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

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 ($\Lambda$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 $r=\sqrt{x^2+y^2}$. 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.

Contents
1.	Introduction	4
2.	Generalities on Einstein’s Gravity, Perfect Fluids and Scalar Fields	11
	2.1. Dimensional Aspects	11
	2.2. Conventions About Spacetime, the Einstein Equations, and the Gravitational Constant	11
	2.3. Einstein’s Gravity with Perfect Fluids and a Scalar Field	12
3	FLRW Cosmologies with Perfect Fluids and a Scalar Field	14
	3.1. Spacetime Structure	14
	3.2. Conditions for Timelike or Lightlike Geodesic Completeness: Nonsingular FLRW Spacetimes	15
	3.3. Hubble Parameter	16
	3.4. The Perfect Fluids	16
	3.5. The Scalar Field	17
	3.6. The Total Stress–Energy Tensor	17
	3.7. Einstein Equations	17
	3.8. Klein–Gordon Equation	18
	3.9. Conservation Law for the n-th Fluid	18
	3.10. Summary of the Evolution Equations	18
	3.11. Curvature Density; Normalized Densities	18
	3.12. Stationary Points of the Scale Factor	19
	3.13. Energy Conditions	19
	3.14. The Scale Factor at a Special Time	20
	3.15. Dimensionless Formalism	20
4.	Analysis of the Evolution Equations	24
	4.1. Determining the Fluids’ Densities	24
	4.2. The Fluids’ Densities in the Linear Case	26
	4.3. The Final Form of the Einstein and Klein–Gordon Equations	26
	4.4. Lagrangian Formalism; Zero-Energy Constraint	27
	4.5. Again on the Scale Factor at a Special Time	28
	4.6. Maximal Solutions	29
5.	The Case with a Constant Potential for the Scalar Field: General Results on the Evolution of $\mathit{a}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ and $\mathit{\varphi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	29
	5.1. Basic Setting	29
	5.2. The Constant of Motion $\Pi _{\ast }$	30
	5.3. Reduced Lagrangian	30
	5.4. Zero-Energy Solutions of the Reduced System	31
	5.5. Time Evolution of the Scalar Field	32
	5.6. On Maximal Solutions	32
	5.7. Hubble Parameter; Densities and Pressures	33
	5.8. The Scale Factor at a Special Time	33
	5.9. Comparison with the Literature	34
6.	Again on the Case of $\mathit{V}\mathbf{\left(}\mathit{\varphi }\mathbf{\right)}$ = Constant: Big Bounce from a Phantom Scalar	34
	6.1. Introducing a More Specific Setting	34
	6.2. The Case of $\Pi _{\ast }=0$ : Recovering the Standard Model of Cosmology	36
	6.3. Introducing the Analysis of Cases with $\Pi _{\ast }\ne 0$	40
	6.4. Preparing the Analysis of the Phantom Case with $\Pi _{\ast }\ne 0$	40
	6.5. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }⩾0$: Analysis of the Potential $\mathcal{V}\left(a\right)$ and the Associated Function $\mathcal{H}\left(a\right):=-\mathcal{V}\left(a\right)/a^2$	41
	6.6. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }⩾0$ : Analysis of the Zero-Energy Solutions and Classicality Condition	45
	6.7. The Phantom Case with $\Pi _{\ast }\ne 0$  (Small), ${\Omega }_{k\ast }=0$ and Specific Values for All the Parameters	54
	6.8. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }<0$: Sketching the Basic Results	57
7.	Polar and Cartesian Coordinates for Phantom Cosmologies with a Periodic Field Potential	62
	7.1. Polar Coordinates (Under Periodicity Assumptions for the Field Potential)	62
	7.2. Cartesian Coordinates	63
8.	An Explicitly Solvable Phantom Model with Dust and a Trigonometric Field	63
	8.1. Some Introductory Considerations	63
	8.2. Introducing the Solvable Model	64
	8.3. The Case $W_i={\lambda }_i^2>0$ ($i=1,2$)	66
	8.4. Cosmologies with ${\lambda }_1={\lambda }_2$	66
	8.5. Cosmologies with ${\lambda }_1$ and ${\lambda }_2$ Arbitrary	69
	8.6. The Case $W_i=-{\omega }_i^2<0$ ($i=1,2$): Lissajous Cosmologies	72
	8.7. Lissajous Cosmologies with ${\omega }_1={\omega }_2$	72
	8.8. Lissajous Cosmologies with ${\omega }_2=2{\omega }_1$	75
	8.9. On General Lissajous Cosmologies	83
	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 Remarks	93
Appendix A.	Riemannian Manifolds, Spacetimes, and Dimensional Analysis	93
Appendix B.	FLRW Spacetimes and Generalizations	94
Appendix C.	Time Orientation of a Spacetime: The Case of a Generalized FLRW Spacetime	95
Appendix D.	A Review on Geodesics and Several Associated Notions of Completeness, Mainly in Riemannian Manifolds and in Spacetimes. Nonsingular Spacetimes	96
Appendix E.	Geodesics and Geodesic Completeness of Generalized FLRW Spacetimes	98
Appendix F.	A Review the Energy Conditions	106
Appendix G.	On the Determination of the Fluids’ Densities	106
Appendix H.	Comparing the Zero Energy Motions of Two One-Dimensional Lagrangians	107
Appendix I.	The Descartes’ Rule of Signs	107
Appendix J.	Two Statements on a Class of Polynomials	108
Appendix K.	Proof of Statements (185) and (186) on the Potential $\mathcal{V}$ of the Standard Cosmological Model	110
Appendix L.	Proof of Statements (a)–(d) in Section 6.5 about the Function $\mathcal{V}$ of Equation (201)	111
Appendix M.	Proof of Statements (f)–(i) in Section 6.5 about the Function $\mathcal{H}$ of Equation (220)	114
Appendix N.	Proof of Equation (250) (Including Convergence of an Integral Therein)	116
Appendix O.	Proof of Some Statements about ${\mathit{a}}_{\mathit{\varphi }\mathit{r}\mathbf{-}}$ in Equation (271)	117
Appendix P.	A Usefulf Inequality	118
Appendix Q.	Estimates on the Time ${\mathit{t}}_{\mathit{i}}$ in Equation (244)	119
Appendix R.	Further Estimates on the Time ${\mathit{t}}_{\mathit{i}}$ in Equation (244)	122
Appendix S.	On the Quantity $\mathbf{\Delta }\mathit{\varphi }$ in Equation (292)	125
Appendix T.	A More Detailed Description of the Model in Section 6.7	128
Appendix U.	Proof of Statements ($\mathit{\alpha }$-$\mathit{\zeta }$) in Section 6.8 about the Function $\mathcal{V}$ of Equation (201)	128
Appendix V.	Derivation of Equations (363)–(366)	130
Appendix W.	Derivation of Equations (401) and (402)	132
Notes		132
References		136

