., decreasing) on [ 0 , + ) (resp., on ( , 0 ] ), and a ( t ) + (exponentially) for t . See Figure 9 again.
  • In all the cases considered above, we have a cosmology with the time domain ( , + ) , in which the scale factor admits a strictly positive infimum. This is nonsingular in all senses considered in Section 3.2 and Section 3.15 (let us recall again Equation (69) and the related comments).
  • Since the very beginning of the present study on cosmological models with matter, radiation, and a scalar field with constant self-potential, we have focused on the zero-energy solutions t a ( t ) of the reduced system fulfilling at some time t the condition a ( t ) = 1 , a ˙ ( t ) = 1 , which is indeed equivalent to a ( t ) = 1 , a ˙ ( t ) > 0 : see the discussion after Equation (177). Such a condition can either hold or fail for the solutions described above. For example, in the case (327), the requirement that a ( t ) = 1 , a ˙ ( t ) > 0 at some time t is fulfilled by the zero-energy solutions of the first type if and only if 1 < a 3 / 2 , and the zero-energy solutions of the second type if and only if a 5 / 2 < 1 .
It should be pointed out that, for cases (324) or (327) to occur, one should assume that the basic parameters of the model values are sensibly different from those currently accepted for the standard model of cosmology; this raises doubts on the physical plausibility of these cases.

7. Polar and Cartesian Coordinates for Phantom Cosmologies with a Periodic Field Potential

In the present Section 7 (and in the subsequent Section 8) we consider a phantom scalar field, so that
σ = 1 .

7.1. Polar Coordinates (Under Periodicity Assumptions for the Field Potential)

Let us consider the Lagrangian L of Equations (121) and (122) with σ = 1 , and pass from the coordinates a > 0 and ϕ to the new coordinates r > 0 and θ , defined by
a = ( C d r ) 2 / d , ϕ = θ C d , C d : = d 2 2 .
The Lagrangian as a function of the new coordinates, again indicated with L, is
L ( r , θ , r ˙ , θ ˙ ) = 1 2 r ˙ 2 + 1 2 r 2 θ ˙ 2 + U ( r ) + r 2 W ( θ ) ,
U ( r ) : = n = 1 N ( C d r ) 2 F n ( C d r ) 2 + Ω k ( C d r ) 2 ( d 2 ) d , W ( θ ) : = C d 2 V ( θ / C d )
(the F n ’s are as in Section 4.1 and Section 4.2). Let us note that ( 1 / 2 ) r ˙ 2 + ( 1 / 2 ) r 2 θ ˙ 2 is the kinetic energy of a particle (of unit mass) in a Euclidean plane, equipped with polar coordinates; the other summands in the Lagrangian (336) can be interpreted as potential energy terms, depending on r and θ .
As a matter of fact, for a genuine interpretation of ( r , θ ) as polar coordinates it is required that29
θ R / ( 2 π Z ) ;
due to the relation between θ and ϕ in (335), (338) holds if and only if
ϕ R / ( P d Z ) , P d : = 2 π C d = 4 2 π d .
This makes some difference with respect to the general setting employed up to now, where ϕ R is the dimensionless form of the scalar field Φ F , and these two objects are connected by the relation (73) Φ = ϕ / 2 γ d G d . Indeed, to make (339) compatible with (73), we must slightly modify the general setting for the scalar field in Section 2.3, Section 3.1, Section 3.2, Section 3.3, Section 3.4 and Section 3.5, and assume30:
Φ F / ( F d Z ) , F d : = P d 2 γ d G d = 4 π d γ d G d F .
Of course, to be consistent with the viewpoint (339) and (340), we must assume that the dimensioned field potential and its dimensionless version are (smooth) functions
V : F / ( F d Z ) D , V : R / ( P d Z ) R ,
still connected through the relation (74) V ( Φ ) = ( H 2 / 2 γ d G d ) V ( ϕ ) . In the same vein, Equation (337) defines a function
W : R / ( 2 π Z ) R .
It should be noted that a function V as in Equation (341) can be identified with a periodic function F D of period F d ; similarly, the functions V , W in Equations (341) and (342) can be identified with periodic functions R R of periods P d and 2 π , respectively.
In the present Section 7 and in the subsequent Section 8, we will stick to the setting (338)–(342). The transformation ( r , θ ) ( a , ϕ ) defined by Equation (335) is a smooth diffeomorphism between R + × R / ( 2 π Z ) and R + × R / ( P d Z ) . The Lagrangian (336) is a function of the variables ( r , θ , r ˙ , θ ˙ ) R + × R / ( 2 π Z ) × R × R , and depends on the smooth functions U : R + R , W : R / ( 2 π Z ) R defined by Equation (337).

7.2. Cartesian Coordinates

Maintaining the setup of the previous Section 7.1, we can pass from the polar coordinates ( r , θ ) to the Cartesian coordinates
x : = r cos θ , y : = r sin θ .
Obviously enough, the above transformation ( r , θ ) ( x , y ) is a smooth diffeomorphism between R + × R / ( 2 π Z ) and R 2 { ( 0 , 0 ) } , whose inverse map can be described via the following equations:
r = x 2 + y 2 , cos θ = x x 2 + y 2 , sin θ = y x 2 + y 2 .
The Lagrangian (336) (337) becomes, in Cartesian coordinates,
L ( x , y , x ˙ , y ˙ ) = 1 2 x ˙ 2 + 1 2 y ˙ 2 + U ( x 2 + y 2 ) + ( x 2 + y 2 ) W ( θ ) | cos θ = x / x 2 + y 2 , sin θ = y / x 2 + y 2 .

8. An Explicitly Solvable Phantom Model with Dust and a Trigonometric Field Potential

8.1. Some Introductory Considerations

In Section 5 and Section 6, we have shown that the evolution equations of the cosmologies considered in this paper can be reduced to quadratures when the potential of the scalar field is constant (or, equivalently, in the presence of a cosmological constant). Of course, one would like to individuate solvable cosmological models with a nonconstant field potential.
In the current Section 8, we will present a phantom model with dust and a trigonometric field potential, which is explicitly solvable. A nice feature of this model is that the evolution equations, when written in Cartesian coordinates ( x , y ) following Section 7, describe two uncoupled one-dimensional systems, each one interpretable as a harmonic repulsor, a harmonic oscillator, or a free particle. In the case of two harmonic oscillators, the trajectories of the cosmological model in the ( x , y ) plane are the Lissajous curves [63,64], mainly popular for non-cosmological reasons.
In Section 8.2, we give a detailed presentation of the model, including the explicit form of the potential for the phantom scalar (see Equations (350) and (351)); admittedly, the main motivation for this choice of the potential is to produce a system with the beautiful mathematical features outlined before. In the final paragraph of the same section, we make comparisons with the previous literature. In the subsequent Section 8.3, Section 8.4, Section 8.5, Section 8.6, Section 8.7, Section 8.8, Section 8.9, Section 8.10 and Section 8.11, we analyze the solutions of the model and their physical meaning, in a number of qualitatively different cases.

8.2. Introducing the Solvable Model

Let us stick to the general framework of Section 7. The setting introduced therein is specialized in the following way:
  • There is only one perfect fluid of the dust type, i.e., with zero pressure. Thus,
    N = 1 , F 1 ( α ) = Ω m α f o r a l l α R + , ( Ω m R + )
    (to determine the expression of F 1 , we used Equation (114) with w n = 0 ); in the sequel we refer to this fluid with the generic denomination of “matter”. We assume the spatial curvature vanishes, so
    Ω k = 0 .
    Due to (346) and (347), the functions U, U of Equations (122) and (337) are constant: U ( a ) = Ω m , U ( r ) = Ω m for all a , r R + . Thus, the Lagrangian in ( a , ϕ ) coordinates and in polar coordinates ( r , θ ) (Equation (121) with σ = 1 and Equation (336)) reads, respectively, as follows:
    L ( a , ϕ , a ˙ , ϕ ˙ ) : = a d 2 a ˙ 2 + a d ϕ ˙ 2 + a d V ( ϕ ) + Ω m ,
    L ( r , θ , r ˙ , θ ˙ ) = 1 2 r ˙ 2 + 1 2 r 2 θ ˙ 2 + r 2 W ( θ ) + Ω m .
  • The function W of Equation (337), giving the potential of the scalar field in terms of the coordinate θ , is assumed to have the form
    W ( θ ) = 1 2 W 1 cos 2 θ + 1 2 W 2 sin 2 θ for   all   θ R / ( 2 π Z ) ( W 1 , W 2 R )
    (i.e., W ( θ ) = ( 1 / 4 ) ( W 1 + W 2 ) + ( 1 / 4 ) ( W 1 W 2 ) cos ( 2 θ ) for all θ ). Due to the relation between W and V in Equation (337), the prescription (350) is equivalent to
    V ( ϕ ) = 1 2 W 1 C d 2 cos 2 ( C d ϕ ) + 1 2 W 2 C d 2 sin 2 ( C d ϕ ) ;
    a further equivalent, in terms of the dimensioned potential V , is readily obtained from Equations (73) and (74).
Due to Equations (349) and (350), the Lagrangian (336) reads
L ( r , θ , r ˙ , θ ˙ ) = 1 2 r ˙ 2 + 1 2 r 2 θ ˙ 2 + 1 2 W 1 r 2 cos 2 θ + 1 2 W 2 r 2 sin 2 θ + Ω m ;
passing to the Cartesian coordinates (343), we have
L ( x , y , x ˙ , y ˙ ) = 1 2 x ˙ 2 + 1 2 y ˙ 2 + 1 2 W 1 x 2 + 1 2 W 2 y 2 + Ω m .
The corresponding energy function reads
E ( x , y , x ˙ , y ˙ ) = 1 2 x ˙ 2 + 1 2 y ˙ 2 1 2 W 1 x 2 1 2 W 2 y 2 Ω m ,
and the Lagrange equations are
x ¨ = W 1 x , y ¨ = W 2 y .
Equation (355) concerns two uncoupled systems. The x-system can be described as a harmonic repulsor if W 1 > 0 , a harmonic oscillator if W 1 < 0 , and a free particle if W 1 = 0 ; a similar description, depending on the sign of W 2 , can be given for the y-system.
In the sequel, we frequently use the vector variable
r : = ( x , y ) R 2 ,
so that the radius r = x 2 + y 2 coincides with usual norm | r | ; we will generically indicate with boldface symbols the elements of R 2 and set 0 : = ( 0 , 0 ) .
Let us recall that we are interested in the solutions t r ( t ) = ( x ( t ) , y ( t ) ) of the Lagrange equations taking values in R 2 { 0 } and fulfilling the energy constraint E = 0 (see Section 4.4). For any such curve, the instantaneous distance between the point r ( t ) and the origin gives the radius r ( t ) and, consequently, the scale factor a ( t ) via Equation (335). The angle θ ( t ) and, consequently, the field ϕ ( t ) can be obtained from Equations (344) and (335), but the related calculations will not be reproduced in the forthcoming examples.
Let us return to the constraint E = 0 . Due to Equation (354), this reads
Ω m = 1 2 x ˙ 2 + 1 2 y ˙ 2 1 2 W 1 x 2 1 2 W 2 y 2 ;
the right-hand side of (357) is automatically constant along any solution of the Lagrange equations (355), and we can use (357) to determine Ω m for any given solution. However, we must keep in mind our requirement Ω m > 0 . If W 1 < 0 and W 2 < 0 (two harmonic oscillators), the right-hand side of (357) is positive along any solution of the Lagrange equations with values in R 2 { 0 } . The positivity of the right-hand side is not granted for different choices of the signs of W 1 , W 2 ; in these cases, the positivity requirement selects a distinguished subclass of solutions of (355).
In the forthcoming sections, we will present several examples of the above general scheme, corresponding to specific choices for W 1 and W 2 . None of the cases treated in the sequel will include an explicit analysis of the energy conditions; however, in most cases, there are violations of such conditions of some of the types in Equations (97) and (98).31
Connections with the previous literature. The solvable model with a phantom scalar and dust proposed in the present Section 8 was considered by Capozziello, Piedipalumbo, Rubano, and Scudellaro [25] in the special case W 1 = 0 (and d = 3 )32. Our present model with arbitrary W 1 and W 2 produces a larger variety of behaviors in the solutions of the model, allowing, e.g., for the Lissajous cosmologies mentioned before. Since the choice W 1 = 0 has already been treated, we will not reconsider it in the sequel (and we will not even discuss the very similar case W 2 = 0 ).
A system decoupling in two harmonic repulsors/oscillators/free particles also arises from certain FLRW cosmological models with a canonical scalar field and (possibly) dust, where the self-interaction potential for the scalar field has an exponential form; these were analyzed by de Ritis, Marmo, Platania, Rubano, Scudellaro and Stornaiolo [26] (without dust); Dereli and Tucker [77] (again, with no dust); Rubano and Scudellaro [28]; Piedipalumbo, Scudellaro, Esposito and Rubano [27], Fré; and Sagnotti and Sorin [30] (without dust: see Section 3.2.1 in this reference), as well as in subsequent works by some of us: the Ph.D. thesis [31] and the paper [32] with D. Fermi. In all these works, the separation coordinates yielding the above decoupling have a different geometrical meaning in comparison with the coordinates in our Equation (355): here, we use two orthonormal Cartesian coordinates in a Euclidean plane, while the separation coordinates in the cited works must be interpreted as orthonormal coordinates in flat, two-dimensional Minkowski space33,34.
For completeness, let us mention that some (not explicitly solvable) models with two coupled repulsors or oscillators have been considered by Castagnino, Giacomini, and Lara [79]; Carroll, Hoffman, and Trodden [20]; and Faraoni [80] as exact or approximate descriptions for FLRW cosmologies with canonical or phantom scalar fields. While repeating that the x and y subsystems in our Equation (355) are uncoupled, let us confirm that they describe exactly an FLRW cosmology with a phantom scalar (and dust).
To continue, we refer to the paper by Chervon and Panina [40] about solutions of FLRW cosmological models with a phantom scalar (and no type of matter); here, the authors present exact solutions for a list of self-interaction potentials, individuated by the superpotential method (see the Introduction). The list in [40] contains potentials of exponential and of other types but does not include the trigonometric potential analyzed in the present Section 8.
A trigonometric self-interaction potential was considered by Ivanov [38] within an FLRW cosmological model with a canonical scalar field (and no matter content); an exact solution of this model was derived in the cited work (and reconsidered in the book [37] via the generating function method; see again the Introduction).

8.3. The Case W i = λ i 2 > 0 ( i = 1 , 2 )

If W 1 and W 2 are both positive we can represent them as indicated in the title, with λ i R + ; in this case, Equation (355) describes two harmonic repulsors.
In the forthcoming Section 8.4, we will treat the case λ 1 = λ 2 ; this gives once more a model with a constant potential for the phantom field (i.e., with a cosmological constant), which deserves consideration since it admits a particularly nice geometric interpretation. In the subsequent Section 8.5, we will treat the general case with λ 1 and λ 2 considered arbitrary, which gives a nonconstant field potential whenever λ 1 λ 2 .

8.4. Cosmologies with λ 1 = λ 2

Let
W 1 = W 2 = λ 2 , λ R + .
This prescription corresponds to a constant and positive field potential since Equations (350), (351), and (358) give
W ( θ ) = const . = 1 2 λ 2 , V ( ϕ ) = const . = 1 2 λ C d 2 .
The case with a constant field potential was discussed in general in Section 5, for arbitrary choices of the matter fluids and of the curvature; according to Equations (130)–(132) at the beginning of the cited section, the present form (359) of V corresponds to a positive cosmological constant
Λ = d ( d 1 ) λ 2 H 2 4 C d 2 = 2 ( d 1 ) λ 2 H 2 d .
Of course, we could solve the cosmological model under analysis specializing the framework of Section 5 to the present subcase with dust and zero curvature; however, the same conclusions can be obtained in the present setting in a simpler and more geometrical way. For this purpose, we write the evolution equations (355) and the expression (357) for the density parameter with W i = λ 2 ( i = 1 , 2 ), using the vector variable r : = ( x , y ) and the standard norm | | of R 2 . In this way, we obtain
r ¨ = λ 2 r ,
Ω m = 1 2 | r ˙ | 2 1 2 λ 2 | r | 2 .
(Let us repeat that | r | coincides with the usual radius r.) As recalled in Appendix V, a function t r ( t ) is a maximal solution of Equation (361) if and only if one of the following cases (i)–(iii) occurs:
(i)
We have
r ( t ) = H e λ t for   all   t R ( H R 2 ) ;
(ii)
We have
r ( t ) = K e λ t for   all   t R ( K R 2 ) ;
(iii)
Up to a time translation t t + const., we have
r ( t ) = A cosh ( λ t ) i + B sinh ( λ t ) j for   all   t R ,
where
A , B R , A , B 0 ; i , j R 2 , | i | = | j | = 1 , i j = 0
( is the standard inner product of R 2 ; the previous conditions on ( i , j ) indicate that this pair is an orthonormal basis of R 2 ).
In the sequel, we will not consider the trivial cases (i) and (ii) and we will fix the attention on case (iii), which we will discuss with the nondegeneracy assumption
A , B > 0 .
Assuming (367), let us write Equation (365) as
r ( t ) = ξ ( t ) i + η ( t ) j for   all   t R , ξ ( t ) : = A cosh ( λ t ) , η ( t ) : = B sinh ( λ t ) ) ;
for all t R we have
ξ 2 ( t ) A 2 η 2 ( t ) B 2 = 1 , ξ ( t ) > 0 ,
and we recognize that the point r ( t ) describes a branch of a hyperbola (see Figure 10). Let us remark that r ( t ) 0 for all t R .
To continue, let us note that Equations (362) and (365) imply
Ω m = 1 2 ( B 2 A 2 ) λ 2 ;
we want Ω m > 0 , which is equivalent to
B > A .
Due to (365) and (366), the radius r ( t ) = | r ( t ) | is such that
r ( t ) = A 2 cosh 2 ( λ t ) + B 2 sinh 2 ( λ t ) = A 2 + ( A 2 + B 2 ) sinh 2 ( λ t ) for   all   t R .
From here, it is evident that t = 0 is a point of absolute minimum for the function t r ( t ) , with
r ( 0 ) = A ;
it is also evident that r ( t ) = r ( t ) and that the function r ( ) is strictly decreasing on ( , 0 ] , strictly increasing on [ 0 , + ) (see Figure 11). Finally,
r ( t ) 1 2 A 2 + B 2 e λ t for   t .
The above features of the radius function are easily rephrased in terms of the scale factor a ( t ) = ( C d r ( t ) ) 2 / d (recall Equation (335)). Thus, the function a ( ) has an absolute minimum at t = 0 , with a ( 0 ) = ( C d A ) 2 / d ; this function decreases strictly on ( , 0 ] , increases strictly on [ 0 , + ) , and grows exponentially for t .
To summarize, the present cosmological model presents a Big Bounce at t = 0 ; note that the minimum value a ( 0 ) (or r ( 0 ) ) can be made arbitrarily small by choosing a sufficiently small A. Needless to say, the strict positivity of the minimum a ( 0 ) ensures this cosmological model to be nonsingular, in all senses considered in Section 3.2 and Section 3.15 (let us recall again Equation (69) and the related comments).

8.5. Cosmologies with λ 1 and λ 2 Arbitrary

We now discuss the general case
W i = λ i 2 , λ i R + ( i = 1 , 2 )
(this will also give an alternative treatment of the special case λ 1 = λ 2 , already discussed in the previous section by different means). Equation (355) reads
x ¨ = λ 1 2 x , y ¨ = λ 2 2 y ;
a function t r ( t ) = ( x ( t ) , y ( t ) ) is a maximal solution of these equations if and only if
x ( t ) = A 1 e λ 1 t + B 1 e λ 1 t , y ( t ) = A 2 e λ 2 t + B 2 e λ 2 t for   all   t R ,
where
A 1 , B 1 , A 2 , B 2 R .
For simplicity, from now on, we just consider the nondegenerate case
A i 0 , B i 0 ( i = 1 , 2 ) .
Let us investigate conditions under which x ( t ) , y ( t ) or r ( t ) vanish. For any t R , we readily find the following equivalences:
x ( t ) = 0 B 1 A 1 < 0 , t = 1 2 λ 1 ln B 1 A 1 ;
y ( t ) = 0 B 2 A 2 < 0 , t = 1 2 λ 2 ln B 2 A 2 .
Clearly, the function r ( ) has a zero if and only if the conditions on the parameters A i , B i indicated above hold, and the above zeroes of x ( ) and y ( ) coincide; the last condition means equality of the times ( 1 / 2 λ i ) ln B i / A i = ( 1 / 2 ) ln ( B i / A i ) 1 / λ i ( i = 1 , 2 ). Thus,
t R such   that r ( t ) = 0 B 1 A 1 < 0 , B 2 A 2 < 0 , B 1 A 1 1 / λ 1 = B 2 A 2 1 / λ 2 .
From now on, we assume
B 1 A 1 > 0 , o r B 2 A 2 > 0 , o r B 1 A 1 < 0 , B 2 A 2 < 0 , B 1 A 1 1 / λ 1 B 2 A 2 1 / λ 2 ;
this means that we are denying the conditions on A i , B i in (382), so that
r ( t ) 0 for   all   t R .
To continue, let us note that Equation (357) with W i = λ i 2 and Equation (377) give
Ω m = 2 ( λ 1 2 A 1 B 1 + λ 2 2 A 2 B 2 ) ;
we want Ω m > 0 , so from now on we also assume
λ 1 2 A 1 B 1 + λ 2 2 A 2 B 2 < 0 .
Let us proceed to the analysis of the (never vanishing) radius r ( t ) = | r ( t ) | . Equation (377) gives
r ( t ) = A 1 2 e 2 λ 1 t + A 2 2 e 2 λ 2 t + B 1 2 e 2 λ 1 t + B 2 2 e 2 λ 2 t + 2 ( A 1 B 1 + A 2 B 2 ) for   t R ;
from here, we infer that
r ˙ ( t ) = R ( t ) r ( t ) for   t R , R ( t ) : = λ 1 A 1 2 e 2 λ 1 t + λ 2 A 2 2 e 2 λ 2 t λ 1 B 1 2 e 2 λ 1 t λ 2 B 2 2 e 2 λ 2 t .
The function R : R R defined above has derivative R ˙ ( t ) = 2 ( λ 1 2 A 1 2 e 2 λ 1 t + λ 2 2 A 2 2 e 2 λ 2 t + λ 1 2 B 1 2 e 2 λ 1 t + λ 2 2 B 2 2 e 2 λ 2 t ) > 0 , and R ( t ) for t . Thus, R is strictly increasing with range ( , + ) ; therefore, there is a unique
t 0 R such   that   R ( t 0 ) = 0 ,
and R ( t ) < 0 for t < t 0 , R ( t ) > 0 for t > t 0 .
Since sign r ˙ ( t ) = sign R ( t ) for all real t, we conclude the following: the radius function t r ( t ) admits t 0 as a point of absolute minimum, is strictly decreasing on ( , t 0 ] and strictly increasing on [ t 0 , + ) (we know that r ( t ) > 0 for all t, this holds, in particular, for the minimum value r ( t 0 ) ). Finally, let us remark that Equation (387) implies
r ( t ) B e λ t for   t , r ( t ) A e λ t for   t + ,
where λ , A , B are defined as follows: If λ 1 > λ 2 , then λ : = λ 1 , A : = | A 1 | , and B = | B 1 | ; if λ 1 < λ 2 , then λ : = λ 2 , A : = | A 2 | , and B = | B 2 | ; if λ 1 = λ 2 , then, λ : = λ 1 = λ 2 , A : = A 1 2 + A 2 2 and B : = B 1 2 + B 2 2 .
The above results on the radius function r ( t ) imply similar statements on the scale factor a = a ( t ) given by (335). Thus, the function a ( ) has a (strictly positive) absolute minimum at t = t 0 , decreases strictly before t 0 , increases strictly after t 0 , and diverges exponentially for t . Again, we have a model with a Big Bounce; this is nonsingular in all senses considered in Section 3.2 and Section 3.15, due to Equation (69) and related comments.
In Figure 12 and Figure 13, we report, respectively, the range of the curve t r ( t ) and the graph of the radius t r ( t ) for a special choice of the parameters, compatible with (383) and (386).

8.6. The Case W i = ω i 2 < 0 ( i = 1 , 2 ): Lissajous Cosmologies

We now address the case where W 1 and W 2 are both negative and represent them as
W i = ω i 2 , ω i R + ( i = 1 , 2 ) .
Then, Equation (355) reads
x ¨ = ω 1 2 x , y ¨ = ω 2 2 y .
and describes two uncoupled harmonic oscillators with pulsations ω i . A function t r ( t ) = ( x ( t ) , y ( t ) ) is a maximal solution of (392) if and only if, up to a time translation t t + const., it can be expressed as follows:
x ( t ) = A 1 sin ( ω 1 t ) , y ( t ) = A 2 sin ( ω 2 t + ψ ) for   all   t R ,
where
A 1 , A 2 R , A 1 , A 2 0 , ψ R / ( 2 π Z ) .
The curves in R 2 corresponding to the solutions of (392) are the familiar Lissajous curves; as is well known, these curves exhibit a variety of different qualitative behaviors, depending mainly on the ratio ω 2 / ω 1 and on the phase ψ (see the already cited references [63,64]). In the sequel, for particular values of ω 2 / ω 1 , we will represent the solutions of (392) in specific forms, different from (393), although equivalent to it, which will be more useful to describe the cases in consideration.
Let us recall that, in view of our cosmological model, we are interested in curves (or portions of such curves) with r ( t ) 0 . Let us also remark that Equation (357) with W i = ω i 2 and Equation (393) imply the following expression for the density parameter:
Ω m = 1 2 ω 1 2 A 1 2 + 1 2 ω 2 2 A 2 2 .
In the forthcoming Section 8.7 and Section 8.8, we will discuss as examples the cases ω 1 = ω 2 and ω 2 = 2 ω 1 ; in Section 8.9, we will present some general facts about Lissajous cosmologies (and we will also mention another example).