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 $\Phi$ is the scalar field and a self-interaction potential $\mathcal{V}(\Phi )$ 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 $-(1/2)\xi R{\Phi }^2$ under the integral in our Equation (6), where R is the scalar curvature, and $\xi$ is a numerical parameter. It is well known [8] that the evolution equation for the scalar field has properties of conformal invariance if $\xi =(d-1)/\left(4d\right)$, 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 $\xi =0$; 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 $\tau$ and the (positive, dimensionless) scale factor $a\left(\tau \right)$.

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 $\mathcal{V}(\Phi )$ 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 $-(\sigma /2){\nabla }_{\mu }\Phi {\nabla }^{\mu }\Phi$ in Equation (6) with $\sigma =+1$ and $\sigma =-1$, 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 $-(\sigma /2){\nabla }_{\mu }\Phi {\nabla }^{\mu }\Phi$ in the action functional (6) ($\sigma =\pm 1$) with one of the form $-(1/2)\omega (\Phi ){\nabla }_{\mu }\Phi {\nabla }^{\mu }\Phi$, where $\Phi \mapsto \omega (\Phi )$ is a given real-valued function, with sign depending on $\Phi$. 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 ($\Lambda$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 $\tau \mapsto a\left(\tau \right)$, well defined for all $\tau \in (-\infty ,+\infty )$, 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, $f\left(R\right)$ 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 $p_s=-(1/3){\rho }_s$ and $p_w=-(2/3){\rho }_w$ (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 $p_{ph}=w_{ph}{\rho }_{ph}$, involving constants $w_{ph}<-1$ (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 $d\in \{2,3,4,\dots \}$. The n-th fluid, not interacting directly with the other actors, has an arbitrary equation of state $p_n={\mathcal{P}}_n({\rho }_n)$, 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 $\tau$ and the scale factor $a=a\left(\tau \right)$, with an arbitrary value for the constant sectional curvature in the spatial part of the metric; the fluids’ densities depend only on $\tau$, 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 $H_{\ast }$; 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 $t=H_{\ast }\tau$ (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 $\varphi$ (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 $L(a,\varphi ,\dot{a},\dot{\varphi })$ (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 $L(a,\varphi ,\dot{a},\dot{\varphi })$ (see again Equations (121) and (122)) is now independent of $\varphi$; so, the canonically conjugate momentum $\Pi _{\ast } :=\partial L/\partial \dot{\varphi }$ 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 $\dot{a}$; the latter can be ultimately put into the form $\mathcal{L}(a,\dot{a})={\dot{a}}^2-\mathcal{V}\left(a\right)$, where $\mathcal{V}$ 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 $\Pi _{\ast }$ (see Equations (141) and (142); the zero-energy constraint is now a condition on this reduced system). Formally, $\mathcal{L}$ 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 $a=a\left(t\right)$ from the graph of $\mathcal{V}$, and to compute the function $a\left(t\right)$ 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 $\Pi _{\ast }$ vanishes, the (canonical or phantom) scalar field is constant and the scale factor $a=a\left(t\right)$ evolves as in the standard ($\Lambda$CDM) model of cosmology, as illustrated in Section 6.2. For $\Pi _{\ast }\ne 0$, 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 $\Pi _{\ast }$ 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 $a\left(t\right)$ attains a minimum value $a_0>0$ at $t=0$, decreases for $t⩽0$, increases for $t⩾0$ and diverges exponentially for $t\to \mp \infty$ (indeed the model is time-symmetric: $a(-t)=a\left(t\right)$); the energy conditions are analyzed, and their violations are indicated. A reasonable choice for the value of $\Pi _{\ast }$ 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 $\Pi _{\ast }$ 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 $(a,\varphi )$ 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=$$\mathrm{const}. \times r^{2/d}$ and $\varphi =$$\mathrm{const}. \times \theta$, the Lagrangian $L(a,\varphi ,\dot{a},\dot{\varphi })$ of the system (see once more Equations (121) and (122)) becomes a function $L(r,\theta ,\dot{r},\dot{\theta })$ (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 $(r,\theta )$, in the presence of forces with a potential depending on r and $\theta$. 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 $\theta$ represents the scalar field. Of course, an equivalent description can be given in terms of the Cartesian coordinates $x=rcos\theta$, $y=rsin\theta$ (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 $\theta$, is a function of the form $\theta \mapsto$$(1/2)W_1{cos}^2\theta +(1/2)W_2{sin}^2\theta$ 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 $\ddot{x}=W_{1}x$, $\ddot{y}=W_{2}y$ (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 $t\mapsto \left(x\right(t),y(t\left)\right)$ 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 $\mathbb{R}$ of real numbers. Lengths, times, and masses are described as elements of appropriate real, one-dimensional, oriented vector spaces $\mathbb{L},\mathbb{T},\mathbb{M}$, 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 $\mathbb{D}=\mathbb{M}{\mathbb{L}}^{-d}$. If $\mathbb{X}$ is any one of the spaces $\mathbb{R},\mathbb{L},\mathbb{T},\mathbb{M},\mathbb{D}$, and so on, we will write ${\mathbb{X}}^{+}$ (respectively, ${\mathbb{X}}_{+}$) for the subset of positive (respectively, non-negative) elements of $\mathbb{X}$ (${\mathbb{X}}_{+}={\mathbb{X}}^{+}\cup \left\{0\right\}$). Throughout the paper, we identify a time duration $\tau$ with the length $c\tau$, where c is the speed of light. Thus, $\mathbb{T}\simeq \mathbb{L}$, 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 $C^{\infty }$”. We frequently refer to Riemannian manifolds or to spacetimes (i.e., manifolds with a Lorentzian metric, of signature $(-,+,\dots ,+)$); 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 ${\mathcal{M}}^{d+1}$ 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 ${\mathcal{M}}^{d+1}$ is generically written as $\left(x^{\mu }\right)$ (with Greek indices); of course, $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. 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 ∇ , $R_{\mu \nu }$, and R. The Einstein equations are written as

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=d(d-1){\gamma }_d{G}_d{T}_{\mu \nu },$$

(1)

where $T_{\mu \nu }$ is the stress–energy tensor; here,

$$G_d\in {\left({\mathbb{L}}^{d-2}{\mathbb{M}}^{-1}\right)}^{+}={\left({\mathbb{L}}^{-2}{\mathbb{D}}^{-1}\right)}^{+}$$

(2)

is the gravitational constant, and ${\gamma }_d$ 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

$${\gamma }_d:={\displaystyle \frac{{\pi }^{d/2}}{(d-2)\Gamma (d/2+1)}}\mathrm{for}d⩾3,{\gamma }_2:=\mathrm{any} \mathrm{positive} \mathrm{real} \mathrm{number}.$$

(3)

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 ${\gamma }_2$ is not specified. Finally, let us remark that

$${\gamma }_3={\displaystyle \frac{4}{3}}\pi ;d(d-1){\gamma }_d{G}_d{|}_{d=3}=8\pi G,G\equiv G_3\mathrm{the} \mathrm{usual} \mathrm{gravitational} \mathrm{constant}.$$

(4)