8.7. Lissajous Cosmologies with ω 1 = ω 2

Let us consider the case
ω 1 = ω 2 ω R + .
This corresponds to a constant and negative field potential since Equations (350), (351), (391), and (396) give
W ( θ ) = const . = 1 2 ω 2 , V ( ϕ ) = const . = 1 2 ω C d 2 .
Let us repeat that the case with a constant field potential was discussed in general in Section 5, for arbitrary choices of the matter fluids and of the curvature; according to Equations (130)–(132) at the beginning of the cited section, the present form (397) of V corresponds to a negative cosmological constant
Λ = d ( d 1 ) ω 2 H 2 4 C d 2 = 2 ( d 1 ) ω 2 H 2 d .
Apart from the negative sign of the potential, the situation we are considering here is conceptually similar to that discussed in Section 8.4 from which we reproduce the following comment: we could solve the cosmological model under analysis by specializing the framework of Section 5 to the present subcase with dust and zero curvature, but the same conclusions can be obtained in a simpler and more geometrical way from the present setting. For this purpose, we write the evolution equations (355) and the energy constraint (357) with W i = ω 2 for i = 1 , 2 , using the vector variable r : = ( x , y ) (and the standard norm | | of R 2 ); in this way, the cited equations become
r ¨ = ω 2 r ,
Ω m = 1 2 | r ˙ | 2 + 1 2 ω 2 | r | 2 .
As recalled in Appendix W, a function t r ( t ) is a maximal solution of Equation (399) if and only if, up to a time translation t t + const., it has the form
r ( t ) = A cos ( ω t ) i + B sin ( ω t ) j for   all   t R ,
involving the parameters
A , B R s . t . 0 A B ; i , j R 2 s . t . | i | = | j | = 1 , i j = 0
(again, denotes the standard inner product of R 2 ; the above conditions on ( i , j ) indicate that this pair is an orthonormal basis of R 2 ). It is evident that the function t r ( t ) in Equation (401) is periodic of period 2 π / ω .
From now on, we consider solutions of the form (401) and (402) with the nondegeneracy property
0 < A B .
Let us rephrase Equation (401) as
r ( t ) = ξ ( t ) i + η ( t ) j for   t R , ξ ( t ) : = A cos ( ω t ) , η ( t ) = B sin ( ω t ) .
It is evident that
ξ 2 ( t ) A 2 + η 2 ( t ) B 2 = 1 ,
so we have an ellipse: the minor semiaxis has direction i and length A, while the major semiaxis has direction j and length B (see Figure 14). Of course, r ( t ) 0 for all t R .
The radius r ( t ) = | r ( t ) | is given by
r ( t ) = A 2 cos 2 ( ω t ) + B 2 sin 2 ( ω t ) = A 2 + ( B 2 A 2 ) sin 2 ( ω t ) for   all   R ;
this is a periodic function of time, with period π / ω (see Figure 15). Clearly, we have
min t R r ( t ) = A ; i f A < B , r ( t ) = A t = n π ω ( n Z ) .
max t R r ( t ) = B ; i f A < B , r ( t ) = B t = 1 2 + n π ω ( n Z ) .
Let us recall that the scale factor a = a ( t ) is related to r ( t ) through Equation (335); we have again a periodic function of period π / ω , with
min t R a ( t ) = ( C d A ) 2 / d , max t R a ( t ) = ( C d B ) 2 / d .
Of course, this cosmological model is nonsingular in all senses considered in Section 3.2 and Section 3.15, due to Equation (69) and related comments.
Choosing appropriately A (resp., B), we can make the minimum of a ( ) arbitrarily small (resp., the maximum of a ( ) arbitrarily large); moreover, the period π / ω can be made arbitrarily long by choosing a sufficiently small ω .
Concerning the mass density parameter Ω m , from Equations (400)–(402), we readily infer
Ω m = 1 2 ω 2 ( A 2 + B 2 ) .
The special case A = B . In this case, the ellipse described by Equation (405) becomes a circumference of radius A; thus, r ( t ) = const. = A and a ( t ) = const. = ( C d A ) 2 / d . Consequently, the spacetime line element in Equation (17) becomes d s 2 = d τ 2 + ( C d A ) 4 / d d 2 ; let us recall that we are assuming zero spatial curvature, so d 2 is the line element of the flat d-dimensional Euclidean space. Due to these facts, spacetime is the flat, ( d + 1 ) -dimensional Minkowski space. According to the Einstein equations, the total stress–energy tensor must be zero in this case; to obtain this result, contributions to stress–energy tensor from dust and the phantom scalar must counterbalance.

8.8. Lissajous Cosmologies with ω 2 = 2 ω 1

Let us consider the case
ω 1 = ω , ω 2 = 2 ω , ω R + .
The maximal solutions t r ( t ) = ( x ( t ) , y ( t ) ) of (392) can be represented as in Equation (393) with the above values for the pulsations. We will consider only nondegenerate cases with A 1 , A 2 > 0 and we will employ the reparametrizations A 1 = 2 R A , A 2 = A ; thus,
x ( t ) = 2 R A sin ( ω t ) , y ( t ) = A sin ( 2 ω t + ψ ) for   t R ,
where
A , R R , A , R > 0 , ψ R / ( 2 π Z ) .
For future use, we also introduce the parameter
ϑ : = 2 π arccos ( R 2 ) ( 0 , 1 ) i f R ( 0 , 1 ) ,
whose relevance will appear in the sequel (with the above limitation on R). The function t r ( t ) in Equation (412) is clearly periodic, of period 2 π / ω . As is well known, the shape of the Lissajous curve described by r ( t ) is very sensitive to the phase ψ . Figure 16 describes the familiar curves corresponding to the choices ψ = π / 2 , π / 4 , 0 , π / 4 ( m o d . 2 π ); by a reflection y y we also obtain the curves with ψ = π / 2 , 3 π / 4 , π , 5 π / 4 ( m o d . 2 π ), since y ( t ; ψ + π ) = y ( t ; ψ ) . For any ψ , Equation (395) for the mass density parameter gives
Ω m = 2 ω 2 ( 1 + R 2 ) A 2 .
Hereafter, we will consider the cases ψ = π / 2 , 0 ( m o d . 2 π ) as examples, which will be treated in detail.
The case ψ = π / 2 ( m o d . 2 π ). We have
x ( t ) = 2 R A sin ( ω t ) , y ( t ) = A cos ( 2 ω t ) for   t R
(with A , R > 0 , as assumed before); the functions t x ( t ) , y ( t ) never vanish simultaneously, so
r ( t ) 0 for   all   R .
To continue, let us remark that y ( t ) / A = cos ( 2 ω t ) = 2 sin 2 ( ω t ) 1 = 2 ( x ( t ) / 2 R A ) 2 1 ; so,
y ( t ) = x 2 ( t ) 2 R 2 A A , 2 R A x ( t ) 2 R A .
Thus, the Lissajous curve we are considering is a parabolic segment with the y axis as a symmetry axis, vertex v and endpoints e , where
v : = ( 0 , A ) , e : = ( 2 R A , A ) , e + : = ( 2 R A , A )
(see Figure 17 and Figure 19). Let us allow t to range in the time interval [ π / ( 2 ω ) , 3 π / ( 2 ω ) ] , whose length corresponds to one period of the function r ( ) ; then the point r ( t ) describes twice the parabolic segment, from e to e + and back to e . In particular,
r π 2 ω = e , r 0 = v , r π 2 ω = e + , r π ω = v , r 3 π 2 ω = e .
Of course, a similar description of the behavior of r ( t ) can be given for t ranging in any time interval [ ( 2 n 1 / 2 ) π / ω , ( 2 n + 3 / 2 ) π / ω ] ( n Z ).
Due to (416), the (never vanishing) radius r ( t ) is given by
r ( t ) = 4 R 2 sin 2 ( ω t ) + cos 2 ( 2 ω t ) A = 2 R 2 2 R 2 cos ( 2 ω t ) + cos 2 ( 2 ω t ) A ;
this is a periodic function of time, of period π / ω , and r ( t ) = r ( t ) . It is readily found that, for all t R ,
r ˙ ( t ) = 2 ω sin ( 2 ω t ) r ( t ) R 2 cos ( 2 ω t ) A 2 .
The sign of r ˙ ( t ) is readily studied, and one concludes the following on the behavior of r ( t ) over one period, e.g., on the interval [ π / ( 2 ω ) , π / ( 2 ω ) ] :
Case R 1 : r ( t ) behaves as follows on [ π / ( 2 ω ) , π / ( 2 ω ) ] . The times π / ( 2 ω ) are absolute maximum points, 0 is an absolute minimum point, and
r π 2 ω = 1 + 4 R 2 A , r 0 = A .
The function t r ( t ) is strictly decreasing on [ π / ( 2 ω ) , 0 ] and strictly increasing on [ 0 , π / ( 2 ω ) ] . From a geometrical viewpoint, the above results indicate that the points of the parabolic segment maximizing the distance from the origin are r ( π / ( 2 ω ) ) = e ; the point minimizing the distance from the origin is r ( 0 ) = v . See Figure 17 and Figure 18.
Case 0 < R < 1 : Let us consider the times
1 2 ω arccos ( R 2 ) = ϑ π 4 ω ,
with ϑ as in Equation (414); then,
π 2 ω < ϑ π 4 ω < 0 < ϑ π 4 ω < π 2 ω ,
and the function t r ( t ) behaves as follows over one period [ π / ( 2 ω ) , π / ( 2 ω ) ] . The times π / ( 2 ω ) are absolute maximum points and 0 is a relative maximum point, with r ( π / ( 2 ω ) ) , and r ( 0 ) as in (423). The times ϑ π / ( 4 ω ) are absolute minimum points, with
r ϑ π 4 ω = R 2 R 2 A .
The function t r ( t ) is strictly decreasing on [ π / ( 2 ω ) , ϑ π / ( 4 ω ) ] and on [ 0 , ϑ π / ( 4 ω ) ] , strictly increasing on [ ϑ π / ( 4 ω ) , 0 ] and on [ ϑ π / ( 4 ω ) , π / ( 2 ω ) ] .
From a geometrical viewpoint, the above results indicate that the points of the parabolic segment maximizing the distance from the origin are e = r ( π / ( 2 ω ) ) . The points minimizing the distance from the origin are
m : = r ϑ π 4 ω = ( 2 R 1 R 2 A , R 2 A )
(the above explicit expressions for m follow from Equations (416) and (424) for x ( ) , y ( ) and for the times involved). See Figure 19 and Figure 20. Both for R 1 and for 0 < R < 1 , the behavior of the function t r ( t ) on any interval [ ( 1 / 2 + n ) π / ω , ( 1 / 2 + n ) π / ω ] ( n Z ) is of course similar to that described above for n = 0 . One can choose A and R so that the absolute minimum of r ( t ) is arbitrarily small and the absolute maximum of r ( t ) is arbitrarily large. Also, the periods involved can be made arbitrarily large by choosing ω sufficiently small.
Of course, analogous conclusions hold for the scale factor a = a ( t ) given by (335): this is a periodic function of period π / ω , with maxima and minima at the times indicated above for the function r = r ( t ) . Equation (69) applies once more to the model under examination and ensures that this cosmology is nonsingular in all senses considered in Section 3.2 and Section 3.15.
The case ψ = 0 ( m o d . 2 π ). Equation (412) gives
x ( t ) = 2 R A sin ( ω t ) , y ( t ) = A sin ( 2 ω t ) for   t R
(with A , R > 0 , as assumed before); the corresponding Lissajous curve is described in Figure 21 and Figure 24. For all t R , we have
r ( t ) = r ( t ) ;
moreover,
r ( t ) = 0 t = t k ( k Z ) , t k : = k π ω .
The existence of zeroes of r ( ) for A , R > 0 is an exceptional feature of the present case ψ = 0 ( m o d . 2 π ), which also appears for ψ = π ( m o d . 2 π ): on the contrary, for any A , R > 0 and for any ψ R / ( 2 π Z ) { 0 , π ( m o d . 2 π ) } , the solution described by Equation (412) is such that r ( t ) 0 for all t R 35.
Keeping in mind (430), we build the present cosmological model restricting the solution r ( ) to a maximal interval J such that r ( t ) 0 for all t J ; we can take J = ( t k , t k + 1 ) for any k Z . The choice of k is immaterial, and we will set k = 0 , i.e.,
J : = 0 , π ω ;
this will be our time domain from now on.
Due to (428), the radius r ( t ) is given by
r ( t ) = 4 R 2 sin 2 ( ω t ) + sin 2 ( 2 ω t ) A = 1 + 2 R 2 2 R 2 cos ( 2 ω t ) cos 2 ( 2 ω t ) A ;
we have
r ( t ) 0 f o r t 0 , π ω .
It is readily found that, for all t J ,
r ˙ ( t ) = 2 ω sin ( 2 ω t ) r ( t ) R 2 + cos ( 2 ω t ) A 2 .
The sign of r ˙ ( t ) is readily studied, and one concludes the following on the behavior of r ( t ) over the interval J :
Case R 1 : t = π / ( 2 ω ) is an absolute maximum point and
r π 2 ω = 2 R A .
The function t r ( t ) is strictly increasing on ( 0 , π / ( 2 ω ) ] and strictly decreasing on [ π / ( 2 ω ) , π / ω ) ; see Figure 22. More geometrically, considering the curve { r ( t ) | t J } (see again Figure 21), we can say that the point of this curve maximizing the distance from the origin is
r π 2 ω = ( 2 R A , 0 ) .
With appropriate choices of A and R, the maximum r ( π / ( 2 ω ) ) can be made arbitrarily large.
Case 0 < R < 1 : Let us consider the times
π 2 ω 1 2 ω arccos ( R 2 ) = 1 ϑ 2 π 2 ω ,
with ϑ as in Equation (414). Then,
0 < 1 ϑ 2 π 2 ω < π 2 ω < 1 + ϑ 2 π 2 ω < π ω ,
and the function t r ( t ) behaves as follows on J . The times ( 1 ϑ / 2 ) π / ( 2 ω ) are absolute maximum points and π / ( 2 ω ) is a relative minimum point, with r ( π / ( 2 ω ) ) as in (435) and
r 1 ϑ 2 π 2 ω = ( 1 + R 2 ) A .
The function t r ( t ) is strictly increasing on ( 0 , ( 1 ϑ / 2 ) π / ( 2 ω ) ] and on [ π / ( 2 ω ) , ( 1 + ϑ / 2 ) π / ( 2 ω ) ] , strictly decreasing on [ ( 1 ϑ / 2 ) π / ( 2 ω ) , π / ( 2 ω ) ] and on [ ( 1 + ϑ / 2 ) π / ( 2 ω ) , π / ω ) ; see Figure 23.
More geometrically, considering the curve { r ( t ) | t J } (see again Figure 24), we can say that the points of this curve maximizing the distance from the origin are
p ± : = r 1 ϑ 2 π 2 ω = 2 R 1 + R 2 A , ± 1 R 4 A ;
the latter distance has a relative minimum point at r ( π / ( 2 ω ) ) as in (436).
With appropriate choices of A and R, the maximum r ( ( 1 ϑ / 2 ) π / ( 2 ω ) ) can be made arbitrarily large, and the relative minimum r ( π / ( 2 ω ) ) arbitrarily small. Finally, the time interval J can be made arbitrarily long by choosing a sufficiently small ω .
Our conclusions about the radius r = r ( t ) have obvious analogs for the scale factor a = a ( t ) , given by (335). Both for R 1 and for 0 < R < 1 , a ( t ) vanishes for t 0 , π / ω ; so, the evolution of this model starts with a Big Bang at t = 0 , and ends with a Big Crunch at t = π / ω . If R 1 , a ( ) has an absolute maximum point at time π / ( 2 ω ) ; if 0 < R < 1 , a ( ) has two absolute maximum points at times ( 1 ϑ / 2 ) π / ( 2 ω ) , and a relative minimum point at π / ( 2 ω ) . The occurrence of a Big Bang and a Big Crunch is strictly related to the existence of zeroes of r ( ) and therefore reflects the exceptional feature of the case ψ = 0 ( m o d . 2 π ) mentioned after Equation (430).
Let us remark that the model under consideration is singular in any one of the senses considered in Section 3.2 and Section 3.15: in fact, any one of conditions (65)–(68) is violated in this case (needless to say, the times t appearing in the cited equations are 0 and π / ω in the present case).

8.9. On General Lissajous Cosmologies

Let us spend a few words on the general case (391) W i = ω i 2 ( i = 1 , 2 ), where ω 1 , ω 2 are chosen arbitrarily in R + . We know that the maximal solutions t r ( t ) = ( x ( t ) , y ( t ) ) of the evolution equations (392) have the form (393), depending on the constants A 1 , A 2 , ψ as in (394). Here, we only consider the nondegerate case A 1 , A 2 0 , so that
x ( t ) = A 1 sin ( ω 1 t ) , y ( t ) = A 2 sin ( ω 2 t + ψ ) for   all   R , ( A 1 , A 2 > 0 , ψ R / ( 2 π Z ) ) .
Let us also recall that the zero-energy constraint gives the expression (395) for the mass density parameter.
Periodicity conditions for the function r ( ) . From Equation (441), it follows that the function t r ( t ) is periodic if and only if ω 2 / ω 1 is rational, i.e., if and only if
ω 2 ω 1 = N 2 N 1 N 1 , N 2 { 1 , 2 , 3 , . } ;
in this case,
r ( t + T ) = r ( t ) for   all   R , T : = 2 N 1 π ω 1 = 2 N 2 π ω 2 .
In the same case, we have a periodic cosmological model provided that r ( t ) 0 for all t R ; whether or not this happens depends on the phase ψ , as illustrated in the next paragraph.
Of course, the cases considered in Section 8.7 and Section 8.8 fit condition (442). As a further example, let
ω 1 = 3 ω , ω 2 = 4 ω , ( ω R + ) ;
Equation (442) clearly holds with N 1 = 3 , N 2 = 4 , and Equation (443) ensures the function R t r ( t ) to be periodic of period T : = 2 π / ω . In Figure 25, we illustrate the range of this function for several choices of the phase ψ (assuming for simplicity A 1 = A 2 ); we will return to this example in the final paragraph of the present section.
If ω 2 / ω 1 is irrational, it can be shown that the range of the (non-periodic) function R t r ( t ) densely fills a rectangle centered at the origin in the ( x , y ) plane [64]: see Figure 26. Again, the function r ( ) is never vanishing or possesses zeroes, depending on the phase ψ . If r ( ) never vanishes, the radius r ( t ) = | r ( t ) | (hence, the scale factor a ( t ) in (335)) is always positive but becomes arbitrarily small at suitable times, due to the above-mentioned density result; it should be mentioned that the functions R t r ( t ) , r ( t ) , a ( t ) are almost periodic, in the rigorous sense given usually to this expression (see [81], page 1).
When does r ( t ) vanish? We will discuss the problem for arbitrary ω 1 and ω 2 , with ω 2 / ω 1 either rational or irrational; for this purpose, we choose any real representative ψ of the phase ψ , so that
ψ R , ψ = ψ ( m o d . 2 π ) .
Let us first individuate the times t at which x ( t ) and y ( t ) vanish separately. From Equation (441) (with ψ replaced by ψ ), one easily infers the following, for any t R :
x ( t ) = 0 t = t ˜ h ( h Z ) , t ˜ h : = h π ω 1 ;
x y ( t ) = 0 t = t ^ ( Z ) , t ^ : = π ω 2 ψ ω 2 .
This implies the following statement about r ( t ) = ( x ( t ) , y ( t ) ):
t R s . t . r ( t ) = 0 , h Z s . t . t ^ = t ˜ h , h Z s . t . ψ = ζ h ,
where
ζ h : = h ω 2 ω 1 π for , h Z .
We can rephrase the result (448) in terms of the phase ψ R / ( 2 π Z ) , which is represented by ψ ; after noting that ζ h (mod. 2 π ) equals ζ 0 h (mod. 2 π ) or ζ 1 h (mod. 2 π ) for Z even or odd, respectively, we conclude that
t R s . t . r ( t ) = 0 ψ Z , Z : = { ζ 0 h , ζ 1 h ( m o d . 2 π ) | h Z } .
Let us note that, for any choice of ω 1 , ω 2 , we have Z 0 ( m o d . 2 π ) (if ψ = 0 ( m o d . 2 π ) , it is evident from (441) that r ( 0 ) = 0 ). To continue our discussion on the zeroes of r ( ) , we must distinguish the cases with ω 2 / ω 1 rational or irrational.
The case of rational ω 2 / ω 1 : In this case, which is described by Equation (442), Z is a finite subset of R / ( 2 π Z ) 36. For each phase ψ Z , the periodicity of r ( ) (see Equation (443)) implies that there are infinitely many times t R such that r ( t ) = 0 .
The case of irrational ω 2 / ω 1 : In this case, the subset Z is infinite and dense in R / ( 2 π Z ) ; however, Z is countable, so its Lebesgue measure is zero. For each phase ψ Z , there is a unique time t R such that r ( t ) = 0 37.
A final comment: We have just noted that, depending on the rational or irrational nature of ω 2 / ω 1 , the phases ψ giving rise to zeroes for r ( ) form a finite or an infinite but countable, zero-measure subset of R / ( 2 π Z ) . In both cases, these phases must be regarded as exceptional.
Nonsingular Lissajous cosmologies. Given ω 1 , ω 2 R + , let us consider a phase ψ outside the exceptional set Z of Equation (450) and any two amplitudes A 1 , A 2 R + . Then, r ( t ) 0 for all t R , and we have a cosmological model with time domain ( , + ) , in which a ( t ) given by Equation (335) is nonvanishing for all real t.
We claim that this cosmological model is nonsingular, in all senses considered in Section 3.2 and Section 3.15. Indeed, if ω 2 / ω 1 is rational, the function r ( ) on R is periodic (let us recall again Equation (443)), and this implies periodicity of the functions r ( ) , a ( ) : R R + . Thus, inf t R a ( t ) = min t R a ( t ) a 0 > 0 , and this suffices to infer all nonsingularity conditions in the above sections (let us recall once more Equation (69) and the related comments).
If ω 2 / ω 1 is irrational, the functions r ( ) , r ( ) , a ( ) on R are not periodic, and inf t R r ( t ) = inf t R a ( t ) = 0 (recall the statements on the irrational case after Equation (444)); nevertheless, all nonsingularity conditions (65)–(68) are satisfied, since it can be checked directly that all integrals in the cited equations are divergent38.
Singular Lissajous cosmologies. Given ω 1 , ω 2 R + , let us consider a phase ψ in the exceptional set Z of Equation (450), and any two amplitudes A 1 , A 2 R + . Then, according to the discussion after Equation (450), the function R t r ( t ) has infinitely many zeroes if ω 2 / ω 1 is rational and unique zero if ω 2 / ω 1 is irrational. To establish a cosmological model, we must confine the time variable to a maximal interval J not containing zeroes. In the rational case, we will have J = ( t , t ) , where < t < t < + are two consecutive zeroes; in the irrational case, J = ( t × , + ) or J = ( , t × ) , where t × R is the unique zero. The scale factor a ( ) : J R + , t a ( t ) determined by (335) vanishes for t t or for t t × . So, in the rational case, the model has a Big Bang at t and a Big Crunch at t , whereas in the irrational case, there is either a Big Bang or a Big Crunch at t × .
With reference to the formal notions of singular or nonsingular spacetime of Section 3.2 and Section 3.15, we can state the following:
  • In rational case, where J = ( t , t ) , the model is singular in any one of the senses considered in Section 3.2 and Section 3.15: in fact, any one of conditions (65)–(68) is violated (of course the times t in the cited equations coincide with t and t , respectively).
  • In the irrational case with J = ( t × , + ) , the model is past timelike and lightlike singular, since conditions (65) and (66) are violated, and future timelike and lightlike nonsingular, since conditions (67) and (68) are fulfilled (of course, the cited equations must be applied with t = t × and t + = + )39.
  • In the irrational case with J = ( , t × ) , the model is future timelike and lightlike singular, since conditions (67) and (68) are violated, and past timelike and lightlike nonsingular, since conditions (65) and (66) are fulfilled (the cited equations must be applied with t = and t + = t × ).
Regarding an example mentioned previously. Let us return to the example (444) ω 1 = 3 ω , ω 2 = 4 ω ( ω R + ; needless to say, A 1 , A 2 > 0 , as in (441)). Let us repeat that Equation (442) holds with N 1 = 3 , N 2 = 4 and that, due to (443), the function R t r ( t ) is periodic of period T : = 2 π / ω .
Using Equations (449) and (450), we see that the set of exceptional phases giving rise to zeroes for the function t r ( t ) is Z = { 2 π / 3 , π / 3 , 0 , π / 3 , 2 π / 3 , π ( m o d . 2 π ) } ; for all the other phases ψ , we have nonsingular, periodic Lissajous cosmologies. We have already mentioned Figure 25, related to this case; the phases ψ considered therein include the exceptional choice ψ = 0 ( m o d . 2 π )40.

8.10. The Case W 1 > 0 , W 2 < 0

We finally address the case
W 1 = λ 2 , W 2 = ω 2 , ( λ , ω R + ) .
Equation (355) reads
x ¨ = λ 2 x , y ¨ = ω 2 y ,
thus describing a repulsor and an oscillator. A function t r ( t ) = ( x ( t ) , y ( t ) ) is a maximal solution of these equations if and only if it can be expressed as follows, up to a time translation t t + const.:
x ( t ) = A e λ t + B e λ t , y ( t ) = C sin ( ω t ) for   all   R ,
where
A , B , C R , C 0 .
In the rest of the present Section 8.10, we direct our attention to the nondegenerate case
A , B 0 , C > 0 .
Let us investigate conditions under which x ( t ) , y ( t ) or r ( t ) vanish. For any t R , we readily find the following equivalences:
x ( t ) = 0 B A < 0 , t = 1 2 λ ln B A ;
y ( t ) = 0 t = n π ω for   some   n Z .
Clearly, the function r ( ) has a zero if and only B / A < 0 , and the zero of x ( ) indicated in (456) coincides with one of the zeroes of y ( ) described by (457); the last condition holds if an only if ( ω / 2 π λ ) ln B / A = n for some n Z . Thus,
t R s u c h t h a t r ( t ) = 0 B A < 0 , ω 2 π λ ln B A Z .
From here to the end of the present Section 8.10, we assume
B A > 0 , o r B A < 0 , ω 2 π λ ln B A Z ;
since we are denying the conditions on A , B in (458), we have
r ( t ) 0 for   all   R .
To continue, let us note that Equation (357) with W 1 = λ 2 , W 2 = ω 2 and Equation (453) give
Ω m = 2 λ 2 A B + 1 2 ω 2 C 2 ;
we want Ω m > 0 , so, from now on, we also assume
C 2 > 4 λ 2 ω 2 A B .
Let us proceed to the analysis of the radius r ( t ) = | r ( t ) | ; Equation (453) gives
r ( t ) = A 2 e 2 λ t + B 2 e 2 λ t + 2 A B + C 2 sin 2 ( ω t ) for   t R .
Of course,
r ( t ) | B | e λ t for   t , r ( t ) | A | e λ t for   t + .
Moreover, if ω λ , we can expect an oscillatory behavior of the radius in an interval around the time t = 0 (with r ( t ) > 0 for each t R , due to (460)); oscillations should be particularly evident if | A | , | B | C .
Similar statements hold for the scale factor a = a ( t ) ( t R ), related to r via Equation (335); in a few words, in this cosmological model, the scale factor diverges exponentially for t and oscillates (but never vanishes) during a time interval centered at t = 0 . In particular, min t R a ( t ) exists and is strictly positive; so, due to Equation (69) and related comments, the present cosmological model is nonsingular in all senses considered in Section 3.2 and Section 3.15.
The situation under analysis is presented in Figure 27, for a special choice of the parameters; Figure 28 represents the curve t r ( t ) for the same values of the parameters.

8.11. Again on the Case W 1 > 0 , W 2 < 0

As in the previous section, let us make the assumption (451) W 1 = λ 2 and W 2 = ω 2 (with λ , ω R + ) . Let us recall the evolution equations (452) and the representation (453) and (454) for their (maximal) solutions t r ( t ) = ( x ( t ) , y ( t ) ) , depending on three real parameters A , B and C 0 .
In the present Section 8.11, we consider some interesting choices for the parameters A , B , C that were previously ruled out by the conditions (455) and (459).
Case A > 0 , B = 0 , C > 0 . This choice violates condition (455). Equation (453) gives
x ( t ) = A e λ t , y ( t ) = C sin ( ω t ) for   all   R ;
for all real t, we have x ( t ) 0 and, consequently, r ( t ) 0 . According to Equation (461), the mass density parameter is
Ω m = 1 2 ω 2 C 2 > 0 .
The radius function is given by
r ( t ) = A 2 e 2 λ t + C 2 sin 2 ( ω t ) for   all   R .
We have
r ( t ) = C 2 sin 2 ( ω t ) + o ( 1 ) for   t , r ( t ) A e λ t for   t +
( o ( 1 ) means, as usual, a function vanishing in the limit considered above). Thus, the radius r ( t ) oscillates (but never vanishes) for t 0 and grows exponentially for t 0 ; similar conclusions hold for the scale factor a = a ( t ) ( t R ), which is once more determined by Equation (335). The present cosmological model is nonsingular in all senses considered in Section 3.2 and Section 3.15, since it fulfills all conditions (65)–(68)41.
In Figure 29 and Figure 30, we plot the curve t r ( t ) and radius t r ( t ) for a specific choice of the parameters involved.
Case A = 0 , B > 0 , C > 0 . Again, we have a choice violating condition (455). Equation (453) gives
x ( t ) = B e λ t , y ( t ) = C sin ( ω t ) for   all   R ;
for all real t, we have x ( t ) 0 , hence r ( t ) 0 . Equation (466) holds again. The radius function is
r ( t ) = B 2 e 2 λ t + C 2 sin 2 ( ω t ) for   all   R ,
and
r ( t ) B e λ t for   t , r ( t ) = C 2 sin 2 ( ω t ) + o ( 1 ) for   t + .
Thus, the radius grows exponentially for t 0 and oscillates (without vanishing) for t 0 ; of course, the scale factor a = a ( t ) ( t R ) determined by Equation (335) behaves similarly. The present cosmological model is, essentially, a time-reversed version of the previous case A > 0 , B = 0 , C > 0 ; it is nonsingular in all senses considered in Section 3.2 and Section 3.15 since it fulfills all conditions (65)–(68).
In Figure 31, we describe the radius function t r ( t ) , for a special choice of the parameters.
Case A = F / 2 , B = F / 2 ( F > 0 ), C > 0 . This choice violates condition (459), since B / A = 1 and ( ω / 2 π λ ) ln ( B / A ) = 0 Z . Equation (453) gives
x ( t ) = F sinh ( λ t ) , y ( t ) = C sin ( ω t ) for   all   R .
For any t R , we have x ( t ) = 0 t = 0 and y ( t ) = 0 t = n π / ω ( n Z ); consequently, r ( t ) = 0 t = 0 . We can accept as a time domain for the present cosmological model a (maximal) interval J where r ( t ) is never vanishing; we take
J = ( 0 , + ) .
Let us also remark that, for the mass density parameter, Equation (461) gives in this case
Ω m = 1 2 ( λ 2 F 2 + ω 2 C 2 ) > 0 .
The radius function is given by
r ( t ) = F 2 sinh 2 ( λ t ) + C 2 sin 2 ( ω t )
= 1 2 F 2 cosh ( 2 λ t ) 1 2 C 2 cos ( 2 ω t ) + 1 2 ( C 2 F 2 ) for   all   t ( 0 , + ) ,
and its derivative reads
r ˙ ( t ) = 1 2 r ( t ) F 2 λ sinh ( 2 λ t ) + C 2 ω sin ( 2 ω t ) = F 2 λ 2 r ( t ) R 2 λ t ; C 2 ω F 2 λ , ω λ for   t ( 0 , + ) ,
where
R ( s ; χ , ϱ ) : = sinh ( s ) + χ sin ( ϱ s ) for   s , χ , ϱ ( 0 , + ) .
Let us remark that R ( s ; χ , ϱ ) sinh ( s ) χ for all s , χ , ϱ , which implies R ( s ; χ , ϱ ) > 0 for all χ , ϱ ( 0 , + ) and all s ( a r c s i n h χ , + ) ; depending on the values of χ and ϱ , the function s R ( s ; χ , ϱ ) does not possess or possesses zeroes in the interval ( 0 , a r c s i n h χ ] . Consequently, for all F , C , λ , ω ( 0 , + ) , the following holds:
r ˙ ( t ) > 0 for   all   t 1 2 λ a r c s i n h C 2 ω F 2 λ , + ,
while the function t r ˙ ( t ) does not possess or possesses zeroes in the interval ( 0 , ( 1 / 2 λ ) a r c s i n h ( C 2 ω / F 2 λ ) ] , depending on the values of the ratios C 2 ω / F 2 λ and ω / λ ; if such zeroes exist, we can say that the radius r ( t ) oscillates for t in a right neighborhood of 0. Finally, let us remark that
r ( t ) 0 + for   t 0 + , r ( t ) F 2 e λ t for   t + .
Similar conclusions hold for the scale factor a = a ( t ) ( t ( 0 , + ) ), given by (335); a ( t ) vanishes for t 0 + , thus presenting a Big Bang, and grows exponentially for t + . The present cosmological model is past timelike and lightlike singular since it violates conditions (65) and (66); it is future timelike and lightlike nonsingular since it fulfills (67) and (68).
In Figure 32 and Figure 33, we describe the curve t r ( t ) and the radius t r ( t ) for a specific choice of the parameters giving rise to zeroes of R , i.e., to initial oscillations of the radius.

9. Concluding Remarks

In the present work, we discussed the FLRW cosmologies with perfect fluids and a scalar field, paying special attention to some exactly solvable models with a phantom scalar. In spite of the rich literature on FLRW cosmologies with scalar fields, we think there are chances to further individuate exactly solvable models or, at least, to classify more exhaustively the already known solvable cases, especially in the presence of a phantom scalar; in the phantom case with a periodic self-potential, the mechanical analogy with a particle in a Euclidean plane, emphasized in Section 7, might offer a guidance in these investigations.

Author Contributions

M.C., M.G. and L.P. contributed equally to all aspects of this article. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by: INdAM, Gruppo Nazionale per la Fisica Matematica; INFN, projects MMNLP and BELL; MUR, project PRIN 2020 “Hamiltonian and dispersive PDEs”; Università degli Studi di Milano.

Data Availability Statement

The original contributions presented in this study are included in the article, further inquiries can be directed to the corresponding author.

Acknowledgments

We are grateful to the funding institutions mentioned above, and to Davide Fermi (Politecnico di Milano) for encouraging us to undertake the present investigation. We are also grateful to all the reviewers of the present work; their comments yielded improvements in the presentation of our results.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. Riemannian Manifolds, Spacetimes, and Dimensional Analysis

Let us repeat the convention Section 2.2: all manifolds considered in this paper are real, smooth, Hausdorff, connected and paracompact.
Let S d denote any manifold of dimension d { 1 , 2 , 3 , } . A Riemannian metric on S d is usually described as a smooth map h assigning for each S S d an inner product h S : T S S d × T S S d R . In the present paper, for consistency with the dimensional analysis aspects introduced in Section 2.1, we must accept a minimal change in the conventional viewpoint and assume h S : T S S d × T S S d L 2 where L is the real, one-dimensional, oriented vector space of lengths (thus, each tangent vector X T S S d has a squared norm h S ( X , X ) h ( X , X ) L 2 and a norm h ( X , X ) in the semi-space L + of non-negative lengths)42. The pair ( S d , h ) will be indicated as a Riemannian manifold. Let us recall the traditional denomination of “squared line element” for the map d 2 sending a tangent vector X (at any point) to d 2 ( X ) : = h ( X , X ) ; for any coordinate system ( x i ) i = 1 , , d of S d in which h has coefficients h i j , we can write d 2 = h i j d x i d x j .
Let us now consider a manifold M d + 1 of dimension d + 1 , where d { 1 , 2 , 3 , } . Our definition of a Lorentzian metric g on M d + 1 takes into account the same dimensional aspects: so, g is a smooth map assigning for each A M d + 1 a symmetric nondegenerate bilinear form g A : T A M d + 1 × T A M d + 1 L 2 of signature ( , + , , + ) . The pair ( M d + 1 , g ) will be referred to as a Lorentzian manifold, or a spacetime; d will be called the spatial dimension. A tangent vector X (at any point A M d + 1 ) will be called timelike, lightlike or spacelike in the three cases g ( X , X ) < 0 , g ( X , X ) = 0 or g ( X , X ) > 0 (of course, g ( X , X ) is short for g A ( X , X ) ). We will employ the traditional denomination of “squared line element” for the map d s 2 sending a tangent vector X (at any point) to d s 2 ( X ) : = g ( X , X ) ; of course d s 2 = g μ ν d x μ d x ν for any coordinate system ( x μ ) of M d + 1 .
Remark: In the present appendix and in the subsequent Appendix B, Appendix C, Appendix D and Appendix E, we primarily use an intrinsic language, in which index notation has just an ancillary role; this style is suitable for the treatment of some technical issues. The use of indices is preferred in Appendix F and in the main text of the paper; however, many formulas therein can be interpreted via the abstract index notation of Penrose [45,82], which is in fact equivalent to the intrinsic, index-free language.

Appendix B. FLRW Spacetimes and Generalizations

For subsequent use, let us recall from Section 2.1 that the real, one-dimensional, oriented vector space T is identified with L in our setting where the speed of light is c = 1 .
The spacetime manifold and its metric. For d { 2 , 3 , 4 , } , let us consider the product manifold
M d + 1 = T × S d ,
where T T ( = L ) is an open interval with its natural coordinate τ , and S d is any manifold of dimension d. A coordinate system of S d is generically indicated with ( x i ) i = 1 , , d (with Latin indexes).
We assume that S d carries a Riemannian metric h, and that a (smooth) function a : T R + , τ a ( τ ) is assigned. With these ingredients, we can define a Lorentzian metric g on M d + 1 in the following way: for each A = ( τ , S ) M d + 1 = T × S d and for any pair of tangent vectors X = ( X 0 , X ) , Y = ( Y 0 , Y ) in the tangent space T A M d + 1 T × T S S d , one has
g A ( X , Y ) : = X 0 Y 0 + a 2 ( τ ) h S ( X , Y )
(in the sequel, we often reproduce this equation by omitting the subscripts A and S). A more traditional description, in terms of the squared line elements d s 2 , d 2 corresponding to g and h, is
d s 2 = d τ 2 + a 2 ( τ ) d 2 .
A spacetime of this kind is introduced in Section 3.1 (see Equations (15) and (17)) with specific prescriptions for the Riemannian manifold ( S d , h ) , which is assumed to be complete (see Appendix D) and with constant sectional curvature; in this case, as in Section 3.1, we refer to an FLRW spacetime.
On the contrary, in the present appendix, the Riemannian manifold ( S d , h ) is unspecified; in the absence of this specification, a spacetime of the form (A1) and (A2) is referred to as a generalized FLRW spacetime.
Let us consider any coordinate system ( x i ) i = 1 , , d of S d ; this induces the following coordinate system on the generalized FLRW spacetime M d + 1 :
( τ , ( x i ) i = 1 , , d ) ( x μ ) μ = 0 , , d
(here and in the sequel, Greek indices always range from 0 to d). In these coordinates, the spacetime metric g has coefficients g μ ν , where
g 00 = 1 , g i j = a 2 h i j , g 0 i = g i 0 = 0 ( i , j = 1 , , d )
(these facts were already noted in Section 3.1 for usual FLRW spacetimes).
Curves in a generalized FLRW spacetime. In the sequel, we frequently consider curves in a spacetime of this kind; it is convenient to fix some notations and write down some equations to be cited subsequently. A (parametrized) curve in a generalized FLRW spacetime is a map
ξ : P R M d + 1 = T × S d , p ( ϑ ( p ) , η ( p ) ) ,
P   an   open   interval , ϑ : P T   and   η : P S d ( both   smooth ) .
The velocity of ξ at any p P is
d ξ d p ( p ) = d ϑ d p ( p ) , d η d p ( p ) .
Due to (A2), we have
g d ξ d p ( p ) , d ξ d p ( p ) = d ϑ d p ( p ) 2 + a 2 ( ϑ ( p ) ) h d η d p ( p ) , d η d p ( p ) ,
and one easily infers from here when the velocity is timelike, lightlike or spacelike.
Christoffel symbols, Ricci tensor, and scalar curvature in a generalized FLRW spacetime. Given a spacetime of this kind, we consider the connection induced by the metric g, and we employ a coordinate system as in Equation (A4).
Let us compute in these coordinates the Christoffel symbols of the metric connection; the standard rule Γ μ ν λ   = ( 1 / 2 ) g λ α ( μ g α ν + ν g α μ α g μ ν ) and Equation (A5) give
Γ 00 0 = 0 , Γ 0 i 0 = Γ i 0 0 = 0 , Γ i j 0 = a a h i j ,
Γ 00 k = 0 , Γ 0 i k = Γ i 0 k = a a δ i k , Γ i j k = γ i j k ( i , j , k = 1 , , d ) ,
where : = d / d τ (as in Section 3.1), and γ i j k are the Christoffel symbols in coordinates ( x i ) of the connection on S d induced by h (in Appendix E, we will use the above expressions to discuss geodesic curves in a spacetime of this kind).
We now consider the Ricci tensor R μ ν and the scalar curvature R of g; the usual prescriptions R μ ν = α Γ μ ν α ν Γ μ α α + Γ α β α Γ μ ν β   Γ ν β α Γ μ α β and R = R μ μ give in the present case
R 00 = d a a , R i j = a a + ( d 1 ) a 2 h i j + r i j , R 0 i = R i 0 = 0 ( i , j = 1 , , d ) ,
R = 2 d a a + d ( d 1 ) a 2 a 2 + r a 2 ,
where r i j and r are the Ricci tensor and the scalar curvature of the metric h.
The case of a usual FLRW spacetime. In particular, we have [71]
r i j = ( d 1 ) k h i j , r = d ( d 1 ) k
for   a   usual   FLRW   space   time   ( h   has   constant   sectional   curvature   k ) .
Equations (A10)–(A12) yield Equations (35) and (36) in the main text.

Appendix C. Time Orientation of a Spacetime: The Case of a Generalized FLRW Spacetime

Let us consider an arbitrary spacetime ( M d + 1 , g ) , i.e., a ( d + 1 ) -dimensional manifold M d + 1 ( d = 1 , 2 , ) equipped with a Lorentzian metric g.
For any A M d + 1 , the set of timelike tangent vectors at A has two connected components, each one formed by the opposites of vectors in the other component. A time orientation of M d + 1 is a rule F assigning smoothly for each A M d + 1 one of these two components, denoted with F A and called the future at A ; the other connected component (i.e., the set of opposites F A ) is called the past at A . The expression “smoothly” used before must be intended as follows: for each A 0 M d + 1 , there is a smooth vector field X defined on an open neighborhood U of A 0 , such that X ( A ) F A for each A U .
Depending on the spacetime ( M d + 1 , g ) considered, it may be possible or impossible to assign F A smoothly for all A M d + 1 [12]; in these two cases, we speak, respectively, of a time-orientable or non-time-orientable spacetime. In the orientable case, there are just two time orientations: if F is one of them, the other one is the correspondence F : A F A .
A time-oriented spacetime is a spacetime ( M d + 1 , g ) , equipped with a time orientation F . Given such a structure, a timelike tangent vector at A M d + 1 is said to be future-directed or past-directed, if it belongs to F A or to F A ; a lightlike tangent vector is said to be future-directed or past-directed, if it belongs to the boundary of F A , or to the boundary of F A (in the natural topology of the tangent space at A ). To continue, let us consider in M d + 1 a (parametrized) curve, i.e., a smooth map
ξ : P R M d + 1 , p ξ ( p )
with P an open interval. Assume that, for each p P , the velocity ( d ξ / d p ) ( p ) T ξ ( p ) M d + 1 is timelike or lightlike; in this case we define ξ to be future-directed, or past-directed, if such feature is possessed by the velocity at any p P .
The case of a generalized FLRW spacetime. Let us consider a generalized FLRW spacetime M d + 1 = T × S d ( d 2 ), for which we keep all notations of Appendix B; in particular, T indicates an open interval in the one-dimensional, oriented vector space of times T and a tangent vector at A = ( τ , S ) M d + 1 is represented as a pair X = ( X 0 , X ) T × T S S d .
This spacetime carries a natural time orientation, in which the future at any point A is
F A : = { X = ( X 0 , X ) T A M d + 1 | X is   timelike   and X 0 > 0 } ;
the boundary of the future is
F A = { X = ( X 0 , X ) T A M d + 1 | X is   lightlike and X 0 > 0 , o r X = ( 0 , 0 ) } .
The past F A and its boundary ( F A ) have similar descriptions, with the inequality X 0 > 0 replaced by X 0 < 0 .
We now consider a (parametrized) curve in this spacetime, for which we use the representation (A6) ξ : P R M d + 1 = T × S d , p ( ϑ ( p ) , η ( p ) ) ( P is an open interval); this has velocity as in Equation (A7). On the grounds of Equation (A14), if the velocity at p P is timelike, we have the equivalence
d ξ d p ( p ) future - directed d ϑ d p ( p ) > 0 ;
if the velocity at p is lightlike, due to (A15), we have
d ξ d p ( p ) future - directed d ϑ d p ( p ) > 0 , or d ϑ d p ( p ) = 0 and d η d p ( p ) = 0 .

Appendix D. A Review on Geodesics and Several Associated Notions of Completeness, Mainly in Riemannian Manifolds and in Spacetimes. Nonsingular Spacetimes

Geodesics in a manifold with connection, and the associated notions of completeness. Let us consider a manifold N m (of any dimension m); a (parametrized) curve in N m is a smooth map
ξ : P R N m , p ξ ( p ) ,
with P an open interval. From now on, we assume N m to carry a connection and denote with ∇ the corresponding covariant derivative. Given a curve ξ as above, we can define the velocity
d ξ d p : P T N m , p d ξ d p ( p ) T ξ ( p ) N m ,
and the acceleration
d p d ξ d p : P T N m , p d p d ξ d p ( p ) T ξ ( p ) N m .
If ( x μ ) μ = 1 , , m is a coordinate system of N m , the curve has a coordinate description ( ξ μ ) : p ( ξ μ ( p ) ) : = ( x μ ( ξ ( p ) ) 43. Of course, the velocity of ξ has components d ξ μ / d p ; the acceleration has the following components:
d p d ξ d p λ = d 2 ξ λ d p 2 + Γ μ ν λ ( ξ ) d ξ μ d p d ξ ν d p ,
where Γ μ ν λ are the Christoffel symbols of the connection. ξ is said to be a (parametrized) geodesic if
d p d ξ d p = 0
(everywhere on the interval P ).
A geodesic ξ : P R N m is said to be maximal if it cannot be extended to a geodesic ξ : P R N m , defined on an open interval P that contains P properly. A geodesic ξ : P N m is said to be complete if P = ( , + ) (which, of course, implies ξ to be maximal).
The manifold with connection N m is said to be geodesically complete if every maximal geodesic is complete.
The case of a Riemannian manifold. All the notions introduced in the previous paragraph apply, in particular, to the case of a Riemannian manifold ( S d , h ) , with the connection induced by the metric h; it is well known that
h d ζ d p , d ζ d p = constant for   each   geodesic ζ : P R S d .
Moreover, the completeness of ( S d , h ) in the geodesic sense of the previous paragraph can be shown to be equivalent to the usual completeness of S d as a metric space, with the distance induced by h44.
The case of a spacetime. We now consider a spacetime ( M d + 1 , g ) (i.e., a manifold equipped with a Lorentzian metric), and the connection induced by g. A result analogous to (A23) holds, namely
g d ξ d p , d ξ d p = constant for   each   geodesic ξ : P R M d + 1 .
In the present framework, one can refine the notion of geodesic completeness by distinguishing among timelike, lightlike and spacelike geodesics45; further refinements are possible if ( M d + 1 , g ) is equipped with a time orientation, allowing us to distinguish between future-directed and past-directed geodesics of the timelike or lightlike type.
Assuming a time orientation is given, let ξ denote a future-directed, timelike, or lightlike geodesic in M d + 1 , defined on an open interval P = ( p , p + ) R . We say that ξ is past complete if p = + , and future complete if p + = .
The time-oriented spacetime ( M d + 1 , g ) is said to be:
  • Past timelike complete if each maximal, future-directed timelike geodesic is past complete;
  • Past lightlike complete if each maximal, future-directed lightlike geodesic is past complete;
  • Past complete if it is both past timelike and past lightlike complete.
The notions of future timelike complete, future lightlike complete, and future complete spacetime are defined similarly, requiring the future completeness of maximal, future-directed geodesics. A spacetime is said to be timelike complete, lightlike complete, or complete if it possesses the previous properties both in the past and the future (i.e., ( p , p + ) = ( , + ) for the maximal geodesics of the corresponding types).
The adjective nonsingular is often used as an equivalent for “complete. So, we can say that a spacetime is past timelike nonsingular, past lightlike nonsingular, ⋯, timelike nonsingular, lightlike nonsingular, nonsingular.
Needless to say, the terms incomplete and singular will be used as negations of “complete” and “nonsingular”.