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

We consider a general model living in a $(d+1)$-dimensional spacetime ${\mathcal{M}}^{d+1}$ (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 $n\in \{1,\dots ,N\}$, we suppose that the n-th fluid has a positive mass–energy density ${\rho }_n$ and a pressure $p_n$. We have ${\rho }_n:{\mathcal{M}}^{d+1}\to {\mathbb{D}}^{+}$ and $p_n:{\mathcal{M}}^{d+1}\to \mathbb{D}$ where $\mathbb{D}$ 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:

  $$p_n={\mathcal{P}}_n\left({\rho }_n\right),{\mathcal{P}}_n:{\mathbb{D}}^{+}\to \mathbb{D}.$$

  (5)
  The above-mentioned perfect fluids can include, for example, dust ($p_n=0$) and a radiation gas ($p_n=(1/d){\rho }_n$).3
• A classical scalar field $\Phi$, minimally coupled to gravity and self-interacting with potential $\mathcal{V}(\Phi )$. We have $\Phi :{\mathcal{M}}^{d+1}\to \mathbb{F}$, where $\mathbb{F}$ is an appropriate real, one-dimensional, oriented vector space. For dimensional reasons that will soon be clarified, we require ${\mathbb{F}}^2=\mathbb{M}{\mathbb{L}}^{2-d}=\mathbb{D}{\mathbb{L}}^2$ and $\mathcal{V}:\mathbb{F}\to \mathbb{D}$. 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

$$\mathcal{S}:={\int }_{{\mathcal{M}}^{d+1}}\mathrm{d}v\left[{\displaystyle \frac{R}{2d(d-1){\gamma }_d{G}_d}}-\sum _{n=1}^N{\rho }_n-{\displaystyle \frac{\sigma }{2}}{\nabla }_{\mu }\Phi {\nabla }^{\mu }\Phi -\mathcal{V}(\Phi )\right].$$

(6)

In the above, $\mathrm{d}v$ is the pseudo-Riemannian volume element corresponding to the spacetime metric g ($\mathrm{d}v=\sqrt{-det\left(g_{\mu \nu }\right)}{\prod }_{\mu }\mathrm{d}{x}^{\mu }$ in any coordinate system $(x^{\mu }$)); moreover, $G_d$ and ${\gamma }_d$ 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, $\Phi$ being a scalar function, ${\nabla }_{\mu }\Phi$ coincides with the usual derivative ${\partial }_{\mu }\Phi$ in any coordinate system. Finally, $\sigma \in \{\pm 1\}$ is a sign parameter: in compliance with the standard nomenclature, we shall call $\Phi$ a canonical scalar field (or an ordinary scalar field) if $\sigma =+1$, and a phantom scalar field if $\sigma =-1$ (in applications to cosmology, the terms quintessence and phantom quintessence are also employed for $\sigma =+1$ and $\sigma =-1$, respectively).

Note that all the summands between square brackets on the right-hand side of Equation (6) take values in the space $\mathbb{D}$; thus, $\mathcal{S}$ takes values in the space $\mathbb{D}{\mathbb{L}}^{d+1}=\mathbb{M}\mathbb{L}$ (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 $\mathcal{S}$ of Equation (6) must be viewed as depending on the spacetime metric g, on the fluids’ histories and on the scalar field $\Phi$; the fluids’ histories are defined via the world lines of the fluids’ particles and $\mathcal{S}$ depends on such histories through the densities ${\rho }_n$, see, e.g.,  [12]4. The evolution equations for the model can be derived requiring the action $\mathcal{S}$ 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

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=d(d-1){\gamma }_d{G}_d\left(\sum _{n=1}^N{T}_{\mu \nu }^n+T_{\mu \nu }^{\Phi }\right);$$

(7)

these involve the stress–energy tensors of the fluids and the scalar field, defined, respectively, as follows:

$$\begin{array}{c}{T}_{\mu \nu }^n:=(p_n+{\rho }_n)U_{\mu }{U}_{\nu }+p_n{g}_{\mu \nu }\mathrm{for}n\in \{1,\dots ,N\},\end{array}$$

(8)

$$\begin{array}{c}{T}_{\mu \nu }^{\Phi }:=\sigma \left({\nabla }_{\mu }\Phi {\nabla }_{\nu }\Phi -{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\nabla }_{\lambda }\Phi {\nabla }^{\lambda }\Phi \right)-\mathcal{V}(\Phi )g_{\mu \nu },\end{array}$$

(9)

with $U^{\mu }$ indicating the common $(d+1)$-velocity vector field of all the fluids ($U_{\mu }{U}^{\mu }=-1$)5.

Secondly, the stationarity of $\mathcal{S}$ under variations in the fluids’ histories leads to the separate conservation laws

$${\nabla }^{\mu }{T}_{\mu \nu }^n=0(n=1,\dots ,N)$$

(10)

(these manifest our assumption, codified in $\mathcal{S}$, that no direct interaction exists between any two fluids or between any fluid and the scalar field).

Finally, the stationarity condition for $\mathcal{S}$ under variations in the field $\Phi$ yields the Klein–Gordon equation

$$-\sigma {\nabla }_{\mu }{\nabla }^{\mu }\Phi +{\mathcal{V}}^{\prime }(\Phi )=0,$$

(11)

where ${\mathcal{V}}^{\prime }$ is the derivative of the function $\mathcal{V}$. 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:

$$\mathcal{V}(\Phi )=\mathrm{const}.\equiv {\mathcal{V}}_{\ast }.$$

(12)

Then, the summand $d(d-1){\gamma }_d{G}_d{T}_{\mu \nu }^{\Phi }$ in the right-hand side of the Einstein Equation (7) contains (according to (9)) a term of the form $-d(d-1){\gamma }_d{G}_d{\mathcal{V}}_{\ast }{g}_{\mu \nu }$; moving this term from the right-hand side to the left-hand side of Equation (7), we obtain

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }=d(d-1){\gamma }_d{G}_d\left(\sum _{n=1}^N{T}_{\mu \nu }^n+{\mathcal{T}}_{\mu \nu }^{\Phi }\right),$$

(13)

$$\Lambda :=d(d-1){\gamma }_d{G}_d{\mathcal{V}}_{\ast },{\mathcal{T}}_{\mu \nu }^{\Phi }:=\sigma \left({\nabla }_{\mu }\Phi {\nabla }_{\nu }\Phi -{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\nabla }_{\lambda }\Phi {\nabla }^{\lambda }\Phi \right).$$

(14)

Equation (13) includes the Einstein equations with a cosmological constant $\Lambda$, and the term ${\mathcal{T}}_{\mu \nu }^{\Phi }$ on their right-hand side is the stress–energy tensor of a free, massless scalar field; note that $\Lambda$ has the same sign as ${\mathcal{V}}_{\ast }$.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

$${\mathcal{M}}^{d+1}=\mathcal{T}\times {\mathfrak{S}}^d,$$

(15)

where $\mathcal{T}\subset \mathbb{T}(=\mathbb{L})$ is an open time interval with its natural coordinate $\tau$ (the “cosmic time”) and ${\mathfrak{S}}^d$ (the “space”) is a d-dimensional, complete Riemannian manifold of constant sectional curvature $k\in {\mathbb{L}}^{-2}$, often referred to as the “spatial curvature” in the sequel. With the additional assumption of simple connectedness, ${\mathfrak{S}}^d$ 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, ${\mathfrak{S}}^d$ 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 ${\mathfrak{S}}^d$, and $\mathrm{d}{\ell }^2$ for the corresponding, squared line element. A coordinate system of ${\mathfrak{S}}^d$ is generically indicated with ${\left(x^i\right)}_{i=1,\dots ,d}$ (using Latin indexes like $i,j,\dots$); of course, $\mathrm{d}{\ell }^2=h_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j$. It is well known (see again [71]) that ${\mathfrak{S}}^d$ can be covered by (local, $\mathbb{L}$-valued) coordinates $\left(x^i\right)$ in which

$$\mathrm{d}{\ell }^2={\displaystyle \frac{{\displaystyle \sum _{i=1}^d{\left(\mathrm{d}{x}^i\right)}^2}}{{\left({\displaystyle 1+\frac{k}{4}\sum _{i=1}^d{\left(x^i\right)}^2}\right)}^2}}.$$

(16)

The squared line element of the spacetime (15) is supposed to have the following form:

$$\mathrm{d}{s}^2=-\mathrm{d}{\tau }^2+a^2\left(\tau \right)\mathrm{d}{\ell }^2,$$

(17)

where $a:\mathcal{T}\to {\mathbb{R}}^{+}$, $\tau \mapsto a\left(\tau \right)$ is the (smooth, dimensionless) scale factor. Of course, any coordinate system ${\left(x^i\right)}_{i=1,\dots ,d}$ on ${\mathfrak{S}}^d$ induces a spacetime coordinate system

$$(\tau ,{\left(x^i\right)}_{i=1,\dots ,d})\equiv {\left(x^{\mu }\right)}_{\mu =0,\dots ,d}$$

(18)

in which $g_{00}=-1$, $g_{ij}=a^2{h}_{ij}$ and $g_{0i}=g_{i0}=0$ for $i,j=1,\dots ,d$. 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 $\left(X^{\mu }\right)$ 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 $i=1,\dots ,d$, while $x^0=\tau$ gives the proper time; so, the $(d+1)$-velocity of the particle has the following components:

$$U^{\mu }={\delta }_0^{\mu }(\mu =0,\dots ,d).$$

(19)

From now on, we intend

$$\stackrel{\circ }{X}:={\displaystyle \frac{\mathrm{d}}{\mathrm{d}\tau }}$$

(20)

and $\stackrel{\circ \circ }{X}:={\mathrm{d}}^2/\mathrm{d}{\tau }^2$; the more standard dotted notation $\dot{X}$ 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 $-\infty$ (resp., with final endpoint $+\infty$).