Appendix E. Geodesics and Geodesic Completeness of Generalized FLRW Spacetimes

Let us consider a generalized FLRW spacetime M d + 1 = T × S d ( d = 2 , 3 , 4 , ), for which we keep all the notations of Appendix B; in particular, g is the metric of M d + 1 and h the (Riemannian) metric of S d (see Equation (A2)). We equip this spacetime with the time orientation described in Appendix C. In addition, we prescribe the following:
: = the   covariant   derivative   in ( M d + 1 , g ) , D : = the   covariant   derivative   in ( S d , h )
(using in both cases the metric connections).
Curves in M d + 1 : reminders, and computation of the acceleration. Let us consider a (parametrized) curve in M d + 1 ; as already indicated in Appendix C, this is a map of the form (A6), here reproduced,
ξ : P R M d + 1 = T × S d , p ( ϑ ( p ) , η ( p ) )
P an   open   interval , ϑ : P T and η : P S d ( both   smooth ) .
Let us also recall Equations (A7) and (A8) (here rephrased as equalities of functions, holding everywhere on P )46:
d ξ d p = d ϑ d p , d η d p ;
g d ξ d p , d ξ d p = d ϑ d p 2 + a 2 ( ϑ ) h d η d p , d η d p .
In the sequel, g ( d ξ / d p , d ξ / d p ) and h ( d η / d p , d η / d p ) will be referred to as the “squared velocities” of the curves ξ and η . Using the covariant derivative ∇ in (A25) we can also define the acceleration ( / d p ) ( d ξ / d p ) , which is computed hereafter.
Let us consider a coordinate system ( x μ ) μ = 0 , , d = ( τ , ( x i ) i = 1 , , d ) as in Equation (A4); in these coordinates, the curve has a description p ( ξ μ ( p ) ) , given by
ξ 0 ( p ) = ϑ ( p ) , ξ i ( p ) = η i ( p ) ( i = 1 , , d )
where, of course, η i ( p ) : = x i ( η ( p ) ) (for all p such that η ( p ) S d is in the domain of the coordinates ( x i ) ). Equations (A21), (A9) and (A28) give
d p d ξ d p 0 = d 2 ϑ d p 2 + a ( ϑ ) a ( ϑ ) h i j ( η ) d η i d p d η j d p ,
d p d ξ d p k = d 2 η k d p 2 + γ i j k ( η ) d η i d p d η j d p + 2 a ( ϑ ) a ( ϑ ) d ϑ d p d η k d p ( k = 1 , , d ) ,
where (we repeat it) γ i j k are the Christoffel symbols of the metric connection on ( S d , h ) . In intrinsic language, this means that
d p d ξ d p = ( A , B ) ,
A : = d 2 ϑ d p 2 + a ( ϑ ) a ( ϑ ) h d η d p , d η d p , B : = D d p d η d p + 2 a ( ϑ ) a ( ϑ ) d ϑ d p d η d p
(let us recall that D is the covariant derivative in ( S d , h ) , see (A25); needless to say, in the above A is a scalar function on the interval P , while B is a function from the interval P to the tangent bundle of S d , given by the acceleration of the curve η plus a correction depending on ϑ ).
Curves in M d + 1 : new expressions for the squared velocity and for the acceleration, upon a reparametrization of the component in S d . Let us consider any curve ξ = ( ϑ , η ) : P R M d + 1 , for which we keep all notations of the previous paragraph. Following a hint by O’Neill [60] we reparametrize the second component η of ξ introducing the map
χ : P R R , p χ ( p ) : = p p d p a 2 ( ϑ ( p ) ) ( p an   arbitrarily   chosen   point   of P ) .
We note that χ is smooth and, for all p P ,
d χ d p ( p ) = 1 a 2 ( ϑ ( p ) ) > 0 ;
thus, χ ( P ) is an open interval in R , and χ is a diffeomorphism between P and χ ( P ) . Due to these facts, there is a unique smooth map
ζ : χ ( P ) R S d , q ζ ( q ) such   that η ( p ) = ζ ( χ ( p ) )   for   all   p P .
This is clearly a reparametrization of the curve η ; in the sequel, we write the above equation as η = ζ ( χ ) . In the same style, we can write
d η d p = d ζ d q ( χ ) d χ d p = 1 a 2 ( ϑ ) d ζ d q ( χ ) ;
inserting this result into Equation (A27) and into the first equality (A32), we obtain
g ( ξ ) d ξ d p , d ξ d p = d ϑ d p 2 + 1 a 2 ( ϑ ) h d ζ d q ( χ ) , d ζ d q ( χ ) ,
A = d 2 ϑ d p 2 + a ( ϑ ) a 3 ( ϑ ) h d ζ d q ( χ ) , d ζ d q ( χ ) .
Equation (A36) also implies
D d p d η d p = d d p 1 a 2 ( ϑ ) d ζ d q ( χ ) + 1 a 2 ( ϑ ) D d p d ζ d q ( χ )
= 2 a ( ϑ ) a 3 ( ϑ ) d ϑ d p d ζ d q ( χ ) + 1 a 2 ( ϑ ) D d q d ζ d q ( χ ) d χ d p ,
that is, by (A34),
D d p d η d p = 2 a ( ϑ ) a 3 ( ϑ ) d ϑ d p d ζ d q ( χ ) + 1 a 4 ( ϑ ) D d q d ζ d q ( χ ) .
Inserting Equations (A36) and (A39) into the second equality (A32), we obtain
B = 1 a 4 ( ϑ ) D d q d ζ d q ( χ ) .
Let us recall that the pair ( A , B ) gives the acceleration of ξ , see Equation (A31).
Characterization of geodesics in M d . Let us again consider a curve
ξ : P R M d + 1 , p ξ ( p ) .
According to the results in the previous two paragraphs, we can uniquely represent ξ as
ξ ( p ) = ( ϑ ( p ) , ζ ( χ ( p ) ) ) for   p P ,
with χ : P R R as in (A33), and
ϑ : P R T T , p ϑ ( p ) , ζ : χ ( P ) R S d , q ζ ( q )
( ϑ , ζ smooth; we recall that T , P , χ ( P ) are open intervals, and χ is a smooth diffeomorphism). Let us recall that the squared velocity of ξ has the expression (A37), and the acceleration of ξ is given by Equation (A31), with A and B as in Equations (A38) and (A40). That said, we have the following equivalences:
ξ is   a   geodesic   in   ( M d + 1 , g ) 0 = d p d ξ d p = ( A , B )
d 2 ϑ d p 2 + a ( ϑ ) a 3 ( ϑ ) h d ζ d q ( χ ) , d ζ d q ( χ ) = 0 ( everywhere   on P )
and D d q d ζ d q = 0 ( everywhere   on   χ ( P ) , meaning   that   ζ is   a   geodesic   in   ( S d , h ) ) .
On the grounds of general results, already mentioned in Appendix D, the geodesic nature of ξ and η implies the constancy of the corresponding squared velocities, which are related through Equation (A37). Putting together the previous considerations, we obtain the following:
ξ is   a   geodesic   in   ( M d + 1 , g )
ζ is   a   geodesic   in   ( S d , h )   and d 2 ϑ d p 2 + 2 M a ( ϑ ) a 3 ( ϑ ) = 0 , M : = the   constant   value   of 1 2 h d ζ d q , d ζ d q 0
1 2 g d ξ d p , d ξ d p = 1 2 d ϑ d p 2 + M a 2 ( ϑ ) = constant   E .
Let us note that M , E are in the space L 2 of squared lengths (which is the range of h and g; see Appendix A). After introducing for each M L 2 the potential function
V M : T T L L 2 , τ V M ( τ ) : = M a 2 ( τ ) ,
we can rephrase Equation (A45) in the following way:
ξ is   a   geodesic   in   ( M d + 1 , g )
ζ is   a   geodesic   in   ( S d , h )   and d 2 ϑ d p 2 + V M ( ϑ ) = 0 , M : = the   constant   value   of 1 2 h d ζ d q , d ζ d q 0
1 2 d ϑ d p 2 + V M ( ϑ ) = constant E and 1 2 g d ξ d p , d ξ d p = E .
Let us note that
the   geodesic ξ is timelike lightlike spacelike E 0
and that, if ξ is timelike or lightlike, it is future-directed under the conditions (A16) and (A17).
The equations d 2 ϑ / d p 2 + V M ( ϑ ) = 0 and ( 1 / 2 ) ( d ϑ / d p ) 2 + V M ( ϑ ) = E are the law of motion and the conservation law of energy of a fictitious one-dimensional, conservative mechanical system with kinetic energy ( 1 / 2 ) ( d ϑ / d p ) 2 and potential energy V M .
Let us assume the conditions (A47) to be fulfilled by ξ . Then, we have the following statements, which are used in the next paragraph:
ζ = constant M = 0 d 2 ϑ d p 2 = 0 ;
ξ timelike   ( resp . ,   lightlike )   and   ζ   nonconstant E > 0   ( resp . , E = 0 )   and   M > 0 ;
E 0 , M 0 and ( E , M ) ( 0 , 0 ) d ϑ d p 2 = 2 E V M ( ϑ ) = 2 E + M / a 2 ( ϑ ) > 0
sign d ϑ d p = constant s ϑ { ± 1 } and d ϑ d p = s ϑ 2 ( E + M / a 2 ( ϑ ) )
ϑ ( p ) ϑ ( p ) d τ 2 ( E + M / a 2 ( τ ) ) = s ϑ ( p p ) for   all   p , p P .
The maximal geodesics in ( M d + 1 , g ) in terms of their Cauchy data, assuming completeness of ( S d , h ) . We again consider a generalized FLRW spacetime M d + 1 = T × S d ( d 2 ), for which we use all previous notations; in addition, we represent the open interval T T in terms of its endpoints, i.e.,
T = ( τ , τ + ) , τ { } T , τ + T { + } , τ < τ + .
From now on, we assume that
the   Riemannian   manifold   ( S d , h )   is   complete .
Let us define a Cauchy datum (more briefly, a datum) as an ordered set
: = ( p , τ , δ , S , Z ) such   that p R , τ ( τ , τ + ) , δ T , S S d , Z T S S d .
Given such a set, let us denote with
ξ : P ( p , p + ) R M d + 1 , p ξ ( p )
(with p < p + + ) the maximal geodesic in M d + 1 such that
p ( p , p + ) , ξ ( p ) = ( τ , S ) , d ξ d p ( p ) = δ , Z a 2 ( τ ) T × T S S d = T ( τ , S ) M d + 1 .
Of course, any maximal geodesic in ( M d + 1 , g ) has the form ξ , for some datum ♭ as above. On the grounds of the description of the geodesics in the previous paragraph, for each datum ♭, we have
ξ ( p ) = ( ϑ ( p ) , ζ ( χ ( p ) ) ) for   all   p ( p , p + ) ,
where47:
χ : P R the   function   χ   of   Equation ( A 33 ) ,   with   P = ( p , p + )   and   p = p ;
x ζ : R S d , q ζ ( q ) the   maximal   geodesic   in   ( S d , h )   such   that ζ ( 0 ) = S , d ζ d q ( 0 ) = Z ;
ϑ : ( p , p + ) ( τ , τ + ) the   maximal   solution   of   the   Cauchy   problem
d 2 ϑ d p 2 + V ( ϑ ) = 0 , ϑ ( p ) = τ , d ϑ d p ( p ) = δ ,
where V ( τ ) : = M a 2 ( τ ) for τ ( τ , τ + ) , M : = 1 2 h ( Z , Z ) ( M L 2 , M 0 ) .
Again due to the results in the previous paragraph, we have
1 2 d ϑ d p 2 + V ( ϑ ) = 1 2 g d ξ d p , d ξ d p = constant = E , E : = 1 2 δ 2 M a 2 ( τ ) L 2
(the expression for E follows computing the above constant function at p = p ). Of course,
ξ   is timelike lightlike spacelike E 0 .
Referring once more to the previous paragraph, we see that
E 0 , M 0 and ( E , M ) ( 0 , 0 ) sign d ϑ d p = const . = sign ( δ ) 0 ;
if we also take into account Equations (A16) and (A17), we find
ξ timelike ,   future - directed E > 0 and δ > 0 ;
ξ lightlike , future - directed   E = 0 , M > 0   and   δ > 0 ,   or   M = 0   and   δ = 0
(note that M = 0 and δ = 0 ( d ξ / d p ) ( p ) = 0 ξ = constant). In addition,
ζ = constant M = 0 d 2 ϑ d p 2 = 0 ,
ϑ ( p ) = τ + δ ( p p ) for p ( p , p + ) ,
p = p + τ τ δ i f δ > 0 , p = p + τ ± τ δ i f δ < 0 , p = i f δ = 0 ;
ξ timelike ( green resp . ,   lightlike )   and   ζ   nonconstant E > 0 ( resp . ,   E = 0 )   and   M > 0
d ϑ d p =   sign   ( δ ) 2 ( E + M / a 2 ( ϑ ) ) 0
τ ϑ ( p ) d τ 2 ( E + M / a 2 ( τ ) ) = sign ( δ ) ( p p ) for   all   p ( p , p + ) ,
p = p τ τ d τ 2 ( E + M / a 2 ( τ ) ) , p + = p + τ τ + d τ 2 ( E + M / a 2 ( τ ) ) i f δ > 0 ,
p = p τ τ + d τ 2 ( E + M / a 2 ( τ ) ) , p + = p + τ τ d τ 2 ( E + M / a 2 ( τ ) ) i f δ < 0
note   that   1 2 ( E + M / a 2 ( τ ) ) = 1 2 a ( τ ) M + E a 2 ( τ ) .
(In the above, the endpoints p are always determined by requiring ϑ to be maximal among the solutions with values to ( τ , τ + ) ; in cases with δ > 0 described above, maximality is equivalent to requiring τ = lim p p ϑ ( p ) , and a similar statement with τ and τ + interchanged holds in the above cases with δ < 0 ).
Completeness conditions for generalized FLRW spacetimes. The necessary and sufficient conditions for a generalized FLRW spacetime to be lightlike complete were obtained by O’Neill [60] (see Chapter 12, Remark 27). Sanchez derived in [62] the necessary and sufficient conditions for a generalized FLRW spacetime to be complete both in the timelike and in the lightlike sense, on the grounds of more general results by Romero and Sanchez for warped product spaces [61]. Hereafter, we report the results of [60,62], which are accompanied by the corresponding proofs just to make the present exposition self-contained. The proofs presented here follow the arguments in the cited works, with some adaptation to our language (concerning, e.g., our use of the spaces T L of times and lengths).
Throughout this paragraph, we consider a generalized FLRW spacetime M d + 1 = T × S d ( d 2 ), for which we maintain the notations and assumptions of the previous paragraph; in particular, we use the representation (A52) T = ( τ , τ + ) and the completeness assumption (A53) for the Riemannian manifold ( S d , h ) . Appendix D is our reference for all completeness notions mentioned here and in the sequel in relation to ( S d , h ) or to ( M d + 1 , g ) .
Proposition A1.
Assume (A52) and (A53), and choose any time τ 0 ( τ , τ + ) ; then, (i)–(iv) hold.
(i)
( M d + 1 , g ) is past timelike complete if and only if
τ = , τ 0 d τ a ( τ ) 1 + a 2 ( τ ) = + .
If τ > all maximal, future-directed timelike geodesics are past incomplete. If τ = and τ 0 d τ a ( τ ) / 1 + a 2 ( τ ) < + all maximal, future-directed timelike geodesics with nonconstant projection on S d are past incomplete.
(ii)
( M d + 1 , g ) is past lightlike complete if and only if
τ τ 0 d τ a ( τ ) = +
(note that (A69) does not require τ to be ).
If τ τ 0 d τ a ( τ ) < + all maximal, future-directed and nonconstant lightlike geodesics are past incomplete.
(iii)
( M d + 1 , g ) is future timelike complete if and only if
τ + = + , τ 0 + d τ a ( τ ) 1 + a 2 ( τ ) = + .
If τ + < + all maximal, future-directed timelike geodesics are future incomplete. If τ + = + and τ 0 + d τ a ( τ ) / 1 + a 2 ( τ ) < + all maximal, future-directed timelike geodesics with nonconstant projection on S d are future incomplete.
(iv)
( M d + 1 , g ) is future lightlike complete if and only if
τ 0 τ + d τ a ( τ ) = +
(note that (A71) does not require τ + to be + ).
If τ 0 τ + d τ a ( τ ) < + all maximal, future-directed and nonconstant lightlike geodesics are future incomplete.
Proof. 
We will present the proofs of (i) and (ii), organizing them in several steps; the arguments proving (iii) and (iv) are very similar and will not be discussed.
Let us note that the definition of past timelike completeness in Appendix D is equivalent, with the notations of the previous paragraph, to the following statement:
for   each   datum     with   E > 0   and   δ > 0 ,   it   is   p = ;
Similarly, the notion of past lightlike completeness in Appendix D is equivalent to the following statement48:
for   each   datum   with   E = 0 , M > 0   and   δ > 0 ,   it   is   p = .
Step 1. Equation (A72) (past timelike completeness of ( M d + 1 , g ) ) implies Equation (A68). Assuming (A72), we first consider a datum ♭ as in (A54) with p = 0 , δ > 0 and Z = 0 , so that ( ζ = constant and) M = 0 . Then, E = ( 1 / 2 ) δ 2 > 0 , so ♭ has the form considered in (A72). From (A72), we know that p = ; on the other hand, Equation (A66) tells us that p = ( τ τ ) / δ , so
τ = ;
this is the first statement in Equation (A68). In order to prove the second statement in (A68) (i.e., the divergence of the integral therein), we choose a datum ♭ as in (A54) with p = 0 , τ = τ 0 , Z 0 (implying M > 0 ) and δ : = 2 M ( 1 + 1 / a 2 ( τ 0 ) ) ; from the expression of δ and from (A61), we infer E = M . Since E , δ > 0 , Equation (A72) tells us that p = . On the other hand, p has the expression provided by the last lines in Equation (A67) with the present specifications for p , τ , E , and we already know that τ = ; thus
= p = 1 2 τ 0 d τ a ( τ ) M + M a 2 ( τ ) = 1 2 M τ 0 d τ a ( τ ) 1 + a 2 ( τ ) ,
which proves the integral in (A68) to be + .
Step 2. Equation (A68) implies Equation (A72) (past timelike completeness). Assuming Equation (A68), let us consider any datum ♭ with E > 0 and δ > 0 ; our aim is to prove that p = .
Indeed, if M = 0 (i.e., ζ = constant), according to Equation (A66) we have
p = p + τ τ δ =
(since (A68) prescribes τ = ).
If M > 0 , according to the last lines in Equation (A67), we have
p = p 1 2 τ d τ a ( τ ) M + E a 2 ( τ ) p 1 2 max ( M , E ) τ d τ a ( τ ) 1 + a 2 ( τ )
= p 1 2 max ( M , E ) τ 0 d τ a ( τ ) 1 + a 2 ( τ ) + τ 0 τ d τ a ( τ ) 1 + a 2 ( τ ) =
(the above integral from to τ 0 equals + again due to (A68); the other integral converges).
Step 3. If τ > , all maximal future-directed timelike geodesics are past incomplete. Assuming τ > , let us consider a maximal, future-directed timelike geodesic; this will be of the form ξ for some datum ♭ with E > 0 , δ > 0 . Hereafter, we will show that p > .
Indeed, if M = 0 , according to Equation (A66), we have
p = p + τ τ δ > .
If M > 0 , according to the last lines in Equation (A67), we have
p = p 1 2 τ τ d τ a ( τ ) M + E a 2 ( τ ) p 1 2 min ( M , E ) τ τ d τ a ( τ ) 1 + a 2 ( τ ) ,
and since α / 1 + α 2 < 1 for all α [ 0 , + ) , we conclude that
p p 1 2 min ( M , E ) τ τ d τ = p τ τ 2 min ( M , E ) > .
Step 4. If τ = and τ 0 d τ a ( τ ) / 1 + a 2 ( τ ) < + all maximal, future-directed timelike geodesics with nonconstant projection on S d are past incomplete. With the given assumptions, let us consider a maximal, future-directed timelike geodesics with nonconstant projection on S d ; this will be of the form ξ for some datum ♭ with E > 0 , δ > 0 and M > 0 ( M = 0 is impossible since this would imply ζ = constant, against the assumption of nonconstant projection on S d ). Hereafter, we will show that p > .
Indeed, according to the last lines in Equation (A67), we have
p = p 1 2 τ d τ a ( τ ) M + E a 2 ( τ ) p 1 2 min ( M , E ) τ d τ a ( τ ) 1 + a 2 ( τ )
= p 1 2 min ( M , E ) τ 0 d τ a ( τ ) 1 + a 2 ( τ ) + τ 0 τ d τ a ( τ ) 1 + a 2 ( τ ) >
(note that, with our assumptions, both integrals above are convergent).
Step 5. Equation (A73) (past lightlike completeness) implies Equation (A69). Assuming (A73), we choose a datum ♭ as in (A54) with p = 0 , τ = τ 0 , Z 0 (implying M > 0 ) and δ : = 2 M / a ( τ 0 ) ; from the expression of δ and from (A61), we infer E = 0 . Since E = 0 , M > 0 and δ > 0 , Equation (A73) tells us that p = . On the other hand, p has the expression provided by the last lines in Equation (A67) with the present specifications for p , τ , E ; thus,
= p = 1 2 M τ τ 0 d τ a ( τ ) ,
which proves the integral in (A69) to be + .
Step 6. Equation (A69) implies Equation (A73) (past lightlike completeness). Assuming Equation (A69), let us consider any datum ♭ with E = 0 , M > 0 and δ > 0 ; we aim to prove that p = .
Indeed, according to the last lines in Equation (A67), for the datum under consideration, we have
p = p 1 2 M τ τ d τ a ( τ ) = p 1 2 M τ τ 0 d τ a ( τ ) + τ 0 τ d τ a ( τ ) =
(the above integral from τ to τ 0 equals + due to (A69), and the other integral converges).
Step 7. If τ τ 0 d τ a ( τ ) < + all maximal, future-directed and nonconstant lightlike geodesics are past incomplete. Assuming convergence of this integral, let us consider a maximal, future-directed and nonconstant lightlike geodesic; this will be of the form ξ for some datum ♭ with E = 0 , M > 0 and δ > 0 (let us recall Equation (A65); the case M = 0 , δ = 0 mentioned therein is impossible, since it would imply ξ = constant). Hereafter, we will show that p > .
Indeed, applying the last lines in Equation (A67) to the present datum, we obtain
p = p 1 2 M τ τ d τ a ( τ ) = p 1 2 M τ τ 0 d τ a ( τ ) + τ 0 τ d τ a ( τ ) >
(since, with our assumptions, both integrals above are convergent). □

Appendix F. A Review the Energy Conditions

Let us consider a spacetime of dimension d + 1 with d { 2 , 3 , 4 , } ; the present appendix follows [12] for the case d = 3 , and [13] for arbitrary d.
Let us assume that a (symmetric) stress–energy tensor T μ ν has been specified at a spacetime point (which is fixed in all the subsequent discussions). The stress–energy tensor is said to fulfill the weak energy condition (WEC) if
T μ ν X μ X ν 0 for   each   timelike   tangent   vector   X μ .
The stress–energy tensor is said to fulfill the strong energy condition (SEC) if
T μ ν 1 d 1 T g μ ν X μ X ν 0 for   each   timelike   tangent   vector   X μ ( T : = T μ μ ) .
(The following should be noted: if the Einstein equations R μ ν 1 2 R g μ ν = κ d T μ ν hold, with κ d : = d ( d 1 ) γ d G d as in Section 2.2, we have R μ ν = κ d ( T μ ν 1 d 1 T g μ ν ) ; in this case, the SEC is equivalent to R μ ν X μ X ν 0 for each timelike tangent vector X μ , which is the condition typically considered in the singularity theorems of Penrose, Hawking, and Geroch mentioned in the Introduction.)
References [12,13] discuss other energy conditions, the most important being the dominant energy condition (DEC); these will not be considered in the present paper.
To continue, let us recall that a basis of tangent vectors E ( α ) μ ( α = 0 , 1 , , d ) is called orthonormal if E μ ( α ) E ( β ) μ = η α β for α , β = 0 , 1 , , d where η 00 : = 1 , η a a : = 1 for a = 1 , , d and η α β : = 0 for α β 49. That said, assume one can associate an orthonormal basis with the stress–energy tensor T μ ν such that the matrix
T ( α ) ( β ) : = T μ ν E ( α ) μ E ( β ) ν ( α , β = 0 , 1 , , d )
has the diagonal form
T ( 0 ) ( 0 ) = ρ , T ( a ) ( a ) = p a for   a = 1 , , d , T ( α ) ( β ) = 0 for   α β .
In this case, it can be shown that [13]
T μ ν fulfills   WEC ρ 0   and   ρ + p a 0   for   a = 1 , , d ;
T μ ν fulfills   SEC ( d 2 ) ρ + a = 1 d p a 0   and   ρ + p a 0   for   a = 1 , , d .
In particular, assume that the stress–energy tensor has the perfect fluid form
T μ ν = ( p + ρ ) U μ U ν + p g μ ν , U μ U μ = 1 .
In this case, we can set E ( 0 ) μ : = U μ and add to this vector other vectors E ( a ) μ ( a = 1 , , d ) so as to obtain an orthonormal basis. The matrix elements of T μ ν with respect to this basis have the form (A77), with ρ , p as in (A80) and
p a = p for   a = 1 , , d ;
thus, ρ + p a = ρ + p for all a, ( d 2 ) ρ + a = 1 d p a = ( d 2 ) ρ + d p , and the equivalences (A78) and (A79) assume the forms (52) and (53) reported in the main text.

Appendix G. On the Determination of the Fluids’ Densities

Proof of the equivalence (102) (under the assumption (101)). As in Section 4.1, let us choose arbitrarily a smooth function a : J R + , t a ( t ) (with J an open real interval), and consider any smooth function r n : J R + , t r n ( t ) .
If r n = F n ( a d ) for some F n F n , computing r ˙ n and using the expression in (100) for F n , we readily obtain Equation (84) with p n = P n ( r n ) .
Conversely, assume r n fulfills (84) with p n = P n ( r n ) , choose t 0 J and let F n F n be such that F n ( a d ( t 0 ) ) = r n ( t 0 ) (such a function exists due to (101)). Let us provisionally put s n : = F n ( a d ) : J R + . Then, (using again the expression for F n in (100)), we see that s ˙ n + d a ˙ a ( q n + s n ) = 0 , with q n : = P n ( s n ) ; moreover, (by the previous choice for F n ) s n ( t 0 ) = F n ( a d ( t 0 ) ) = r n ( t 0 ) . To summarize, r n and s n fulfill the same ODE and coincide at time t 0 : so, by the standard uniqueness theorem for the Cauchy problem, we have r n = s n = F n ( a d ) .
On the quadrature formula (105) (under the assumption (104)). This quadrature formula arises in an obvious way from the differential equation for F n in (100) and from the specified initial condition F n ( α 0 ) = C n 0 . Let us show that (105) actually individuates a smooth function F n : R + R + .
For this purpose, let us recall that the first condition in Equation (104) reads P n ( r ) + r 0 for all r R + whence, by continuity, s i g n ( P n ( r ) + r ) = constant ζ { ± 1 } . To continue, let us rephrase Equation (105) as
G n ( F n ( α ) ) = ln α α 0 for   all   α R + , where G n : R + R , r G n ( r ) : = C n 0 r d r P n ( r ) + r .
By construction, the function G n is smooth with never vanishing derivative G n ( r ) = 1 / ( P n ( r ) + r ) , of constant sign equal to ζ . Again due to (104), G n ( r ) ζ × ( ) for r 0 + and G n ( r ) ζ × ( + ) for r + ; thus, G n is a smooth diffeomorphism between R + and ( , + ) = R . On the other hand, the map α ln ( α / α 0 ) is also a diffeomorphism between R + and ( , + ) ; so, the condition G n ( F n ( α ) ) = ln ( α / α 0 ) in (A82) actually defines a unique smooth function F n : R + R + .

Appendix H. Comparing the Zero Energy Motions of Two One-Dimensional Lagrangians

Let us consider, in general, two smooth Lagrangians
L , L : R + × R R , ( a , a ˙ ) L ( a , a ˙ ) , L ( a , a ˙ )
related by
L ( a , a ˙ ) = g ( a ) L ( a , a ˙ )
where g : R + R is a smooth, never vanishing function. The corresponding energy functions are E : = ( L / a ˙ ) a ˙ L , E : = ( L / a ˙ ) a ˙ L , and it is readily seen that
E ( a , a ˙ ) = g ( a ) E ( a , a ˙ ) .
Let us now show that the zero-energy solutions of the Lagrange equations induced by L and L coincide. In fact, along any function t a ( t ) , we have
δ L δ a = d d t L a ˙ + L a = d d t a ˙ g ( a ) L + a g ( a ) L = d d t g ( a ) L a ˙ + a g ( a ) L =
= g ( a ) d d t L a ˙ + L a g ( a ) L a ˙ a ˙ L = g ( a ) δ L δ a g ( a ) E ;
this identity, with the relation (A85) between E , E and with the assumption that g never vanishes, ensures the following: a function t a ( t ) fulfills δ L / δ a = 0 , E = 0 if and only if it fulfills δ L / δ a = 0 , E = 0 .

Appendix I. The Descartes’ Rule of Signs

Some statements in the subsequent appendices are proved via Descartes’ rule of signs (see, e.g., [84]); let us summarize this rule, which allows us to evaluate the number of positive roots of a real polynomial.
We consider a real polynomial function Q : R R of arbitrary degree, which we represent in terms of its nonzero coefficients arranging in increasing (or decreasing) order the corresponding powers:
Q ( a ) = i = 1 N Q i a k i for   all a R ,
Q i R { 0 } , k i N for i = 1 , , N , k 1 < k 2 < < k N ( or   k 1 > k 2 > > k N ) .
Let us consider the list of nonzero coefficients ( Q 1 , , Q N ) and the list of their signs ( s 1 , , s N ) (i.e., s i : = sign   Q i { ± 1 } for i = 1 , , N ). Let C be the number of sign changes between consecutive terms in that list (i.e., C : = # { i { 1 , , N 1 } | s i + 1 s i } ).
Then, the number of positive roots of Q (counting each root according to its multiplicity) is C P , for some even integer P such that 0 P C .

Appendix J. Two Statements on a Class of Polynomials

Let
m { 1 , 2 , 3 , } ; K { m + 1 , m + 2 , m + 3 , } finite ,   nonempty ;
ν R , ν > 0 ; P m R , P m > 0 ; P k R for   all   k K , P k > 0 for   some   k K .
We introduce the polynomial function
P : R R , a P ( a ) : = P m ν m + P m a m + k K P k a k
(noting that any polynomial of the form P ( a ) : = ε + P m a m + k K P k a k , with ε > 0 , can be represented as above with ν : = ( ε / P m ) 1 / m ). We also set
K : = { k K | P k 0 } ; K < : = { k K | P k < 0 }
(noting that K is nonempty; from now on, any sum over K < is meant to be zero if K < is empty).
The forthcoming statements (i) and (ii) individuate two special values a = ν θ and a = ν ( 1 ψ ) , with θ , ψ ( 0 , 1 ) suitably defined, such that P ( a ) < 0 and P ( a ) > 0 ; these results will be employed in some of the subsequent appendices to locate the zeroes of certain polynomials of the form (A88) and (A89). Here are the two statements, which are proved in the sequel of the present appendix:
(i)
Let
θ : = 1 + 1 P m k K P k ν k m 1 / m .
Then, θ is well defined, and
0 < θ < 1 ,
P ( ν θ ) < 0 .
(ii)
In addition, assume
k K P k ν k > 0 ,
and put
ψ : = k K P k ν k m m P m + k K k P k ν k m .
Then, ψ is well defined, and
0 < ψ < 1 ,
P ( ν ( 1 ψ ) ) > 0 .
Proof of (i). Let us consider Equation (A91) defining θ , and note that ( 1 / P m ) k K P k ν k m > 0 ; in fact P m > 0 , ν > 0 , P k 0 for all k K and P k > 0 for some k K . Thus, the term between round brackets in Equation (A91) is > 1 ; this suffices to infer that the quantity θ therein is well defined and fulfills (A92).
In order to derive Equation (A93), we first note that50
P ( a ) = P m ν m + P m a m + k K P k a k + k K < P k a k
P m ν m + P m a m + k K P k a k for   all   a R + ;
in particular,
P ( ν θ ) P m ν m + P m ν m θ m + k K P k ν k θ k .
However, for any k K we have θ k < θ m since θ ( 0 , 1 ) and k > m ; since P k ν k 0 for all k K and P k ν k > 0 for some k K , this implies P k ν k θ k P k ν k θ m for all k K , with a strict inequality < for some k. Thus
P ( ν θ ) < P m ν m + P m ν m + k K P k ν k θ m
= P m ν m 1 + 1 + 1 P m k K P k ν k m θ m ,
and the term between square brackets vanishes due to the definition (A91) of θ ; so, the relation (A93) is proved.
Proof of (ii). Let us consider the ratio defining ψ in Equation (A95). The numerator in this ratio is positive due to (A94), and the denominator is positive since m P m > 0 and k P k ν k m 0 for all k K . Thus, ψ is well defined and ψ > 0 . Moreover,
k K P k ν k m = k K P k ν k m + k K < P k ν k m k K P k ν k m
k K k P k ν k m < m P m + k K k P k ν k m
(since m P m > 0 ); thus, ψ < 1 , and the proof of (A96) is concluded.
Let us proceed to prove Equation (A97); for this purpose, we write
P ( ν ( 1 ψ ) ) = P m ν m + P m ν m ( 1 ψ ) m + k K + k K < P k ν k ( 1 ψ ) k .
Since 0 < ψ < 1 , m 1 and k > 1 for all k K , by Bernoulli’s inequality (see, e.g., [85]), we have ( 1 ψ ) m 1 m ψ and ( 1 ψ ) k > 1 k ψ for all k K . Since P m ν m > 0 , P k ν k 0 for all k K and P k ν k > 0 for some k K , we infer
P m ν m ( 1 ψ ) m P m ν m ( 1 m ψ ) ,
P k ν k ( 1 ψ ) k P k ν k ( 1 k ψ ) for   all   k K , P k ν k ( 1 ψ ) k > P k ν k ( 1 k ψ ) for   some   k K .
Let us also note that ( 1 ψ ) k < 1 for k K and P k ν k < 0 for k K < , so that
P k ν k ( 1 ψ ) k > P k ν k for   k K < .
Inserting the inequalities (A103), (A104) into (A102), we obtain
P ( ν ( 1 ψ ) ) > P m ν m + P m ν m ( 1 m ψ ) + k K P k ν k ( 1 k ψ ) + k K < P k ν k
= ψ m P m ν m + k K k P k ν k + k K + k K < P k ν k ,
i.e.,
P ( ν ( 1 ψ ) ) > ν m ψ m P m + k K k P k ν k m + k K P k ν k m ;
the above quantity between square brackets vanishes due to the definition (A95) of ψ , so the relation (A97) is proved.

Appendix K. Proof of Statements (185) and (186) on the Potential V of the Standard Cosmological Model

We consider the potential V of (183), with the assumptions (165) and (176) Ω r , Ω m , Υ > 0 and with an unspecified sign for Ω k ; let us derive the results (185) and (186) about the derivative V . For this purpose, we note that Equation (183) implies
V ( a ) = 2 Ω r a 3 + Ω m a 2 2 Υ a = Q ( a ) a 3 for   a R + ,
where we have introduced the polynomial function
Q : R R , a Q ( a ) : = 2 Ω r + Ω m a 2 Υ a 4 .
Equation (A106) implies that, for all a R + ,
sign V ( a ) = sign Q ( a ) ( in   particular , V ( a ) = 0 Q ( a ) = 0 ) .
Let us discuss the zeroes of V in R + , using their coincidence with the positive roots of the polynomial Q and employing Descartes’ rule of Appendix I. The nonzero coefficients of Q are ( 2 Ω r , Ω m , 2 Υ ) , with signs ( + , + , ) . The number of sign changes in this list is C = 1 ; the number of positive roots of Q is C P with P even and 0 P C , which means 1 0 = 1 . In conclusion, there is a unique a 2 such that
a 2 R + , Q ( a 2 ) = 0 ;
this is also the unique zero of V in R + , described by Equation (185). The function Q is never vanishing on ( 0 , a 2 ) and on ( a 2 , + ) ; hence, by continuity, Q has a constant nonzero sign on each one of these intervals. On the other hand, for a 0 + , we have Q ( a ) 2 Ω r > 0 , while for a + , we have Q ( a ) ; these facts suffice to infer
Q ( a ) > 0 for   a ( 0 , a 2 ) , Q ( a ) < 0 for   a ( a 2 , + ) .
Due to (A108), analogous inequalities hold for V on the above intervals; this proves statements (186).

Appendix L. Proof of Statements (a)–(d) in Section 6.5 about the Function V of Equation (201)

Let us recall the general assumptions (198)–(200) and (204); the proofs of (c) and (d) also require condition (211).
Proof of (a). Equation (201) tells us that
V ( a ) = P ( a ) a 4 for   a R + ,
where we have introduced the polynomial function
P : R R , a P ( a ) = Ω r μ 2 + Ω r a 2 + Ω m a 3 + Ω k a 4 + Υ a 6 .
Of course, (A111) implies that, for all a R + ,
sign V ( a ) = sign P ( a ) ( in   particular , V ( a ) = 0 P ( a ) = 0 ) .
Keeping in mind these facts, let us discuss the positive roots of the polynomial P using Descartes’ rule reviewed in Appendix I. If Ω k > 0 the nonzero coefficients of P are ( Ω r μ 2 , Ω r , Ω m , Ω k , Υ ) with signs ( , + , + , + , + ) ; if Ω k = 0 the nonzero coefficients of P are ( Ω r μ 2 , Ω r , Ω m , Υ ) with signs ( , + , + , + ) . In both cases, the number of sign changes is C = 1 and the unique number of the form C P , with P even and 0 P C , is 1 0 = 1 .
In conclusion, there is a unique a 0 such that
a 0 R + , P ( a 0 ) = 0 ;
this is also the unique zero of V in R + , described by Equation (205). The function P is never vanishing on ( 0 , a 0 ) and on ( a 0 , + ) ; hence, by continuity, P has a constant nonzero sign on each one of these intervals. On the other hand, for a 0 + , we have P ( a ) Ω r μ 2 < 0 , while for a + , we have P ( a ) + ; these facts suffice to infer
P ( a ) < 0 for   a ( 0 , a 0 ) , P ( a ) > 0 for   a ( a 0 , + ) .
Due to (A113), the reversed inequalities hold for V in these two intervals; this proves statements (206).
Finally, let us prove the inequality (207) a 0 < 1 . Due to (A114) and (A115), it will suffice to show that
P ( 1 ) > 0 ;
in fact, Equations (A112) and (200) give51
P ( 1 ) = Ω r μ 2 + Ω r + Ω m + Ω k + Υ = 1 .
Proof of (b). It is evident that the quantities N , F defined by (208) fulfill N > 0 , F > 0 and F μ < 1 . That said, let us derive the lower and upper bounds (209) on a 0 ; for this purpose, we apply the general setting of Appendix J, choosing for P the polynomial function P of Equation (A112). By comparison between Equations (A88), (A89) and (A112), we see that in the present case
ν = μ ; m = 2 , P m = P 2 = Ω r ;
K = { 3 , 4 , 6 } , P 3 = Ω m > 0 , P 4 = Ω k 0 , P 6 = Υ > 0 ; K = K .
Due to (A118), Equations (A91) gives in the present case
θ = 1 + Ω m μ + Ω k μ 2 + Υ μ 4 Ω r 1 / 2 = 1 1 + N μ ,
with N as in (208); the condition (A94) is automatically fulfilled, and Equation (A95) gives
ψ = Ω m μ + Ω k μ 2 + Υ μ 4 2 Ω r + 3 Ω m μ + 4 Ω k μ 2 + 6 Υ μ 4 = F μ ,
with F as in (208).
The inequalities (A93) and (A97) become in this case
P μ 1 + N μ < 0 , P μ ( 1 F μ ) > 0 .
By comparison between Equations (A121), (A114) and (A115), we obtain μ / 1 + N μ < a 0 and a 0 < μ ( 1 F μ ) ; these are just the bounds on a 0 in (209) (and it is evident that the upper bound implies a less refined bound a 0 < μ ).
Proof of (c). Equation (210) tells us that
V ( a ) = Q ( a ) a 5 for a R + ,
where we have introduced the polynomial function
Q : R R , a Q ( a ) : = 4 Ω r μ 2 + 2 Ω r a 2 + Ω m a 3 2 Υ a 6 ;
of course (A122) implies that, for all a R + ,
sign V ( a ) = sign Q ( a ) ( in particular , V ( a ) = 0 Q ( a ) = 0 ) .
Let us discuss the zeroes of V in R + , using their coincidence with the positive roots of the polynomial Q and employing again Descartes’ rule of Appendix I. The nonzero coefficients of Q are ( 4 Ω r μ 2 , 2 Ω r , Ω m , 2 Υ ) , with signs ( , + , + , ) . The number of sign changes in this list is C = 2 ; the number of positive roots of Q is C P with P even and 0 P C , which means either 2 0 = 2 or 2 2 = 0 .
We now show that Q has in fact two distinct, positive roots using the inequality (211) 4 2 Υ / Ω m 1 / 3 μ < 1 , which is assumed in (c). In fact,
Q ( 2 μ ) = 2 2 Ω m μ 3 1 4 2 Υ Ω m μ 3
and the above term between round brackets is positive due to (211), so that
Q ( 2 μ ) > 0 .
On the other hand, from the definition of Q , it appears that
Q ( 0 ) = 4 Ω r μ 2 < 0 , lim a + Q ( a ) = .
Thus, there is a unique pair a 1 , a 2 such that
a 1 , a 2 R + , a 1 < a 2 , Q ( a 1 ) = Q ( a 2 ) = 0 ;
moreover,
a 1 < 2 μ < a 2 ,
Q ( a ) < 0 for a ( 0 , a 1 ) ( a 2 , + ) , Q ( a ) > 0 for a ( a 1 , a 2 ) .
Since V has the same zeroes and the same sign of Q on R + , statements (212) and (213) of (c) about the zeroes and the sign of V are proved. Moreover, Equation (A128) gives the second and the third inequality in (214).
To conclude, let us prove the first inequality in (214), namely the relation
a 0 < a 1 .
Due to the already proved statements (205) and (206) in (a), to obtain the inequality (A130), it suffices to show that
V ( a 1 ) < 0 ;
on the other hand, we know that a 1 < 2 μ < a 2 and that V > 0 on ( a 1 , a 2 ) ; thus, V ( a 1 ) < V ( 2 μ ) and, consequently, (A131) follows if we show that
V ( 2 μ ) < 0 .
But we know from (A113) that sign V = sign P with P as in Equation (A112), so we are reduced to proving that
P ( 2 μ ) > 0 .
Indeed, Equation (A112) gives
P ( 2 μ ) = Ω r μ 2 + 2 2 Ω m μ 3 + 4 Ω k μ 4 + 8 Υ μ 6 ,
so Equation (A133) (and thus, Equation (A131)) is proved.
Proof of (d). It is evident that the quantities M , G defined by (215) fulfill M > 0 , G > 0 (by (211)) and G μ < 1 . That said, let us derive the lower and upper bounds (216) on a 1 ; for this purpose, we apply the general setting of Appendix J, choosing for P the polynomial function Q of Equation (A123). By comparison between Equations (A88), (A89) and (A123), we see that in the present case
ν = 2 μ ; m = 2 , P m = P 2 = 2 Ω r ;
K = { 3 , 6 } , P 3 = Ω m > 0 , P 6 = 2 Υ < 0 ; K = { 3 } .
Due to (A135), Equations (A91) gives in the present case
θ = 1 + Ω m 2 Ω r · 2 μ 1 / 2 = 1 1 + M μ ,
with M as in (215). The condition (A94) is fulfilled, since
k K P k ν k = Ω m ( 2 μ ) 3 2 Υ ( 2 μ ) 6 = 2 2 Ω m μ 3 1 4 2 Υ Ω m μ 3 ,
and the above quantity is positive due to the assumption (211). Equation (A95) gives
ψ = Ω m · 2 μ 2 Υ ( 2 μ ) 4 4 Ω r + 3 Ω m · 2 μ = G μ ,
with G as in (215). The inequality (A93) becomes in this case
Q 2 μ 1 + M μ < 0 .
By comparison between Equations (A138), (A127) and (A129), we obtain that either
2 μ 1 + M μ < a 1 ,
or
a 2 < 2 μ 1 + M μ .
On the other hand, the inequality (A140) cannot hold; in fact, if (A140) were true we would infer a 2 < 2 μ (because M > 0 ), while we know from Equation (214) that a 2 > 2 μ . Having excluded (A140), we conclude that (A139) holds.
We now refer to the inequality (A97), which in the present case becomes
Q 2 μ 1 G μ > 0 .
By comparison between Equations (A141), (A127) and (A129), we obtain
a 1 < 2 μ 1 G μ .
Equations (A139) and (A142) correspond to the bounds (216) for a 1 .

Appendix M. Proof of Statements (f)–(i) in Section 6.5 about the Function H of Equation (220)

Let us recall once more the general assumptions (198)–(200) and (204); the proof of (i) also requires condition (211).
Proof of (f). Equation (225) tells us that, for each a R + ,
H ( a ) = R ( a ) a 7 , R ( a ) : = 6 Ω r μ 2 + 4 Ω r a 2 + 3 Ω m a 3 + 2 Ω k a 4 ;
again for a R + , we have
sign H ( a ) = sign R ( a ) ( in particular , H ( a ) = 0 R ( a ) = 0 ) .
Let us discuss the zeroes of H in R + using their coincidence with the positive roots of the polynomial R , and employing again Descartes’ rule of Appendix I. If Ω k > 0 , the nonzero coefficients of R are ( 6 Ω r μ 2 , 4 Ω r , 3 Ω m , 2 Ω k ) , with signs ( , + , + , + ) ; if Ω k = 0 , the nonzero coefficients of R are ( 6 Ω r μ 2 , 4 Ω r , 3 Ω m ) , with signs ( , + , + ) . In both cases, the number of changes in the list of signs is C = 1 ; the number of positive roots of R is C P with P even and 0 P C , that means 1 0 = 1 .
Thus, there is a unique a 1 / 2 such that
a 1 / 2 R + , R ( a 1 / 2 ) = 0 .
Noting that R ( a ) 6 Ω r μ 2 < 0 for a 0 + and R ( a ) + for a + , we also infer
R ( a ) < 0 for a ( 0 , a 1 / 2 ) , R ( a ) > 0 for a ( a 1 / 2 , + ) .
Equations (A144)–(A146) yield statements (226) and (227) in (f) about the zero and the sign of H . The statements in (f) after Equation (227), yielding Equations (228) and (229), are obvious.
Proof of (g). It is evident that the quantities S , T defined by (230) fulfill S > 0 , T > 0 and T μ < 1 . That said, let us derive the lower and upper bounds (231) on a 1 / 2 ; for this purpose, we apply the general setting of Appendix J, choosing for P the polynomial function R of Equation (A143). By comparison between Equations (A88), (A89) and (A143), we see that in the present case
ν = 3 / 2 μ ; m = 2 , P m = P 2 = 4 Ω r ;
K = { 3 , 4 } , P 3 = 3 Ω m > 0 , P 4 = 2 Ω k 0 ; K = K .
Due to (A147), Equations (A91) gives in the present case
θ = 1 + 3 Ω m · 3 / 2 μ + 2 Ω k ( 3 / 2 μ ) 2 4 Ω r 1 / 2 = 1 1 + S μ ,
with S as in (230); the condition (A94) is automatically fulfilled, and Equation (A95) gives
ψ = 3 Ω m · 3 / 2 μ + 2 Ω k ( 3 / 2 μ ) 2 8 Ω r + 9 Ω m · 3 / 2 μ + 8 Ω k ( 3 / 2 μ ) 2 = T μ ,
with T as in (230).
The inequalities (A93) and (A97) become in this case
R 3 / 2 μ 1 + S μ < 0 , R 3 / 2 μ 1 T μ > 0 .
By comparison between Equations (A150), (A145) and (A146), we obtain the inequalities (231) about a 1 / 2 ; the comment in (g) after Equation (231) is obvious.
Proof of (h). We must derive the bounds (232) and (233) on H ( a 1 / 2 ) , on the grounds of the bounds (231) on a 1 / 2 . For this purpose, let us rephrase the definition (220) of H for all a R + as follows:
H ( a ) = Ω r a 4 μ 2 a 2 + 1 + Ω m Ω r a + Ω k Ω r a 2 + Υ Ω r a 4 .
This holds, in particular, for a = a 1 / 2 (note that, for any a > a 0 and in particular for a = a 1 / 2 , we have H ( a ) > 0 due to (224), and this ensures the positivity of the factor between square brackets in the right-hand side of Equation (A151)). On the other hand, due to (231), we can write
4 9 μ 4 ( 1 T μ ) 4 < 1 ( a 1 / 2 ) 4 < 4 ( 1 + S μ ) 2 9 μ 4
and
2 3 ( 1 + S μ ) + 1 < μ 2 ( a 1 / 2 ) 2 + 1 < 2 3 ( 1 T μ ) 2 + 1 ,
i.e.,
1 3 2 3 S μ < μ 2 ( a 1 / 2 ) 2 + 1 < 1 3 4 T μ 3 ( 1 T μ ) 2 + 2 T 2 μ 2 3 ( 1 T μ ) 2 ;
again due to (231), we have
3 / 2 μ 1 + S μ k < ( a 1 / 2 ) k < 3 / 2 μ 1 T μ k ( k = 1 , 2 , 4 ) .
Using Equation (A151) with a = a 1 / 2 , and inserting therein the inequalities (A152)–(A154) we readily obtain the bounds (232) on H ( a 1 / 2 ) , with X and Y as in (233).
Proof of (i). Assume (211) holds. Our aim is to infer the inequality (235) a 1 / 2 < a 1 under the supplementary condition (234) M μ < 1 3 . For this purpose, we use the known inequalities
a 1 / 2 < 3 / 2 μ , 2 μ 1 + M μ < a 1
(see the comment after Equation (231) and Equation (216)). Due to these relations, we have a 1 / 2 < a 1 if
3 / 2 μ < 2 μ 1 + M μ ;
on the other hand, (A156) is clearly equivalent to (234).

Appendix N. Proof of Equation (250) (Including Convergence of an Integral Therein)

Let us keep the assumptions (198)–(200), (204), (211) and (234) of Section 6.6; as in Equation (247), we refer to the time t R + such that a ( t ) = 1 . From now on, we consider any time t such that
t t
(so that a ( t ) 1 ). Using, e.g., Equation (148) we can write
1 a ( t ) d a V ( a ) = t t .
On the other hand,
ln a ( t ) = 1 a ( t ) d a a = 1 a ( t ) d a 1 a Υ V ( a ) + 1 a ( t ) d a Υ V ( a ) .
In the last expression above, the second integral equals Υ ( t t ) due to (A158); the first integral will be denoted with I ( t ) . To summarize,
ln a ( t ) = I ( t ) + Υ ( t t ) , I ( t ) : = 1 a ( t ) d a 1 a Υ V ( a ) .
Of course, this implies
a ( t ) = e I ( t ) e Υ ( t t ) .
We now consider the limit t + ; then a ( t ) + , and
I ( t ) I , I : = 1 d a 1 a Υ V ( a ) .
The integral I is convergent; in fact, from the expression (201) of V , we see that
1 a Υ V ( a ) = 1 a 1 a 1 + Ω k Υ a 2 + Ω m Υ a 3 + Ω r Υ a 4 Ω r μ 2 Υ a 6 1 / 2 ,
and for a + , we infer
1 a Υ V ( a ) = 1 a 1 a 1 + O 1 a 2 1 / 2 = 1 a 1 a 1 + O 1 a 2 = O 1 a 3 .
From the above considerations and from Equations (A161) and (A162), we see that, for t + , we have
a ( t ) e I e Υ ( t t ) = K e Υ t , K : = exp I Υ t .
On the other hand, making explicit the definition (A162) of I we see that the present result (A165) coincides with the thesis (250).