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 $(-\infty ,+\infty )$).

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 $\mathcal{T}\subset \mathbb{T}$ as

$$\mathcal{T}=({\tau }_{-},{\tau }_{+}),{\tau }_{-}\in \{-\infty \}\cup \mathbb{T},{\tau }_{+}\in \mathbb{T}\cup \{+\infty \},{\tau }_{-}<{\tau }_{+},$$

(21)

and let us choose any instant ${\tau }_0\in \mathcal{T}$. Then, the given, time-oriented spacetime is:

• Past timelike complete, if and only if

  $${\tau }_{-}=-\infty ,{\int }_{-\infty }^{{\tau }_0}\mathrm{d}\tau {\displaystyle \frac{a\left(\tau \right)}{\sqrt{1+a^2\left(\tau \right)}}}=+\infty ;$$

  (22)
• Past lightlike complete, if and only if

  $${\int }_{{\tau }_{-}}^{{\tau }_0}\mathrm{d}\tau a\left(\tau \right)=+\infty$$

  (23)
  (without requiring ${\tau }_{-}=-\infty$);
• Future timelike complete, if and only if

  $${\tau }_{+}=+\infty ,{\int }_{{\tau }_0}^{+\infty }\mathrm{d}\tau {\displaystyle \frac{a\left(\tau \right)}{\sqrt{1+a^2\left(\tau \right)}}}=+\infty ;$$

  (24)
• Future lightlike complete, if and only if

  $${\int }_{{\tau }_0}^{{\tau }_{+}}\mathrm{d}\tau a\left(\tau \right)=+\infty$$

  (25)
  (without requiring ${\tau }_{+}=+\infty$).

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 ${\mathfrak{S}}^d$ or the geodesic itself are constant).

As an example, an FLRW spacetime with

$$\mathcal{T}=(-\infty ,+\infty )\equiv \mathbb{T},\underset{\tau \in \mathcal{T}}{inf}a\left(\tau \right)=a_0>0$$

(26)

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

$$\mathcal{T}=(-\infty ,+\infty ),a\left(\tau \right)=e^{H_{\ast }\tau }\mathrm{for} \tau \in \mathcal{T}(H_{\ast }\in \mathbb{T},H_{\ast }\ne 0)$$

(27)

violates conditions (22) and (23) if $H_{\ast }>0$ and conditions (24) and (25) if $H_{\ast }<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

$$H:={\displaystyle \frac{\stackrel{\circ }{a}}{a}};$$

(28)

at each time $\tau \in \mathcal{T}$, $H\left(\tau \right)\in {\mathbb{T}}^{-1}={\mathbb{L}}^{-1}$ 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 $a\left(\tau \right)\mathrm{d}\ell$).

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 $n=1,\dots ,N$, the $(d+1)$-velocity appearing in the stress–energy tensors (8) has the form (19), in any coordinate system ${\left(x^{\mu }\right)}_{\mu =0,\dots ,d}$ 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:

$${\rho }_n\equiv {\rho }_n\left(\tau \right).$$

(29)

Consequently, the same happens for the pressure: $p_n=p_n\left(\tau \right)={\mathcal{P}}_n\left({\rho }_n\left(\tau \right)\right)$ (recall Equation (5)).

3.5. The Scalar Field

Again to comply with the homogeneity hypothesis, we assume that the scalar field $\Phi$ depends only on cosmic time:

$$\Phi \equiv \Phi \left(\tau \right).$$

(30)

In this case, the stress–energy tensor (9) also has the perfect fluid form with a suitable density ${\rho }_{\Phi }$ and pressure $p_{\Phi }$, namely

$$T_{\mu \nu }^{\Phi }=(p_{\Phi }+{\rho }_{\Phi })U_{\mu }{U}_{\nu }+p_{\Phi }{g}_{\mu \nu },$$

(31)

$${\rho }_{\Phi }:={\displaystyle \frac{\sigma }{2}}{\stackrel{\circ }{\Phi }}^2+\mathcal{V}(\Phi ),p_{\Phi }:={\displaystyle \frac{\sigma }{2}}{\stackrel{\circ }{\Phi }}^2-\mathcal{V}(\Phi )$$

(32)

(note that ${\rho }_{\Phi },p_{\Phi }$ actually take values in the space of densities $\mathbb{D}$, due to the dimensional assumptions about $\Phi$ and $\mathcal{V}$ in Section 2.3).