Appendix O. Proof of Some Statements about a ϕ r in Equation (271)

Equation (271) involves the polynomial function
S : R R , a S ( a ) : = Ω r μ 2 + Ω r a 2 + Υ a 6 ,
and states that S has a unique root in R + , indicated with a ϕ r ; the same equation states that, for a R + , S ( a ) 0 if and only if a a ϕ r . The subsequent Equations (272) and (273) contain inequalities about a ϕ r . All these claims are proved hereafter.
Proof that S has a unique root in R + . We again use Descartes’ rule of Appendix I. The nonzero coefficients of the polynomial S are ( Ω r μ 2 , Ω r , Υ ) , with signs ( , + , + ) . The number of sign changes in this list is C = 1 ; the number of positive roots of S is C P with P even and 0 P C , which means 1 0 = 1 . To summarize, there is a unique a ϕ r such that
a ϕ r R + , S ( a ϕ r ) = 0 .
The sign of S on R + . Clearly, S has a constant nonzero sign on each one of the intervals ( 0 , a ϕ r ) and ( a ϕ r , + ) . Noting that S ( a ) Ω r μ 2 < 0 for a 0 and S ( a ) + for a + , we conclude that
S ( a ) < 0 for a ( 0 , a ϕ r ) , S ( a ) > 0 for a ( a ϕ r , + ) .
Proof of the relation (272) a ϕ r > a 0 . Let us consider besides S the polynomial function P of Equation (A112). We have
P ( a ) = S ( a ) + Ω m a 3 + Ω k a 4 > S ( a ) for all a R + ;
from here and from S ( a ϕ r ) = 0 , we obtain
P ( a ϕ r ) > 0 .
Equations (A170), (A114) and (A115) yield the thesis a ϕ r > a 0 .
Proof of the bounds (273) on a ϕ r . We apply the general setting of Appendix J, choosing for P the polynomial function S of Equation (A166). By comparison between Equations (A88), (A89) and (A166), we see that in the present case
ν = μ ; m = 2 , P m = P 2 = Ω r ;
K = { 6 } , P 6 = Υ > 0 ; K = K .
Due to (A171), Equation (A91) gives in the present case
θ = 1 1 + Υ μ 4 / Ω r ;
the condition (A94) is automatically fulfilled, and Equation (A95) gives
ψ = Υ μ 4 2 Ω r + 6 Υ μ 4 .
The inequalities (A93) and (A97) become in this case
S μ 1 + Υ μ 4 / Ω r < 0 , S μ 1 Υ μ 4 2 Ω r + 6 Υ μ 4 > 0 .
By comparison between Equations (A174), (A167) and (A168), we obtain the inequalities (273).

Appendix P. A Usefulf Inequality

The inequality presented hereafter will be employed in Appendix R and Appendix S; it states that
1 ( x + δ ) γ > 1 x γ max ( 1 , γ ) δ ϑ x γ + ϑ for γ , x , δ R + and ϑ [ 0 , 1 ] .
In the sequel, we prove (A175); our argument will be divided into two steps.
Step 1. One has
1 ( 1 + z ) γ > 1 max ( γ , 1 ) z ϑ for γ , z , R + and ϑ [ 0 , 1 ] .
To prove this, let γ R + and ϑ [ 0 , 1 ] . By the Bernoulli inequality (see again [85]), for any z ( 1 , + ) { 0 } , we have 1 / ( 1 + z ) γ = ( 1 + z ) γ > 1 γ z . In addition, z z ϑ if z ( 0 , 1 ] ; thus,
1 ( 1 + z ) γ > 1 γ z ϑ for z ( 0 , 1 ] .
To continue, we note that 1 1 / ( 1 + z ) γ < 1 for all z ( 1 , + ) . If z [ 1 , + ) we also have 1 z ϑ , so that 1 1 / ( 1 + z ) γ < z ϑ ; equivalently, we can say that
1 ( 1 + z ) γ > 1 z ϑ for z [ 1 , + ) .
Finally, γ z ϑ , z ϑ max ( γ , 1 ) z ϑ for z R + , i.e.,
γ z ϑ , z ϑ max ( γ , 1 ) z ϑ for z R + .
Equations (A177)–(A179) yield the thesis (A176).
Step 2. Derivation of Equation (A175). Let γ , x , δ R + , ϑ [ 0 , 1 ] , and let us apply (A176) with z = δ / x ; this gives
x γ ( x + δ ) γ > 1 max ( γ , 1 ) δ ϑ x ϑ ,
and dividing both sides by x γ , we derive the thesis (A175).

Appendix Q. Estimates on the Time t i in Equation (244)

Let us again refer to the potential V of Equation (201), under the assumptions (200) and (204). We consider a time t i by admitting the integral representation (244)
t i = a 0 a i d a V ( a ) ,
where a 0 > 0 is the unique zero of V , and the final extreme a i in the integral is any real number such that a i > a 0 .
In the sequel, we will derive rigorous upper and lower bounds for t i , and we will present situations in which they are useful to estimate t i .
Statement of the bounds on t i . Let us define the constants
Z : = a i a 0 1 ,
U : = 1 + 3 Ω m 2 Ω r a 0 + 2 Ω k Ω r a 0 2 + 3 Υ Ω r a 0 4 , W : = 1 + 3 Ω m Ω r a i + 6 Ω k Ω r a i 2 + 15 Υ Ω r a i 4
(noting that Z > 0 and U , W > 1 ); moreover, let us introduce the functions
I 0 , I 1 , I 2 : R + R + ,
I 0 ( z ) : = 0 1 d s 1 + z s 2 = arcsinh z z , I 1 ( z ) : = 0 1 d s s 2 1 + z s 2 = z ( 1 + z ) arcsinh z 2 z 3 / 2 ,
I 2 ( z ) : = 0 1 d s s 4 1 + z s 2 = z ( 1 + z ) ( 2 z 3 ) + 3 arcsinh z 8 z 5 / 2 for z R + .
That said, our bounds on t i are as follows:
t i t i t i + ,
t i : = 2 Z U Ω r a 0 2 I 0 W Z 2 U + 2 Z I 1 W Z 2 U + Z 2 I 2 W Z 2 U ,
t i + : = 2 Z Ω r a 0 2 I 0 Z 2 + 2 Z I 1 Z 2 + Z 2 I 2 Z 2 .
The inequalities in (A183) have a general validity, but of course they deserve a special interest when t i and t i + are close. From the explicit expressions of t i , it is clear that t i are close if U 1 , W 1 and Z is not large, which happens if a 0 , a i are both small and a i / a 0 is not large.
The forthcoming three paragraphs in this appendix will be devoted to the proof of (A183). The subsequent paragraph will present coarser bounds on t i , following from (A183) when lower and upper bounds are available for a 0 and a i . The final two paragraphs will describe a number of applications; these include the justification of Equations (285)–(287) in the main text, describing the asymptotic behavior of certain times when the parameter μ of Section 6.4, Section 6.5 and Section 6.6 is sent to zero.
A preliminary to the derivation of (A183): a new integral representation for t i . According to Equations (A111) and (A112), for all a R + , we have
V ( a ) = P ( a ) a 4 , P ( a ) = Ω r μ 2 + Ω r a 2 + Ω m a 3 + Ω k a 4 + Υ a 6 ;
from Equations (A114) and (A115), we know that P ( a 0 ) = 0 and P ( a ) > 0 for all a > a 0 . Equation (244) for t i can be rephrased as
t i = a 0 a i d a a 2 P ( a ) .
Let a [ a 0 , a i ] . By the Taylor formula, we have
P ( a ) = P ( a 0 ) + P ( a 0 ) ( a a 0 ) + 1 2 P ( c a ) ( a a 0 ) 2 , c a [ a 0 , a ] [ a 0 , a i ] .
But
P ( a 0 ) = 0 , P ( a 0 ) = 2 Ω r a 0 + 3 Ω m a 0 2 + 4 Ω k a 0 3 + 6 Υ a 0 5 = 2 U Ω r a 0
with U as in Equation (A180), and
1 2 P ( c a ) = Ω r + 3 Ω m c a + 6 Ω k c a 2 + 15 Υ c a 4 .
These facts will be used hereafter to derive the lower and upper bounds (A183) on t i .
Proof of the lower bound in (A183). Keeping in mind Equations (A184)–(A187) we note that, for a [ a 0 , a i ] ,
1 2 P ( c a ) Ω r + 3 Ω m a i + 6 Ω k a i 2 + 15 Υ a i 4 = W Ω r ,
with W as in Equation (A180). Again for a [ a 0 , a i ] , Equations (A185), (A186) and (A188) give
P ( a ) 2 U Ω r a 0 ( a a 0 ) + W Ω r ( a a 0 ) 2 = 2 U Ω r a 0 ( a a 0 ) 1 + W 2 U a 0 ( a a 0 ) ,
and inserting this inequality in (A184), we obtain
t i 1 2 U Ω r a 0 a 0 a i d a a 2 a a 0 1 + W 2 U a 0 ( a a 0 ) .
We now reexpress the last integral via a change in variable a = a 0 + ( a i a 0 ) s 2 , 0 s 1 . By comparison with the definition (A180) of Z , we can write equivalently a = a 0 ( 1 + Z s 2 ) and this gives
t i 2 Z U Ω r a 0 2 0 1 d s ( 1 + Z s 2 ) 2 1 + W Z 2 U s 2 .
We finally expand ( 1 + Z s 2 ) 2 , and integrate term by term; by comparison with Equation (A182), this gives the lower bound on t i t i in Equation (A183).
Proof of the upper bound in (A183). We refer again to Equations (A184)–(A187). Let a [ a 0 , a i ] ; noting that Equations (A186) and (A187) imply
P ( a 0 ) 2 Ω r a 0 ,
1 2 P ( c a ) Ω r ,
we infer from (A185) that
P ( a ) 2 Ω r a 0 ( a a 0 ) + Ω r ( a a 0 ) 2 = 2 Ω r a 0 ( a a 0 ) 1 + 1 2 a 0 ( a a 0 ) .
Inserting this inequality into Equation (A184), we obtain
t i 1 2 Ω r a 0 a 0 a i d a a 2 a a 0 1 + 1 2 a 0 ( a a 0 ) .
We now reexpress the last integral via a change in variable a = a 0 + ( a i a 0 ) s 2 = a 0 ( 1 + Z s 2 ) , 0 s 1 , which gives
t i 2 Z Ω r a 0 2 0 1 d s ( 1 + Z s 2 ) 2 1 + Z 2 s 2 .
Expanding ( 1 + Z s 2 ) 2 , integrating term by term and comparing with Equation (A182), we finally obtain the upper bound on t i t i + in Equation (A183).
Coarser bounds on t i , following from (A183). We now assume that we have bounds a 0 and a i for a 0 and a i , fulfilling the inequalities
0 < a 0 < a 0 < a 0 + < a i a i a i + .
In this case, let us put
Z : = a i a 0 + 1 , Z + : = a i + a 0 1 ,
U + : = 1 + 3 Ω m 2 Ω r a 0 + + 2 Ω k Ω r ( a 0 + ) 2 + 3 Υ Ω r ( a 0 + ) 4 ,
W + : = 1 + 3 Ω m Ω r a i + + 6 Ω k Ω r ( a i + ) 2 + 15 Υ Ω r ( a i + ) 4
(noting that Z > 0 and U + , W + > 1 ). We claim that the time t i described by Equation (244) admits the bounds
t i < t i < t i + + ,
t i : = 2 Z U + Ω r ( a 0 ) 2 I 0 W + Z + 2 + 2 Z I 1 W + Z + 2 + ( Z ) 2 I 2 W + Z + 2 ,
t i + + : = 2 Z + Ω r ( a 0 + ) 2 I 0 Z 2 + 2 Z + I 1 Z 2 + ( Z + ) 2 I 2 Z 2 ,
where I k : R + R + ( k = 0 , 1 , 2 ) are again the functions in (A182).
In order to prove (A200) we start from the bounds (A183) t i t i t i + , where t i are defined via the functions I k and the constants Z , U , W of Equation (A180). It is readily checked that Z < Z < Z + , 1 < U < U + , W W + , and that I k : R + R + is (strictly) decreasing for k = 0 , 1 , 2 . These facts (and the inequalities ( a 0 ) 2 < a 0 2 < ( a 0 + ) 2 ) imply
t i < t i , t i + < t i + + ;
so the bounds (A200) follow from the bounds (A183), of which they are coarsenings. Needless to say, the coarser bounds (A200) are especially interesting when a 0 and a i in Equation (A197) are sufficiently close to a 0 and a i , respectively.
Applications of the coarser bounds (A200). Let us make the assumptions (200), (204), (211) and (234). The bounds (A200) can be applied to the times t i in Section 6.5, Section 6.6 and Section 6.7 with i = 1 / 2 , 1 , ϕ r , using available bounds a 0 and a i (that are required to fulfill (A197)).
More precisely, we can use the bounds a 0 provided by Equations (208) and (209), and the bounds a i provided by Equations (230) and (231) for i = 1 / 2 , by Equations (215) and (216) for i = 1 , and by Equation (273) for i = ϕ r (i.e.: a 0 = μ / 1 + N μ and a 0 + = μ ( 1 F μ ) , with N , F as in Equation (208); a 1 / 2 = 3 / 2 μ / 1 + S μ with S as in (230), and so on).
Justification of the asymptotic expressions for certain times. Let us consider a family of models as in (217), and the limit μ 0 + . In the main text, it is stated that the times t i ( i = 1 / 2 , 1 , ϕ r ) have the asymptotic expressions (285), (286) and (287) for μ 0 + .
These statements can be justified on the grounds of the bounds (A200). Let us consider, for example, Equation (285) here reproduced:
t 1 / 2 = K 1 / 2 Ω r μ 2 ( 1 + O ( μ ) ) , K 1 / 2 : = 3 4 + arcsinh 1 2 3 2 1 2 0 . 7623 .
This can be justified using Equation (A200) with i = 1 / 2 , a 0 as in (208) and (209), and a 1 / 2 as in (230) and (231); with these prescriptions, both the lower and the upper bounds t 1 / 2 , t 1 / 2 + + in (A200) behave like the right-hand side of Equation (285) for μ 0 + .

Appendix R. Further Estimates on the Time t i in Equation (244)

Let us refer once more to the potential V of Equation (201), under the assumptions (200) and (204); we reconsider the quantity t i = a 0 a i d a / V ( a ) in Equation (244), with a i > a 0 . In the present appendix, we will derive upper and lower bounds on t i , different from those of Appendix Q, and we will illustrate situations in which the new bounds are useful.
Statement of the bounds on t i . We begin our construction by introducing the constants U as in (A180), and
U : = 1 + 3 Ω m a 0 + 6 Ω k a 0 2 + 15 Υ a 0 4 Ω r ,
A : = Ω m + 4 Ω k a 0 + 20 Υ a 0 3 Ω r , B : = Ω k + 15 Υ a 0 2 Ω r , C : = Υ Ω r
(note that U , U > 1 and A , B , C > 0 ); from these objects, we build the function
R : [ 0 , 1 ] R + ,
R ( s ) : = U + A ( a i a 0 ) s 2 + B ( a i a 0 ) 2 s 4 + 6 C a 0 ( a i a 0 ) 3 s 6 + C ( a i a 0 ) 4 s 8 .
That said, our bounds on t i are
θ i t i θ i + ,
θ i : = 2 ( a i a 0 ) 2 Ω r 0 1 d s s 3 R ( s ) 2 3 U 2 U a 0 Ω r ( a i a 0 ) 3 / 2 ,
θ i + : = 2 ( a i a 0 ) 2 Ω r 0 1 d s s 3 R ( s ) + 2 a 0 U Ω r 2 3 ( a i a 0 ) 3 / 2 + a 0 a i a 0 .
The inequalities in (A204) have a general validity but, of course, are especially useful when the bounds θ i are close. It is seen from the explicit expressions that θ i and θ i + tend to the same limit if a 0 is sent to 0 + while keeping a i > 0 and the other parameters fixed; this suggests that θ i and θ i + should be close for a 0 small and for appropriate values of a i and of the parameters, a fact that is checked a posteriori in many interesting cases and, in particular, in the applications of the present framework considered in Section 6.7.
In order to employ the bounds (A204), the integral 0 1 d s s 3 / R ( s ) appearing therein must be computed numerically; however, this calculation is more reliable than the direct numerical computation of t i via (244), due to the nonsingular nature of the function s s 3 / R ( s ) on [ 0 , 1 ] .
The forthcoming three paragraphs in this appendix will be devoted to the proof of (A204). The subsequent paragraph will present coarser bounds on t i , following from (A204) when suitable bounds are available for a 0 and a i ; the final paragraph will describe a number of applications.
A preliminary to the derivation of (A204): another integral representation for t i . We start from the representation (A184)
t i = a 0 a i a 2 d a P ( a ) ,
where P is the polynomial function defined in Equation (A112). Having degree 6, the polynomial P coincides with its Taylor expansion of order 6 about any point and, in particular, about a 0 ; thus, for all real a , we have
P ( a ) = k = 0 6 P ( k ) ( a 0 ) k ! ( a a 0 ) k .
From Equation (A186), we know that P ( a 0 ) = 0 and P ( a 0 ) = 2 U Ω r a 0 , with U as in Equation (A180). Moreover, with U , A , B , C as in Equation (A202), we have
1 2 P ( a 0 ) = Ω r U ,
1 3 ! P ( a 0 ) = Ω r A , 1 4 ! P ( 4 ) ( a 0 ) = Ω r B , 1 5 ! P ( 5 ) ( a 0 ) = 6 Ω r C a 0 , 1 6 ! P ( 6 ) ( a 0 ) = Ω r C .
Thus, for all real a ,
P ( a ) = Ω r ( a a 0 ) ×
2 U a 0 + ( a a 0 ) U + A ( a a 0 ) + B ( a a 0 ) 2 + 6 C a 0 ( a a 0 ) 3 + C ( a a 0 ) 4 .
We insert this expression for P ( a ) into Equation (A184), and then we reexpress the integral therein via a change in variable a = a 0 + ( a i a 0 ) s 2 , 0 s 1 . In this way, we obtain
t i = 2 a i a 0 Ω r 0 1 d s a 0 + ( a i a 0 ) s 2 2 2 U a 0 + ( a i a 0 ) s 2 R ( s ) ,
with R ( s ) as in Equation (A203). The integral representation (A208) will be used hereafter to derive the lower and upper bounds (A204) on t i .
Proof of the lower bound in (A204). Our argument will refer to the inequality
1 x + δ > 1 x δ x for all x , δ R + ;
this is just the relation (A175) with γ = ϑ = 1 / 2 .
That said, let us return to Equation (A208) for t i ; for any s ( 0 , 1 ) , using (A209) with δ = 2 U a 0 and x = ( a i a 0 ) s 2 R ( s ) , we obtain
1 2 U a 0 + ( a i a 0 ) s 2 R ( s ) > 1 ( a i a 0 ) s 2 R ( s ) 2 U a 0 ( a i a 0 ) s 2 R ( s ) .
On the other hand, from the definition (A203) of R , it is evident that R ( s ) > U ; thus, for s ( 0 , 1 ) , we have
1 2 U a 0 + ( a i a 0 ) s 2 R ( s ) > 1 ( a i a 0 ) s 2 R ( s ) 2 U a 0 U ( a i a 0 ) s 2 .
Again for s ( 0 , 1 ) , we have of course a 0 + ( a i a 0 ) s 2 > ( a i a 0 ) s 2 ; inserting this inequality and the relation (A210) into Equation (A208) for t i , we infer
t i 2 a i a 0 Ω r 0 1 d s ( a i a 0 ) s 2 2 1 ( a i a 0 ) s 2 R ( s ) 2 U a 0 U ( a i a 0 ) s 2 )
= 2 ( a i a 0 ) 2 Ω r 0 1 d s s 3 R ( s ) 2 U 2 U a 0 Ω r ( a i a 0 ) 3 / 2 0 1 d s s 2 ;
this is just the lower bound on t i θ i in Equation (A204).
Proof of the upper bound in (A204). Let us expand the square a 0 + ( a i a 0 ) s 2 2 in Equation (A208); in this way, we obtain
t i = 2 a i a 0 Ω r a 0 2 0 1 d s 2 U a 0 + ( a i a 0 ) s 2 R ( s ) + 2 a 0 ( a i a 0 ) 0 1 d s s 2 2 U a 0 + ( a i a 0 ) s 2 R ( s ) +
( a i a 0 ) 2 0 1 d s s 4 2 U a 0 + ( a i a 0 ) s 2 R ( s ) .
Now, we insert in the first two integrals above the bound 2 U a 0 + ( a i a 0 ) s 2 R ( s ) 2 U a 0 , while we put in the third integral the bound 2 U a 0 + ( a i a 0 ) s 2 R ( s ) ( a i a 0 ) s 2 R ( s ) . In this way, we obtain the inequality
t i 2 a i a 0 Ω r a 0 3 / 2 2 U 0 1 d s + 2 a 0 U ( a i a 0 ) 0 1 d s s 2 + ( a i a 0 ) 3 / 2 0 1 d s s 3 R ( s ) ,
which in fact coincides with the upper bound t i θ i + in (A204).
Coarser bounds on t i , following from (A204). We now assume that we have bounds a 0 + and a i , fulfilling the inequalities
0 < a 0 < a 0 + < a i a i a i + .
In this case, we consider the constants U + as in (A199), C as in (A202) and
U + : = 1 + 3 Ω m a 0 + + 6 Ω k ( a 0 + ) 2 + 15 Υ ( a 0 + ) 4 Ω r ,
A : = Ω m Ω r , A + : = Ω m + 4 Ω k a 0 + + 20 Υ ( a 0 + ) 3 Ω r ,
B : = Ω k Ω r , B + : = Ω k + 15 Υ ( a 0 + ) 2 Ω r
(noting that U + , U + > 1 and A , B + , C > 0 , B 0 ); we also introduce the functions
R : [ 0 , 1 ] R + ,
R ( s ) : = 1 + A ( a i a 0 + ) s 2 + B ( a i a 0 + ) 2 s 4 + C ( a i a 0 + ) 4 s 8 ,
R + ( s ) : = U + + A + a i + s 2 + B + ( a i + ) 2 s 4 + 6 C a 0 + ( a i + ) 3 s 6 + C ( a i + ) 4 s 8 .
We now claim that the time t i in Equation (244) admits the bounds
θ i < t i < θ i + + ,
θ i : = 2 ( a i a 0 + ) 2 Ω r 0 1 d s s 3 R + ( s ) 2 3 2 U + a 0 + Ω r ( a i + ) 3 / 2 ,
θ i + + : = 2 ( a i + ) 2 Ω r 0 1 d s s 3 R ( s ) + 2 a 0 + Ω r 2 3 ( a i + ) 3 / 2 + a 0 + a i + .
The comment following Equation (A204) must be rephrased here in the following way: the integrals 0 1 d s s 3 / R ( s ) appearing in (A215) require a numerical calculation which however is more reliable than the direct numerical computation of t i via (244), due to the nonsingular nature of the functions s s 3 / R ( s ) on [ 0 , 1 ] .
In order to prove (A215), we start from the bounds (A204) θ i t i θ i + , where θ i are defined via the constants U , U , A , B , C of Equation (A202) and the function R of Equation (A203). We have the inequalities 1 < U < U + , 1 < U < U + , a i a 0 + < a i a 0 < a i + and R ( s ) < R ( s ) < R + ( s ) for all s [ 0 , 1 ] , which imply
θ i < θ i , θ i + < θ i + + ;
so the bounds (A215) follow from the bounds (A204), of which they are coarsenings. The coarser bounds (A215) are especially useful when a 0 + and a i in Equation (A212) are sufficiently close to a 0 and a i , respectively.
Applications of the coarser bounds (A215). Let us make the assumptions (200), (204), (211) and (234). Some interesting candidates for application of the bounds (A215) are the times t i in Section 6.5, Section 6.6 and Section 6.7 with i = 2 , , ϕ , ϕ r + , ϕ m , ϕ k , r m , r k , m k .
In order to obtain the bounds (A215) for these choices of i , we can use available bounds a 0 + and a i (which are required to fulfill (A212)). More precisely, we should note the following:
  • We can take a 0 + = μ ( 1 F μ ) (see (209)), with F as in (208).
  • a 2 is the second positive solution of the equation V ( a ) = 0 (see (210) and (212)) or of the equivalent algebraic equation Q ( a ) = 0 (see (A122)–(A124)). We can use for a 2 a lower and an upper bound on a 2 obtained by treating this equation numerically.
  • According to (249), the time t corresponds to the value a : = 1 of the scale factor; of course, we can set a : = 1 .
  • For i = ϕ , ϕ m , r m , r k , m k we have the analytic expression of a i (see Equations (270), (278), (279) and (282)–(284)); we can set a i = a i .
  • For i = ϕ r + , ϕ k , the quantity a i is the unique positive solution of an algebraic equation (see (277), (280) and (281)) which can be treated, e.g., numerically. For a i , we can use a lower and an upper bound on a i obtained from the numerical treatment.

Appendix S. On the Quantity Δ ϕ in Equation (292)

Let us refer to the general framework of Section 6.5 and Section 6.6, keeping in mind, in particular, the assumptions (200) and (204). Equation (292) contains the quantity
Δ ϕ : = 2 Ω r μ a 0 + d a a 6 V ( a ) ,
accounting for the variation in the scalar field from t = 0 (the time of the Big Bounce) to t = + , or from t = to t = 0 ; we already noted that the integral in the definition of Δ ϕ is convergent.
Statement of the bounds on Δ ϕ . Let us consider the constants U in (A180), and U , A , B , C in (A202) ( U , U > 1 and A , B , C > 0 ); moreover, let us choose any real number ϑ such that
0 < ϑ < 1 4 .
In the sequel we will prove that
Δ ϕ Δ ϕ Δ + ϕ ,
Δ ϕ : = μ a 0 π 2 max ( U , U ) 2 π Γ ( 1 4 ϑ ) Γ ( 3 / 2 ϑ ) A + B a 0 + 7 C a 0 3 ϑ a 0 ϑ ,
Δ + ϕ : = μ a 0 π 2 min ( U , U ) .
The lower bound Δ ϕ depends on ϑ and involves the Gamma function (factorial function) Γ ; the upper bound Δ + ϕ is independent of ϑ . For small μ , we have a 0 μ , U , U 1 and Δ ϕ π / 2 .
The forthcoming three paragraphs in this appendix will be devoted to the proof of (A218). The subsequent paragraph will present coarser bounds on Δ ϕ , following from (A218) and from estimates on a 0 . The final paragraph will use such bounds to justify Equation (293) of the main text, which describes rigorously the μ 0 + asymptotics of Δ ϕ .
A preliminary to the derivation of (A218): a new integral representation for Δ ϕ . Let us express V as in Equation (A111), with P the polynomial function of Equation (A112); this gives
Δ ϕ = 2 Ω r μ a 0 + d a a P ( a ) .
We expand P ( a ) in terms of powers of a a 0 following Equation (A207), and then we reexpress the integral in (A219) via a change in variable a = a 0 ( 1 + s 2 ) , 0 s < + . This gives
Δ ϕ = 2 2 μ a 0 0 + d s ( 1 + s 2 ) 2 U + U s 2 + a 0 Y ( s ) ,
Y ( s ) : = A s 4 + B a 0 s 6 + 6 C a 0 3 s 8 + C a 0 3 s 10 .
Proof of the lower bound in (A218) on Δ ϕ . Let us recall the inequality (A175), which we apply with γ = 1 / 2 and ϑ as in (A217); this gives
1 x + δ > 1 x δ ϑ x 1 / 2 + ϑ for x , δ R + .
Returning to the expression (A220) for Δ ϕ , and using (A222) with x = 2 U + U s 2 , δ = a 0 Y ( s ) ( 0 < s < + ), we obtain
Δ ϕ 2 2 μ a 0 ( I a 0 ϑ J ) ,
I : = 0 + d s 1 ( 1 + s 2 ) 2 U + U s 2 ,
J : = 0 + d s Y ( s ) ϑ ( 1 + s 2 ) ( 2 U + U s 2 ) 1 / 2 + ϑ .
It should be noted that
Y ( s ) ϑ ( 1 + s 2 ) ( 2 U + U s 2 ) 1 / 2 + ϑ = O 1 s 3 8 ϑ for s + ;
due to condition (A217), we have 3 8 ϑ > 1 , which ensures convergence of the integral J (convergence of I is evident). To continue, we observe that for all s ( 0 , + ) we have
2 U + U s 2 max ( U , U ) ( 2 + s 2 ) ,
so that
I 1 max ( U , U ) 0 + d s 1 ( 1 + s 2 ) 2 + s 2 = 1 max ( U , U ) × π 4 .
Again for s ( 0 , + ) , the relations U , U > 1 give
2 U + U s 2 > 2 + s 2 > 1 + s 2 ,
and the relation s < 1 + s 2 gives
Y ( s ) < A ( 1 + s 2 ) 2 + B a 0 ( 1 + s 2 ) 3 + 6 C a 0 3 ( 1 + s 2 ) 4 + C a 0 3 ( 1 + s 2 ) 5
< ( A + B a 0 + 7 C a 0 3 ) ( 1 + s 2 ) 5 .
Inserting (A229) and (A230) into (A225), we obtain
J ( A + B a 0 + 7 C a 0 3 ) ϑ 0 + d s 1 ( 1 + s 2 ) 3 / 2 4 θ
= ( A + B a 0 + 7 C a 0 3 ) ϑ × π Γ ( 1 4 ϑ ) 2 Γ ( 3 / 2 4 ϑ ) .
Substituting (A228) and (A231) into (A220), we finally obtain the lower bound Δ ϕ Δ ϕ of Equation (A218).
Proof of the upper bound in (A218) on Δ ϕ . We return to Equation (A220) and insert therein the inequalities
2 U + U s 2 min ( U , U ) ( 2 + s 2 ) ,
Y ( s ) > 0 ,
holding for all s ( 0 , + ) . This gives
Δ ϕ 2 2 μ a 0 × 1 min ( U , U ) × 0 + d s 1 ( 1 + s 2 ) 2 + s 2 = 2 2 μ a 0 × 1 min ( U , U ) × π 4 ,
thus yielding the upper bound Δ ϕ Δ + ϕ of Equation (A218).
Coarser bounds on Δ ϕ , following from (A218). We now assume that we have for a 0 lower and upper bounds a 0 , such that
0 < a 0 < a 0 < a 0 + ;
typical choices will be those provided by (209), namely
a 0 = μ 1 + N μ , a 0 + = μ ( 1 F μ ) ,
with N , F as in Equation (208). We also consider the constants U + in (A199), U + , A + , B + in (A213) and C in (A202) ( U + , U + > 1 ; A + , B + , C > 0 ). We claim that
Δ ϕ < Δ ϕ < Δ + + ϕ ,
Δ ϕ : = μ a 0 + π 2 max ( U + , U + ) 2 π Γ ( 1 4 ϑ ) Γ ( 3 / 2 ϑ ) A + + B + a 0 + + 7 C ( a 0 + ) 3 ϑ ( a 0 + ) ϑ ,
Δ + + ϕ : = π 2 μ a 0 ;
note that μ / a 0 = 1 + N μ and μ / a 0 + = 1 / ( 1 F μ ) if we use (A236). The bounds (A237) are coarsenings of the bounds (A218) Δ ϕ Δ ϕ Δ ϕ ; in fact, using the inequalities (A235) and 1 < U < U + , 1 < U < U + , A < A + , B < B + , we readily see that Δ ϕ < Δ ϕ and Δ + ϕ < Δ + + ϕ .
Justification of the asymptotics (293). We now consider a family of models as in (217). The inequalities (A237) can be used to justify the μ 0 + asymptotics (293), i.e.,
Δ ϕ = π 2 ( 1 + O ( μ ϑ ) ) ;
this follows immediately from the explicit expressions of Δ ϕ and Δ + + ϕ , with a 0 as in (A236).

Appendix T. A More Detailed Description of the Model in Section 6.7

Let us refer to the model in the cited section, based on the choices (302) and (308) for the parameters Ω r , Ω m , Ω k , Υ and μ .
Considering the densities r ϕ , r r and r m as functions of the scale factor a or of the cosmic time τ (say, for τ 0 ) and employing the inequalities (270), (271), (277)–(279) and (282), in items (0.)–(4.) at the end of Section 6.7 we obtained a number of results about the sign of r ϕ and the comparison of r ϕ , r r and r m .
A more detailed description, stemming again from the inequalities (270), (271), (277)–(279) and (282), is as follows:
a 0 a < a ϕ r , i . e . , 0 τ < τ ϕ r : r ϕ < 0 and | r ϕ | > r r > r m ; a = a ϕ r , i . e . , τ = τ ϕ r : r ϕ < 0 and | r ϕ | = r r > r m ; a ϕ r < a < a ϕ m , i . e . , τ ϕ r < τ < τ ϕ m + : r ϕ < 0 and r r > | r ϕ | > r m ; a = a ϕ m , i . e . , τ = τ ϕ m : r ϕ < 0 and r r > | r ϕ | = r m ; a ϕ m < a < a ϕ , i . e . , τ ϕ m < τ < τ ϕ : r ϕ < 0 and r r > r m > | r ϕ | ; a = a ϕ , i . e . , τ = τ ϕ : r ϕ = 0 and r r > r m ( > 0 ) ; a ϕ < a < a r m , i . e . , τ ϕ < τ < τ r m : r ϕ > 0 and r r > r m > r ϕ ; a = a r m , i . e . , τ = τ rm : r ϕ > 0 and r r = r m > r ϕ ; a r m < a < a ϕ r + , i . e . , τ r m < τ < τ ϕ r + : r ϕ > 0 and r m > r r > r ϕ ; a = a ϕ r + , i . e . , τ = τ ϕ r + : r ϕ > 0 and r m > r r = r ϕ ; a ϕ r + < a < a ϕ m + , i . e . , τ ϕ r + < τ < τ ϕ m + : r ϕ > 0 and r m > r ϕ > r r ; a = a ϕ m + , i . e . , τ = τ ϕ m + : r ϕ > 0 and r m = r ϕ > r r ; a ϕ m + < a < + , i . e . , τ ϕ m + < τ < + : r ϕ > 0 and r ϕ > r m > r r .

Appendix U. Proof of Statements ( α - ζ ) in Section 6.8 about the Function V of Equation (201)

Let us recall the assumptions (198)–(200), (313), (211), (314) and (315).
Proof of ( α ) and ( β ). The proofs of all statements in ( α ) and ( β ) are identical to those of the corresponding statements in items (c) and (d) of Section 6.5: see Appendix L. The cited items (c), (d), ( α ) and ( β ) refer to the derivative V , which is independent of Ω k (see Equation (210)); so, the fact that Section 6.5 and Appendix L assume Ω k 0 , while Section 6.8 and the present appendix assume Ω k < 0 , is not relevant for the present discussion.
Proof of ( γ ). Let us derive the relation (317) V ( a 1 ) < 0 . For this purpose, we use the already-proved inequalities (316) a 1 < 2 μ < a 2 , recalling that V is strictly increasing on the interval [ a 1 , a 2 ] by (213). Thus
V ( a 1 ) < V ( 2 μ ) = Ω r 2 2 Ω m μ 4 Ω k μ 2 8 Υ μ 4 4 μ 2 < Ω r 4 Ω k μ 2 4 μ 2 < 0 ,
where, in the last two steps, we used the inequalities Ω m μ < 0 , Υ μ 4 < 0 and the assumption (314).
Proof of ( δ ) ( δ 1 ) ( δ 2 ) ( δ 3 ). First of all, let us show that the parameter W defined by Equation (318) is positive; for this purpose, we rephrase Equation (318) as
W = [ Ω r μ 2 + Ω r ( a 2 ) 2 ] + [ Ω m ( a 2 ) 3 + Υ ( a 2 ) 6 ] ( a 2 ) 4 .
Of course Ω m ( a 2 ) 3 + Υ ( a 2 ) 6 > 0 ; moreover, due to the already proved inequality a 2 > 2 μ in (316) we have Ω r μ 2 + Ω r ( a 2 ) 2 > Ω r μ 2 + Ω r ( 2 μ ) 2 = Ω r μ 2 > 0 . In conclusion, W > 0 .
To continue, let us observe that the definitions (201) of V and (318) of W ensure
V ( a 2 ) = W Ω k ;
due to (A240), the equivalences (indicated with ⟺) in items ( δ 1 ) ( δ 2 ) ( δ 3 ) are evident. Recalling that V behaves as in the already proved statements ( α ) and ( β ), it is easy to check all statements in ( δ 1 ) ( δ 2 ) ( δ 3 ) about the zeroes and the sign of V , except perhaps for the positioning of 2 μ in the inequalities (329) of item ( δ 3 ). Concerning this issue, let us recall that a 1 < 2 μ < a 2 by (316); it is clear that a 1 < a 3 / 2 < a 2 . We now proceed to compare the quantities 2 μ , a 3 / 2 ( a 1 , a 2 ) ; we know that V is strictly increasing on [ a 1 , a 2 ] and V ( a 3 / 2 ) = 0 , while V ( 2 μ ) < 0 (see (A238)), so we conclude that 2 μ < a 3 / 2 .
Proof of ( ε ). In anyone of cases ( δ 1 ) ( δ 2 ) ( δ 3 ), we have the following implication for any a R + :
V ( a ) < 0 a 0 < a .
Thus, to prove that a 0 < 1 , it suffices to show that
V ( 1 ) < 0 .
On the other hand, this is true since V ( 1 ) = 1 (see Equation (160) and its review after Equation (177)).
Proof of ( ζ ). Let us refer to the quantities N , F defined by (331). It is evident that N > 0 . The denominator of F in (331) is clearly positive; the numerator is Ω m + Ω k μ + Υ μ 3 > Ω m + Ω k μ > 0 due to (315), so that F > 0 . Moreover F μ = ( Ω m μ + Ω k μ 2 + Υ μ 4 ) / ( 2 Ω r + 3 Ω m μ + 6 Υ μ 4 )   < ( Ω m μ + Υ μ 4 ) / ( 2 Ω r + 3 Ω m μ + 6 Υ μ 4 ) < 1 .
We now proceed to prove the bounds (332) on a 0 , using an argument rather similar to that employed for proving the bounds (209) (see Appendix L, proof of (b)).
First of all, we represent V in the form (A111) V ( a ) = P ( a ) / a 4 for all a R + , where P is the polynomial function (A112). Of course, V and P have opposite signs and the same zeroes in R + , as stated in (A113). To continue, we apply the general setting of Appendix J, with P = P . By comparison between Equations (A88), (A89) and (A112), we see that in the present case
ν = μ ; m = 2 , P m = P 2 = Ω r ;
K = { 3 , 4 , 6 } , P 3 = Ω m > 0 , P 4 = Ω k < 0 , P 6 = Υ > 0 ; K = { 3 , 6 } .
Due to (A243), Equations (A91) gives in the present case
θ = 1 + Ω m μ + Υ μ 4 Ω r 1 / 2 = 1 1 + N μ ,
with N as in (331). The condition (A94) is fulfilled, since
k K P k ν k = Ω m μ 3 + Ω k μ 4 + Υ μ 6 > μ 3 ( Ω m + Ω k μ ) > 0
(the above statement of positivity is due to (315)). Equation (A95) gives
ψ = Ω m μ + Ω k μ 2 + Υ μ 4 2 Ω r + 3 Ω m μ + 6 Υ μ 4 = F μ ,
with F as in (331). The inequalities (A93) and (A97) become in this case
P μ 1 + N μ < 0 , P μ ( 1 F μ ) > 0 ,
i.e., since V and P have opposite signs,
V μ 1 + N μ > 0 , V μ ( 1 F μ ) < 0 .
On the other hand, in anyone of cases ( δ 1 )-( δ 3 ) we can say that, for any a R + ,
V ( a ) > 0 , a < 2 μ a < a 0
(to prove this in case ( δ 3 ), recall that a 0 < 2 μ < a 3 / 2 by (329)). Applying this result with a = μ / 1 + N μ < μ < 2 μ , we obtain μ / 1 + N μ < a 0 . Let us also recall the implication (A241) V ( a ) < 0 a 0 < a ; this can be applied with a = μ ( 1 F μ ) , the conclusion being a 0 < μ ( 1 F μ ) . In conclusion, both the lower and the upper bound (332) on a 0 are proved.

Appendix V. Derivation of Equations (363)–(366)

Let us consider Equation (361) r ¨ = λ 2 r for an unknown function t r ( t ) = ( x ( t ) , y ( t ) ) R 2 , where λ R + ; throughout the present appendix, the term “solution” is employed in relation to (361) to mean a maximal solution. In this appendix, we will show that a function t r ( t ) is a solution of Equation (361) if and only if one of the following cases (i)–(iii) occurs:
(i)
We have
r ( t ) = H e λ t for all t R ( H R 2 ) ;
(ii)
We have
r ( t ) = K e λ t for all t R ( K R 2 ) ;
(iii)
We have
r ( t ) = A cosh ( λ ( t t 0 ) ) i + B sinh ( λ ( t t 0 ) ) j for all t R ,
where
t 0 R ; A , B R , A , B 0 ; i , j R 2 , | i | = | j | = 1 , i j = 0
( | | , are the standard norm and inner product of R 2 ).
Equations (A250) and (A251) reproduce Equations (363) and (364) in the main text. Equations (365) and (366) in the main text are obtained from (A252) and (A253) renaming t the difference t t 0 . In order to prove that the solutions of (361) coincide with functions of type (i), (ii) or (iii), We proceed in several steps.
Step 1. A function t r ( t ) is a solution of Equation (361) if and only if it admits the representation
r ( t ) = H e λ t + K e λ t for all t R , ( H , K R 2 ) .
The above statement is well known (it is just a vector description of the general solutions of equations x ¨ = λ 2 x , y ¨ = λ 2 y ).
Step 2. The solution of Equation (361) with arbitrary initial data r ( t 0 ) = r 0 , r ˙ ( t 0 ) = r ˙ 0 ( t 0 R , r 0 , r ˙ 0 R 2 ) is unique, and given by
r ( t ) = r 0 cosh ( λ ( t t 0 ) ) + r ˙ 0 λ sinh ( λ ( t t 0 ) ) for all t R .
This is another well-known fact.
Step 3. Let t r ( t ) ( t R ) be any solution of (361); then, the function t r ( t ) fits case (i) or case (ii), or the following holds:
(iii’) there is t 0 R such that ( d r 2 / d t ) ( t 0 ) = 0 (with r ( t ) : = | r ( t ) | = r ( t ) r ( t ) ).
In order to prove the above statements, we start from the representation (A254) of r ( ) , depending on two vectors H , K R 2 . If K = 0 , case (i) holds; if H = 0 , case (ii) holds.
We now assume H , K 0 , and infer (iii’). For this purpose, we note that, for all t R , Equation (A254) implies
r 2 ( t ) = r ( t ) r ( t ) = H 2 e 2 λ t + K 2 e 2 λ t + 2 H K ,
where H : = | H | and K : = | K | . Therefore,
d r 2 d t ( t ) = 2 λ ( H 2 e 2 λ t K 2 e 2 λ t ) ,
and this implies
d r 2 d t ( t 0 ) = 0 , t 0 : = 1 2 λ ln K H
( t 0 is well defined, since H , K > 0 ). To summarize, (iii’) holds.
Step 4. Let t r ( t ) ( t R ) be any solution of (361); then (i), (ii) or (iii) holds. To prove this we refer to the third step, ensuring that (i),(ii) or (iii’) holds. We now show that (iii’) implies (iii).
For this purpose, let us consider the derivative ( d r 2 / d t ) ( t ) = 2 r ( t ) r ˙ ( t ) . Let t 0 R be the time mentioned in (iii’), and let us put r 0 : = r ( t 0 ) , r ˙ 0 : = r ˙ ( t 0 ) ; then,
0 = d r 2 d t ( t 0 ) = 2 r 0 r ˙ 0 .
So, we can express r ( t ) as in (A255), with r 0 and r ˙ 0 orthogonal; due to orthogonality, we can write
r 0 = A i , r ˙ 0 = λ B j , A , B R , A , B 0 , ( i , j ) orthonormal basis of R 2 .
Substituting (A260) into (A255), we conclude that the solution t r ( t ) can be represented as in (A252) with suitable parameters t 0 , A , B , i , j fulfilling all conditions in (A253). To summarize, (iii) holds.
Step 5. Any function as in (i), (ii) or (iii) is a solution of (361). This is evident.

Appendix W. Derivation of Equations (401) and (402)

Let us consider Equation (399) r ¨ = ω 2 r for an unknown function t r ( t ) = ( x ( t ) , y ( t ) ) R 2 , where ω R + ; throughout the present appendix, the term “solution” is employed in relation to (399) to mean a maximal solution. In this appendix, we will show that a function t r ( t ) is a solution of Equation (399) if and only if it has the form
r ( t ) = A cos ( ω ( t t 0 ) ) i + B sin ( ω ( t t 0 ) ) j for all t R ,
depending on the parameters
t 0 R ; A , B R s . t . 0 A B ; i , j R 2 s . t . | i | = | j | = 1 , i j = 0
( | | , are again the standard norm and inner product of R 2 ).
Equations (A261) and (A262) will justify Equations (401) and (402) in the main text, which are obtained by renaming t the difference t t 0 .
In order to derive Equations (A261) and (A262), we proceed in several steps.
Step 1. The solution of Equation (399) with arbitrary initial data r ( t 0 ) = r 0 , r ˙ ( t 0 ) = r ˙ 0 ( t 0 R , r 0 , r ˙ 0 R 2 ) is unique, and given by
r ( t ) = r 0 cos ( ω ( t t 0 ) ) + r ˙ 0 ω sin ( ω ( t t 0 ) ) for all t R ;
this implies, among others, that any solution of (399) is periodic (of period 2 π / ω ). These statements are well known.
Step 2. Let t r ( t ) ( t R ) be any solution of (399); then the function t r ( t ) can be represented as in (A261), with suitable parameters t 0 , A , B , i , j as in (A262). In order to prove this statement, let us note that the squared radius function t r 2 ( t ) = r ( t ) r ( t ) is a periodic, smooth real-valued function, and consequently possesses infinitely many points of absolute minimum and maximum; this function has derivative ( d r 2 / d t ) ( t ) = 2 r ( t ) r ˙ ( t ) .
Let t 0 R be a point of absolute minimum of the squared radius, and let us put r 0 : = r ( t 0 ) , r ˙ 0 : = r ˙ ( t 0 ) ; then, 0 = ( d r 2 / d t ) ( t 0 ) = 2 r 0 r ˙ 0 . So we can express r ( t ) as in (A263), with r 0 and r ˙ 0 orthogonal; due to orthogonality, we can write
r 0 = A i , r ˙ 0 = ω B j , A , B R , 0 A , B , ( i , j ) orthonormal basis of R 2 .
Substituting (A264) into (A263), we conclude that the solution t r ( t ) can be represented as in (A261) with suitable parameters t 0 , A , B , i , j fulfilling all conditions in (A262), except possibly for the inequality A B ; let us show that the latter inequality holds as well. For this purpose, let us observe that Equation (A261) implies r ( t 0 ) = A i and r ( t 0 + π / ( 2 ω ) ) = B j , whence r 2 ( t 0 ) = A 2 and r 2 ( t 0 + π / ( 2 ω ) ) = B 2 , but t 0 is a point of absolute minimum of the squared radius, so A B .
Step 3. Any function of the form (A261) and (A262) is a solution of (399). This is evident.