3.6. The Total Stress–Energy Tensor

Due to Equations (8) and (31), this has the following form:

$$T_{\mu \nu }=(p+\rho )U_{\mu }{U}_{\nu }+p{g}_{\mu \nu },$$

(33)

$$\rho :=\sum _{n=1}^N{\rho }_n+{\rho }_{\Phi },p:=\sum _{n=1}^N{p}_n+p_{\Phi }.$$

(34)

3.7. Einstein Equations

The components of the Ricci tensor $R_{\mu \nu }$ and the scalar curvature R for the spacetime metric (17) are readily computed in any coordinate system $\left(x^{\mu }\right)$ as in (18); it is found that

$$R_{00}=-d{\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}},R_{ij}=\left[a\stackrel{\circ \circ }{a}+(d-1){\stackrel{\circ }{a}}^2+(d-1)k\right]h_{ij},R_{0i}=R_{i0}=0(i,j=1,\dots ,d),$$

(35)

$$R=2d{\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}}+d(d-1){\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+d(d-1){\displaystyle \frac{k}{a^2}}$$

(36)

(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,

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=d(d-1){\gamma }_d{G}_d\left[(\mathfrak{P}+\mathfrak{R})U_{\mu }{U}_{\nu }+\mathfrak{P}{g}_{\mu \nu }\right],$$

(37)

where the “density” $\mathfrak{R}$ and the “pressure” $\mathfrak{P}$ are given by

$$\mathfrak{R}:={\displaystyle \frac{1}{2{\gamma }_d{G}_d}}\left({\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{k}{a^2}}\right),\mathfrak{P}:=-{\displaystyle \frac{1}{d{\gamma }_d{G}_d}}\left({\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{k}{a^2}}\right).$$

(38)

Due to (33), (37) and (38), the Einstein Equations (7) read as follows:

$${\displaystyle \frac{1}{2{\gamma }_d{G}_d}}\left({\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{k}{a^2}}\right)=\rho ,$$

(39)

$$-{\displaystyle \frac{1}{d{\gamma }_d{G}_d}}\left({\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{k}{a^2}}\right)=p,$$

(40)

with $\rho ={\sum }_{n=1}^N{\rho }_n+{\rho }_{\Phi }$, $p={\sum }_{n=1}^N{p}_n+p_{\Phi }$ as in (34), $p_n={\mathcal{P}}_n\left({\rho }_n\right)$, as in (5), and ${\rho }_{\Phi },p_{\Phi }$ as in (32).

3.8. Klein–Gordon Equation

In the present framework, ${\displaystyle {\nabla }_{\mu }{\nabla }^{\mu }\Phi =-\stackrel{\circ \circ }{\Phi }-d\frac{\stackrel{\circ }{a}}{a}\stackrel{\circ }{\Phi }}$12; thus, the Klein–Gordon Equation (11) for the scalar field reads as follows:

$$\sigma (\stackrel{\circ \circ }{\Phi }+d{\displaystyle \frac{\stackrel{\circ }{a}}{a}}\stackrel{\circ }{\Phi })+{\mathcal{V}}^{\prime }(\Phi )=0$$

(41)

(note that $\stackrel{\circ }{a}/a$ is the Hubble parameter H of Equation (28)).

3.9. Conservation Law for the n-th Fluid

Let $n\in \{1,\dots ,N\}$. In the present setting, one has ${\displaystyle {\nabla }^{\mu }{T}_{\mu \nu }^n=[{\stackrel{\circ }{\rho }}_n+d\frac{\stackrel{\circ }{a}}{a}(p_n+{\rho }_n)]U_{\nu }}$; thus, the conservation law (10) reads as follows:

$${\stackrel{\circ }{\rho }}_n+d{\displaystyle \frac{\stackrel{\circ }{a}}{a}}(p_n+{\rho }_n)=0.$$

(42)

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

$$\mathcal{T}\ni \tau \mapsto a\left(\tau \right)\in {\mathbb{R}}^{+},\Phi \left(\tau \right)\in \mathbb{F},{\rho }_n\left(\tau \right)\in {\mathbb{D}}^{+}(n=1,\dots ,N),$$

(43)

$$\mathcal{T}\subset \mathbb{T}\mathrm{an} \mathrm{open} \mathrm{interval}.$$

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 $a,\Phi ,{\rho }_n$ ($n=1,\dots ,N$) be as in (43). We define ${\rho }_{\Phi }$ as usually, following Equation (32); for a reason that will be clear hereafter, we also define the curvature density

$${\rho }_k:=-{\displaystyle \frac{k}{2{\gamma }_d{G}_d}}{\displaystyle \frac{1}{a^2}}$$