Notes

1
In models with a cosmological constant Λ , the Einstein equations R μ ν ( 1 / 2 ) R g μ ν + Λ g μ ν = κ T μ ν must be rephrased as R μ ν ( 1 / 2 ) R g μ ν = κ T μ ν with T μ ν : = T μ ν ( Λ / κ ) g μ ν ; the energy conditions of the singularity theorems are required to hold for T μ ν . The term Δ T μ ν : = ( Λ / κ ) g μ ν individually fulfills the weak energy condition if and only if Λ 0 , and the strong energy condition if and only if Λ 0 .
2
Such tensorial constructions are described with great detail in two books by Matolcsi [65,66]. A technically different approach is developed in a paper by Janiška, Modugno, and Vitolo [67] and also sketched in a subsequent book by Janiška and Modugno [68]; these authors focus on the notion of one-dimensional semi-vector space, which can be interpreted as the positive part of a one-dimensional, oriented vector space.
3
In the general framework of Section 2, Section 3, Section 4 and Section 5 and Section 7, the fluids are not required to fulfill the usual (e.g., the weak or strong) energy conditions; however this happens in the applications of Section 6 and Section 8, where the fluids are in fact dust or a radiation gas.
4
Let us add some detail. A history of the n-th fluid is a congruence of timelike curves in M d + 1 ; such curves, one through any spacetime point, represent the world lines of the fluid’s particles. A history of the fluid determines a mass density μ n : M d + 1 D + , describing the spacetime distribution of the particles’ masses due to their motions. The fluid also possesses a (constitutive, history-independent) function ϵ n : D + ( 1 , + ) , representing an elastic potential per unit mass; given a history with mass density μ n : M d + 1 D + , the corresponding mass–energy density is ρ n : = μ n ( 1 + ϵ n ( μ n ) ) . Concerning the previous issues, see Section 3.3, Example 4 of the cited book [12] (this reference considers a single fluid, calling ρ the mass density and μ the mass–energy density; here, we interchange the letters used for the two densities. The same reference considers the congruences of timelike curves without using the denomination of histories, which, however, is generally employed in variational calculus).
5
Let us add a comment on the pressures p n appearing in Equation (8). Variational calculations performed as in Section 3.3, Example 4 of [12], give the pressure p n of each fluid as a function of the mass density μ n , determined by the elastic potential ϵ n (see the previous note). More precisely, we have p n = μ n 2 ϵ n ( μ n ) , where ′ is the derivative. We know that ρ n = μ n ( 1 + ϵ n ( μ n ) ) (see again the previous note); assuming that this relation can be inverted, we finally obtain p n as a function of ρ n , as in Equation (5). For example, if ϵ n ( μ n ) = 1 + ( μ n / μ 0 n ) w n with two given constants μ 0 n D + , w n R , we finally obtain p n = w n ρ n .
6
Obviously enough, the previous considerations can be generalized to the case where we have a decomposition V ( Φ ) = V + U ( Φ ) , with V a constant and U any smooth function. In this case, the Einstein equations can be put in the form (13), where the cosmological constant Λ is given again by Equation (14), and T μ ν Φ is the stress–energy tensor of a scalar field with self-potential U ( Φ ) .
7
Appendix B also introduces the notion of generalized FLRW spacetime, which is used in some of the subsequent appendices and allows us to draw links with some of the references cited in the present work. However, in the main text of the present paper we always refer to an ordinary FLRW spacetime, as described in Section 3.1.
8
The notions of timelike and lightlike completeness date back to the pioneering works in this area; see, e.g., [44]. The definition of a nonsingular spacetime adopted in this work, based on the requirements of timelike and lightlike completeness, can be found in [12] (page 258). Other authors (see, e.g., [45] (page 215)) also require spacelike completeness, meaning that the maximal spacelike geodesics have domain ( , + ) ; we briefly return to this point in the subsequent note.
9
The results in the cited papers apply to a class of generalized FLRW spacetimes, described in Appendix B and Appendix E; as already indicated, the main text of the present work will only consider the usual FLRW spacetimes of Section 3.1. The studies [61,62] also prove the following result: if a generalized FLRW spacetime is lightlike complete, it is as well spacelike complete (in the sense of the previous note).
10
Note that the function F : [ 0 , + ) [ 0 , + ) , α F ( α ) : = α / 1 + α 2 is strictly increasing (with F ( 0 ) = 0 and lim α + F ( α ) = 1 ). Under the assumptions in (26), the features of F ensure a ( τ ) / 1 + a 2 ( τ ) a 0 / 1 + a 0 2 > 0 for all τ T = ( , + ) , implying the divergence of the integrals in (22) and (24). Again under the assumptions in (26), the divergence of the integrals in (23) and (25) is obvious.
11
The lightlike incompleteness of an example very close to (27) is indicated in [60] (Chapter 7, Example 41); see also the references mentioned therein.
12
This expression for μ μ Φ and the subsequent one for μ T μ ν n follow easily from the standard rules for the coordinate representation of covariant derivatives, as well as from the expressions for the Christoffel symbols of the spacetime metric reported in Appendix B.
13
The rest of the paper makes more and more clear that there are analogies between the fluids’ densities ρ n and the curvature density ρ k ; this is why, in Equation (46) and in many subsequent formulas, the term corresponding to the curvature density is written immediately after the terms related to the fluids’ densities.
14
The relations (54) and the subsequent statements (55)–(57) generalize the results presented in [46] for d = 3 , for an arbitrary FLRW cosmological model whose stress–energy tensor has the perfect fluid form.
15
Let us recall Equation (26), and the note that accompanies it.
16
For a direct check, one can use the identity 0 = E ˙ + d a ˙ a ( E + A ) + F ϕ ˙ = 1 a d d d t a d E + d a ˙ a A + F ϕ ˙ . Due to this identity, A = 0 and F = 0 a d E = constant; thus, A = 0 , F = 0 and E ( t 0 ) = 0 at some time t 0 E = 0 (at all times).
17
By the standard theory of the Cauchy problem, this solution exists and is unique if we require it to be maximal, in the sense of the subsequent Section 4.6.
18
All figures presented in this paper were produced using Mathematica (version 12.3) by Wolfram Research Inc. (Champaign, Illinois); the same package was employed for all numerical computations.
19
The occurrence of a Big Bang in case (iii) is derived by the methods of Section 5.4 for the qualitative description of one-dimensional conservative systems. The essential reason for the Big Bang is the same holding for the standard model: for small a, V ( a ) decreases to . (As in the standard model, the singularity at a = 0 is attained at a finite time since 1 / V ( a ) vanishes for a 0 + , and thus is integrable in a right neighborhood of zero.) That said, let us remark that the a 0 + asymptotics of V ( a ) is different in the two cases: in this limit, we have V ( a ) Ω r / a 2 in the standard case (see Equation (184)), while V ( a ) Π 2 / ( 2 a 4 ) (due to (168)) in case (iii); these facts imply different laws for the growth of the scale factor immediately after the Big Bang.
20
Let us consider the standard model with positive cosmological constant and, e.g., with nonpositive spatial curvature, displaying a Big Bang at time t = 0 . In this case, we also have a ¨ ( t ) > 0 after a suitable time t 2 ; see Section 6.2. However, in this model, the SEC holds close to t = 0 ; according to the singularity theorems mentioned in the Introduction, this is just the reason why the model presents a Big Bang!
21
One could argue that the condition justifying the classical treatment of gravity is ( 1 / ρ P ) max t R | ρ ( t ) | 1 , where ρ = ρ r + ρ m + ρ ϕ is the total density without the curvature contribution ρ k . The discussion of this issue is nontrivial; regardless, in the application of the classicality condition presented in this paper (see Section 6.7), the spatial curvature vanishes, so that ρ = ρ ˜ ( 0 ) .
22
Of course the idea to put a condition on a cosmological model based on the Planck units, so as to justify the classical treatment of gravity, is not at all new: see, e.g., the condition on the Riemann curvature tensor presented after Equations (2) and (3) in [54].
23
Since the cited Equations (170) and (203) make sense for any a R + , here, we momentarily neglect that a ( t ) a 0 at all times; the bound a a 0 will be recovered in the sequel.
24
Of course, the above expressions of a ϕ m via radicals are obtained noting that the cubes of these quantities fulfill algebraic equations of degree two. We could also express a ϕ r , a ϕ k via radicals, noting that their squares fulfill algebraic equations of degree three and using the Cardano formula.
25
The results (297) can be obtained in an alternative way from Equation (95) with d = 3 , using a number of facts about a ˙ and a ¨ . First of all, a ˙ 2 = V ( a ) ; see (144). Secondly, we have a ¨ = V ( a ) / 2 ; see (143). These two facts also imply a ¨ / a + a ˙ 2 / a 2 = ( 1 / 2 ) V ( a ) / a V ( a ) / a 2   = ( a / 2 ) ( d / d a ) ( V ( a ) / a 2 ) = ( a / 2 ) H ( a ) (recall (220)).
26
The cited Equation (197) also reports the value of a time τ 2 , indistinguishable from the homonymous value in Equation (310). Indeed, after an extremely short time-lapse from the Big Bounce, the present model (with nonzero but extremely small Π ) becomes quantitatively indistinguishable in several aspects from the standard model (corresponding to Π = 0 ).
27
Here, we do not consider the values of a involved in comparisons with the curvature density r k , since the latter vanishes.
28
Let us repeat that all numerical computations mentioned in this paper were performed using Mathematica (version 12.3).
29
Here and in the sequel, Z = { 0 , 1 , 2 , } is the set of relative integers. For each P R { 0 } , we denote with R / ( P Z ) the one-dimensional torus of period P, i.e., the quotient of R with respect to the relation of equivalence mod. P (two real numbers ψ 1 , ψ 2 are equivalent mod. P if there is m Z such that ψ 2 = ψ 1 + P m ). For each ψ 1 R , we often write ψ 1 (mod. P) for the equivalence class containing ψ 1 ; thus, ψ 1 (mod. P) = { ψ 1 + P m | m Z } R / ( P Z ) .
30
For each F F { 0 } , the quotient F / ( F Z ) is defined like the quotient R / ( P Z ) of the previous note, replacing R with F and P with F.
31
Equations (97) and (98) refer to the first and second derivative of the function t a ( t ) . On the other hand, a ( t ) = C d 2 / d r ( t ) 2 / d by (335), whence a ˙ ( t ) = ( 2 / d ) C d 2 / d r ( t ) 2 / d 1 r ˙ ( t ) , a ¨ ( t ) = ( 2 / d ) C d 2 / d r ( t ) 2 / d 1 ( r ¨ ( t ) + ( 2 / d 1 ) r ˙ ( t ) 2 ) . For any time t 0 , we have the equivalence a ˙ ( t 0 ) = 0 r ˙ ( t 0 ) = 0 , and the equivalence a ˙ ( t 0 ) = 0 , a ¨ ( t 0 ) > 0 r ˙ ( t 0 ) = 0 , r ¨ ( t 0 ) > 0 .
32
See, in particular, Equation (19) of [25]; essentially, this can be re-obtained from our Equations (350) and (351) for the field potential setting W 1 = 0 . Let us also mention that our Cartesian coordinates x , y have a role similar to the coordinates u , v of [25] (more precisely, our variables x , y are analogs for const. u and const. v ). In our approach, (polar and) Cartesian coordinates are a rather natural choice in the Lagrangian setting, which subsequently results in individuating the potential (350) and (351) as one giving a solvable model. On the contrary, the potential of [25] and the coordinates u , v used therein were individuated by applying the theory of Nöther symmetries to the general Lagrangian formalism for a phantom scalar in the presence of dust.
33
All works [26,27,28,30,31,32] consider a canonical scalar field with a potential that can be expressed in the dimensionless form V ( ϕ ) = V 1 e K d ϕ + V 2 e K d ϕ , with K d > 0 a suitable normalization constant; V has a slightly different exponential form in [77] (where this model is considered for a peculiar reason: it is a preliminary step in the construction of an FLRW cosmology with degenerate, signature changing spacetime metric). Hereafter, with ( u , v ) , we indicate the separation coordinates in any one of the cited papers; these can be interpreted as orthonormal coordinates on the two-dimensional Minkowski space, in which the line element reads d σ 2 = ( d u 2 d v 2 ) . The scale factor of the corresponding cosmological model is expressed as a = const. ( u 2 v 2 ) 1 / d , and this formula replaces our expression a = const. r 2 / d = const. ( x 2 + y 2 ) 1 / d (see the present Equations (335) and (344)). So, the trajectories in the ( u , v ) -space of [26,27,28,30,31,32,77] and the trajectories in the ( x , y ) -space of the present paper have different cosmological meanings.
34
The model of [77] with a canonical scalar field and an exponential self-interaction also appears in a recent review by Jalalzadeh, Rasouli, and Moniz [78] on supersymmetric quantum cosmology, where it is considered for application of the Wheeler–DeWitt equation; the cited review mentions (with no analysis of the cosmological implications) that the uncoupled harmonic oscillators resulting from the model produce Lissajous curves. For the reasons indicated previously, the Lissajous curves of [78] and those considered in the present work have different cosmological interpretations.
35
See Section 8.9, and the subsequent note on the case ω 2 = 2 ω 1 .
36
For example, if ω 2 = 2 ω 1 , we have ζ 0 h = 2 h π and ζ 1 h = π 2 h π , so that Z = { 0 , π ( m o d . 2 π ) } . This confirms the statements after Equation (430).
37
Here is the proof of the uniqueness of the zero of r ( ) . Let t , t R be two times such that r ( t ) = r ( t ) = 0 . According to Equations (446)–(448), we have t = t ^ = t ˜ h and t = t ^ = t ˜ h for some , h , , h Z . These facts imply t ^ t ^ = t ˜ h t ˜ h , i.e., by the explicit expressions in (446) and (447), ( ) ( π / ω 2 ) = ( h h ) ( π / ω 1 ) . Due to the last equality, and h h are both nonvanishing or both vanishing. If it were 0 and h h 0 , we could infer from the last equality that ω 2 / ω 1 = ( ) / ( h h ) , against the assumption that ω 2 / ω 1 is irrational. Thus, = 0 and h h = 0 ; these equalities imply t ^ = t ^ and t ˜ h = t ˜ h , whence t = t .
38
For example, the divergence of the integral in Equation (65) is checked in the following way. Let us consider the expression (441) for x ( t ) , y ( t ) (which, among others, emphasizes the general assumption A 1 , A 2 > 0 ); this implies
r ( t ) = A 1 2 sin 2 ( ω 1 t ) + A 2 2 sin 2 ( ω 2 t + ψ ) A for   t R , A : = A 1 2 + A 2 2 .
From here, and from Equation (335), we infer
a ( t ) = ( C d ) 2 / d A 1 2 sin 2 ( ω 1 t ) + A 2 2 sin 2 ( ω 2 t + ψ ) 1 / d ( C d A ) 2 / d f o r t R ;
so, for any t 0 R , we have
t 0 d t a ( t ) 1 + a 2 ( t ) ( C d ) 2 / d 1 + ( C d A ) 4 / d t 0 d t A 1 2 sin 2 ( ω 1 t ) + A 2 2 sin 2 ( ω 2 t + ψ ) 1 / d = + .
39
The divergence of the integral in Equation (67) is checked by arguments very similar to those employed in the previous note.
40
For ω 1 = 3 ω , ω 2 = 4 ω and ψ = 0 ( m o d . 2 π ), it turns out that r ( t ) = 0 if and only if t = t k ( k Z ), with t k : = k π / ω . As a domain for the corresponding cosmological model, we can assume any interval J of the form ( t k , t k + 1 ) , for example, the interval J : = ( t 0 , t 1 ) = ( 0 , π / ω ) ; the model has a Big Bang at t = 0 and a Big Crunch at t = π / ω .
41
In the present case, the ends of the time domain are t = , as required by Equations (65) and (67). The verification of the other nonsingularity conditions is simple; the less obvious statement is the divergence of the integral in Equation (65), where t 0 R is arbitrarily chosen. In order to clarify this point, after fixing t 0 , we note that Equation (467) implies
C | sin ( ω t ) | r ( t ) K for   t ( , t 0 ) , K : = A 2 e 2 λ t 0 + C 2 ;
from here, and from Equation (335), we infer
( C d C ) 2 / d sin 2 ( ω t ) 1 / d a ( t ) ( C d K ) 2 / d for   t ( , t 0 ) ,
so that
t 0 d t a ( t ) 1 + a 2 ( t ) ( C d C ) 2 / d 1 + ( C d K ) 4 / d t 0 d t sin 2 ( ω t ) 1 / d = + .
42
Very similar viewpoints are proposed in [65,68].
43
Needless to say, the coordinate description of ξ is defined for p P such that ξ ( p ) is in the domain of the coordinate system. In Equation (A21), Γ μ ν λ ( ξ ) indicates the map p Γ μ ν λ ( ξ ( p ) ) . One could make similar comments on other expressions, appearing in Appendix D and Appendix E.
44
Of course, the distance between two points in the infimum of the lengths of the (piecewise smooth) curves joining the points, defining such lengths via h. The equivalence between geodesic completeness and completeness in the distance induced by h is proved, e.g., in [83].
45
A (parametrized) curve p ξ ( p ) is said to be timelike, lightlike or spacelike if so is its velocity at any p.
46
Of course, the expression a 2 ( ϑ ) in Equation (A26) stands for the function p a 2 ( ϑ ( p ) ) . Similar comments apply to a number of expressions appearing in the sequel; for example, the notation h i j ( η ) in Equation (A29) stands for the map p h i j ( η ( p ) ) , defined for all p such that η ( p ) is in the domain of the coordinates ( x i ) .
47
Concerning the prescriptions in (A59) for ζ and its derivative at 0, and the prescriptions in (A60) for ϑ and its derivative at p , let us note that Equations (A57) and (A58) imply
ξ ( p ) = ( ϑ ( p ) , ζ ( 0 ) ) , d ξ d p ( p ) = d ϑ d p ( p ) , 1 a 2 ( ϑ ( p ) ) d ζ d q ( 0 ) ;
these relations should be compared with the prescriptions (A56).
48
In principle, in Equation (A73), we should ask p = even for any datum with M = 0 and δ = 0 , corresponding to a lightlike geodesic with ξ = constant; however, this requirement is superfluous, since Equation (A66) automatically gives p = if M = 0 and δ = 0 .
49
Here and in the rest of Appendix F, μ , ν should be understood as abstract indexes in the sense of Penrose, while α , β , a are concrete labels.
50
Even though P k < 0 for k K < , we cannot grant a strict inequality in (A98) since K < could be empty.
51
Equivalently, due to (205) and (206), for proving that a 0 < 1 it suffices to show that V ( 1 ) < 0 ; indeed, we already know from (160) that V ( 1 ) = 1 .

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Figure 1. Qualitative plot of the potential a V ( a ) in Equation (183) (corresponding to the standard model of cosmology), under the assumption (187) of nonpositive spatial curvature.
Figure 1. Qualitative plot of the potential a V ( a ) in Equation (183) (corresponding to the standard model of cosmology), under the assumption (187) of nonpositive spatial curvature.
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Figure 2. Qualitative plot of the scale factor t a ( t ) in the standard model of cosmology, under the assumption (187) of nonpositive spatial curvature (see the related Equations (191)–(193)).
Figure 2. Qualitative plot of the scale factor t a ( t ) in the standard model of cosmology, under the assumption (187) of nonpositive spatial curvature (see the related Equations (191)–(193)).
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Figure 3. Qualitative plot of the function a V ( a ) in Equation (201), under the assumptions of Section 6.5.
Figure 3. Qualitative plot of the function a V ( a ) in Equation (201), under the assumptions of Section 6.5.
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Figure 4. Qualitative plot of the function a H ( a ) in Equation (220), under the assumptions of Section 6.5.
Figure 4. Qualitative plot of the function a H ( a ) in Equation (220), under the assumptions of Section 6.5.
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Figure 5. Qualitative plot of the scale factor t a ( t ) , under the assumptions of Section 6.5.
Figure 5. Qualitative plot of the scale factor t a ( t ) , under the assumptions of Section 6.5.
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Figure 6. Qualitative plot of the potential a V ( a ) in Equation (201), under the general conditions of Section 6.8 and the assumption (324) on the curvature.
Figure 6. Qualitative plot of the potential a V ( a ) in Equation (201), under the general conditions of Section 6.8 and the assumption (324) on the curvature.
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Figure 7. Qualitative plot of the potential a V ( a ) in Equation (201), under the general conditions of Section 6.8 and the assumption (327) on the curvature.
Figure 7. Qualitative plot of the potential a V ( a ) in Equation (201), under the general conditions of Section 6.8 and the assumption (327) on the curvature.
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Figure 8. Solutions t a ( t ) of the first type (in blue) and of the second type (in red), under the general conditions of Section 6.8 and the assumption (324) on the curvature. The same assumptions yield a third type of solution, namely a ( t ) = constant = a 2 for all real t.
Figure 8. Solutions t a ( t ) of the first type (in blue) and of the second type (in red), under the general conditions of Section 6.8 and the assumption (324) on the curvature. The same assumptions yield a third type of solution, namely a ( t ) = constant = a 2 for all real t.
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Figure 9. Solutions t a ( t ) of the first type (in blue) and of the second type (in red), under the general conditions of Section 6.8 and the assumption (327) on the curvature.
Figure 9. Solutions t a ( t ) of the first type (in blue) and of the second type (in red), under the general conditions of Section 6.8 and the assumption (327) on the curvature.
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Figure 10. The branch of hyperbola described by r ( t ) , in the framework of Section 8.4.
Figure 10. The branch of hyperbola described by r ( t ) , in the framework of Section 8.4.
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Figure 11. Graph of the radius t r ( t ) , in the framework of Section 8.4.
Figure 11. Graph of the radius t r ( t ) , in the framework of Section 8.4.
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Figure 12. With reference to Section 8.5, let λ 1 : = λ , λ 2 : = 2 λ , A 1 : = 8 A / 3 , B 1 : = 8 A / 3 , A 2 : = A / 5 , B 2 : = 16 A / 5 with arbitrary λ , A > 0 . The curve in blue is the range of r / A R 2 as a function of the variable λ t , for 0.99 λ t 2.06 .
Figure 12. With reference to Section 8.5, let λ 1 : = λ , λ 2 : = 2 λ , A 1 : = 8 A / 3 , B 1 : = 8 A / 3 , A 2 : = A / 5 , B 2 : = 16 A / 5 with arbitrary λ , A > 0 . The curve in blue is the range of r / A R 2 as a function of the variable λ t , for 0.99 λ t 2.06 .
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Figure 13. With reference to Section 8.5, let the parameters be fixed as in Figure 12. The curve in blue is the graph of r / A as a function of the variable λ t , for 0.99 λ t 2.06 .
Figure 13. With reference to Section 8.5, let the parameters be fixed as in Figure 12. The curve in blue is the graph of r / A as a function of the variable λ t , for 0.99 λ t 2.06 .
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Figure 14. The ellipse described by r ( t ) , in the framework of Section 8.7.
Figure 14. The ellipse described by r ( t ) , in the framework of Section 8.7.
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Figure 15. Graph of the radius t r ( t ) , in the framework of Section 8.7.
Figure 15. Graph of the radius t r ( t ) , in the framework of Section 8.7.
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Figure 16. Lissajous curves with ω 2 = 2 ω 1 and several choices of the phase ψ .
Figure 16. Lissajous curves with ω 2 = 2 ω 1 and several choices of the phase ψ .
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Figure 17. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), R 1 (see Section 8.8). The parabolic segment in blue is the range of the map t r ( t ) .
Figure 17. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), R 1 (see Section 8.8). The parabolic segment in blue is the range of the map t r ( t ) .
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Figure 18. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), R 1 (see Section 8.8). Plot of the radius t r ( t ) over a period.
Figure 18. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), R 1 (see Section 8.8). Plot of the radius t r ( t ) over a period.
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Figure 19. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). The parabolic segment in blue is the range of the map t r ( t ) .
Figure 19. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). The parabolic segment in blue is the range of the map t r ( t ) .
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Figure 20. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). Plot of the radius t r ( t ) over a period.
Figure 20. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = π / 2 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). Plot of the radius t r ( t ) over a period.
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Figure 21. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), R 1 (see Section 8.8). The curve in blue is the range of the map t r ( t ) for t in the domain ( 0 , π / ω ) of the cosmological model. The dashed curve in blue is the range of the map t r ( t ) for t ( π / ω , 0 ) .
Figure 21. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), R 1 (see Section 8.8). The curve in blue is the range of the map t r ( t ) for t in the domain ( 0 , π / ω ) of the cosmological model. The dashed curve in blue is the range of the map t r ( t ) for t ( π / ω , 0 ) .
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Figure 22. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), R 1 (see Section 8.8). Plot of the radius t r ( t ) over the domain ( 0 , π / ω ) .
Figure 22. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), R 1 (see Section 8.8). Plot of the radius t r ( t ) over the domain ( 0 , π / ω ) .
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Figure 23. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). Plot of the radius t r ( t ) over the domain ( 0 , π / ω ) .
Figure 23. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). Plot of the radius t r ( t ) over the domain ( 0 , π / ω ) .
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Figure 24. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). The curve in blue is the range of the map t r ( t ) for t in the domain ( 0 , π / ω ) of the cosmological model; p ± are the points maximizing the distance from the origin (see Equation (440)). The dashed curve in blue is the range of the map t r ( t ) for t ( π / ω , 0 ) .
Figure 24. Lissajous cosmology with ω 2 = 2 ω 1 and ψ = 0 ( m o d . 2 π ), 0 < R < 1 (see Section 8.8). The curve in blue is the range of the map t r ( t ) for t in the domain ( 0 , π / ω ) of the cosmological model; p ± are the points maximizing the distance from the origin (see Equation (440)). The dashed curve in blue is the range of the map t r ( t ) for t ( π / ω , 0 ) .
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Figure 25. Lissajous curves with ω 2 / ω 1 = 4 / 3 and several choices of the phase ψ .
Figure 25. Lissajous curves with ω 2 / ω 1 = 4 / 3 and several choices of the phase ψ .
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Figure 26. If ω 2 / ω 1 is irrational (and A 1 , A 2 > 0 ), the range of the map R t r ( t ) densely fills a rectangle.
Figure 26. If ω 2 / ω 1 is irrational (and A 1 , A 2 > 0 ), the range of the map R t r ( t ) densely fills a rectangle.
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Figure 27. With reference to Section 8.10, let the parameters be fixed as in Figure 28. The curve in blue is the graph of r / A as a function of the variable λ t , for 5 λ t 5 .
Figure 27. With reference to Section 8.10, let the parameters be fixed as in Figure 28. The curve in blue is the graph of r / A as a function of the variable λ t , for 5 λ t 5 .
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Figure 28. With reference to Section 8.10, let ω : = 6 λ , B : = A and C : = 40 A , with arbitrary λ , A > 0 . The curves in blue are the ranges of the map t r ( t ) / A R 2 for 5 λ t 0 (dashed curve) and for 0 λ t 5 (continuous curve); we have indicated the points r ( t ) / A for λ t = 5 .
Figure 28. With reference to Section 8.10, let ω : = 6 λ , B : = A and C : = 40 A , with arbitrary λ , A > 0 . The curves in blue are the ranges of the map t r ( t ) / A R 2 for 5 λ t 0 (dashed curve) and for 0 λ t 5 (continuous curve); we have indicated the points r ( t ) / A for λ t = 5 .
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Figure 29. With reference to Section 8.11, let ω : = 6 λ , B : = 0 and C : = 40 A , with arbitrary λ , A > 0 . The curve in blue is the range of the map t r ( t ) / A R 2 for 5 λ t 5 ; we have indicated the point r ( t ) / A for λ t = 5 .
Figure 29. With reference to Section 8.11, let ω : = 6 λ , B : = 0 and C : = 40 A , with arbitrary λ , A > 0 . The curve in blue is the range of the map t r ( t ) / A R 2 for 5 λ t 5 ; we have indicated the point r ( t ) / A for λ t = 5 .
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Figure 30. With reference to Section 8.11, let the parameters be fixed as in Figure 29. The upper picture presents the graph of r / A as a function of the variable λ t , for 5 λ t 5 . The lower picture is a detail of the same graph for λ t in a small interval, confirming that r ( t ) can attain small values, but it does not vanish.
Figure 30. With reference to Section 8.11, let the parameters be fixed as in Figure 29. The upper picture presents the graph of r / A as a function of the variable λ t , for 5 λ t 5 . The lower picture is a detail of the same graph for λ t in a small interval, confirming that r ( t ) can attain small values, but it does not vanish.
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Figure 31. With reference to Section 8.11, let ω : = 6 λ , A : = 0 and C : = 40 B , with arbitrary λ , B > 0 . The curve in blue is a graph of the r / B as a function of λ t , for 5 λ t 5 .
Figure 31. With reference to Section 8.11, let ω : = 6 λ , A : = 0 and C : = 40 B , with arbitrary λ , B > 0 . The curve in blue is a graph of the r / B as a function of λ t , for 5 λ t 5 .
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Figure 32. With reference to Section 8.11, let ω : = 6 λ , A : = F / 2 , B : = F / 2 and C : = 10 F , with arbitrary λ , F > 0 . The curve in blue is the range of the map t r ( t ) / F R 2 for 0 < λ t 4 . 5 . We have indicated the point r ( t ) / F for λ t = 4 . 5 ; also, let us recall that r ( t ) 0 for t 0 + .
Figure 32. With reference to Section 8.11, let ω : = 6 λ , A : = F / 2 , B : = F / 2 and C : = 10 F , with arbitrary λ , F > 0 . The curve in blue is the range of the map t r ( t ) / F R 2 for 0 < λ t 4 . 5 . We have indicated the point r ( t ) / F for λ t = 4 . 5 ; also, let us recall that r ( t ) 0 for t 0 + .
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Figure 33. With reference to Section 8.11, let the parameters be fixed as in Figure 32. The curve in blue is the graph of r / F as a function of the variable λ t , for 0 < λ t 4 . 5 .
Figure 33. With reference to Section 8.11, let the parameters be fixed as in Figure 32. The curve in blue is the graph of r / F as a function of the variable λ t , for 0 < λ t 4 . 5 .
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Cimaglia, M.; Gengo, M.; Pizzocchero, L. Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes. Universe 2024, 10, 467. https://doi.org/10.3390/universe10120467

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Cimaglia M, Gengo M, Pizzocchero L. Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes. Universe. 2024; 10(12):467. https://doi.org/10.3390/universe10120467

Chicago/Turabian Style

Cimaglia, Michela, Massimo Gengo, and Livio Pizzocchero. 2024. "Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes" Universe 10, no. 12: 467. https://doi.org/10.3390/universe10120467

APA Style

Cimaglia, M., Gengo, M., & Pizzocchero, L. (2024). Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes. Universe, 10(12), 467. https://doi.org/10.3390/universe10120467

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