(44)

(taking values in $\mathbb{D}$). That said, it is clear that the first Einstein Equation (39) means ${\displaystyle \frac{1}{2{\gamma }_d{G}_d}\frac{{\stackrel{\circ }{a}}^2}{a^2}=\rho +{\rho }_k}$, i.e.,

$${\displaystyle \frac{H^2}{2{\gamma }_d{G}_d}}=\tilde{\rho },$$

(45)

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

$$\tilde{\rho }:=\rho +{\rho }_k=\sum _{n=1}^N{\rho }_n+{\rho }_k+{\rho }_{\Phi }.$$

(46)

For comparing the different densities considered here, it is customary to introduce the normalized densities

$${\Omega }_n:={\displaystyle \frac{{\rho }_n}{\tilde{\rho }}}(n=1,\dots ,N),{\Omega }_k:={\displaystyle \frac{{\rho }_k}{\tilde{\rho }}},{\Omega }_{\Phi }:={\displaystyle \frac{{\rho }_{\Phi }}{\tilde{\rho }}},$$

(47)

which are well defined and dimensionless at any time $\tau$ such that $\tilde{\rho }\left(\tau \right)\ne 0$; of course, these fulfill by constructing the following condition:

$$\sum _{n=1}^N{\Omega }_n+{\Omega }_k+{\Omega }_{\Phi }=1.$$

(48)

If the first Einstein equation (i.e., Equation (39) or (45)) holds, we have $\tilde{\rho }\left(\tau \right)\ne 0$ if and only if $H\left(\tau \right)\ne 0$, and we can also write

$${\Omega }_n={\displaystyle \frac{2{\gamma }_d{G}_d}{H^2}}{\rho }_n(n=1,\dots ,N),{\Omega }_k={\displaystyle \frac{2{\gamma }_d{G}_d}{H^2}}{\rho }_k=-{\displaystyle \frac{k}{{\stackrel{\circ }{a}}^2}},$$

(49)

$${\Omega }_{\Phi }={\displaystyle \frac{2{\gamma }_d{G}_d}{H^2}}{\rho }_{\Phi }={\displaystyle \frac{2{\gamma }_d{G}_d}{H^2}}\left[{\displaystyle \frac{\sigma }{2}}{\stackrel{\circ }{\Phi }}^2+\mathcal{V}(\Phi )\right].$$

3.12. Stationary Points of the Scale Factor

Let $a,\Phi ,{\rho }_n$ ($n=1,\dots ,N$) be as in (43), and assume the first Einstein Equation (39) or (45) holds; let ${\tau }_0\in \mathcal{T}$. It is readily seen from (45) that ${\tau }_0$ is a stationary point for the scale factor if and only if the total density, including the curvature contribution, vanishes at this time:

$$\stackrel{\circ }{a}\left({\tau }_0\right)=0⟺0=\tilde{\rho }\left({\tau }_0\right)=\left(\sum _{n=1}^N{\rho }_n+{\rho }_k+{\rho }_{\Phi }\right)\left({\tau }_0\right).$$

(50)

We are assuming ${\rho }_n>0$ at all times; moreover, the definition (44) indicates that ${\rho }_k⩾0$ at all times if $k⩽0$. Thus,

$$k⩽0\mathrm{and}\stackrel{\circ }{a}\left({\tau }_0\right)=0⟹{\rho }_{\Phi }\left({\tau }_0\right)<0.$$

(51)

Of course, the density ${\rho }_{\Phi }=(\sigma /2){\stackrel{\circ }{\Phi }}^2+\mathcal{V}(\Phi )$ is more likely to become negative in the phantom case $\sigma =-1$; 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 $\rho$ and pressure p,

$$\mathrm{WEC} \mathrm{holds}⟺\rho ⩾0,\rho +p⩾0;$$

(52)

$$\mathrm{SEC} \mathrm{holds}⟺(d-2)\rho +dp⩾0,\rho +p⩾0.$$

(53)

In the present framework, whenever the Einstein equations (39) and (40) hold, we have14:

$$\rho ={\displaystyle \frac{1}{2{\gamma }_d{G}_d}}\left({\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{k}{a^2}}\right),\rho +p={\displaystyle \frac{1}{d{\gamma }_d{G}_d}}\left(-{\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}}+{\displaystyle \frac{{\stackrel{\circ }{a}}^2}{a^2}}+{\displaystyle \frac{k}{a^2}}\right),$$

(54)

$$(d-2)\rho +dp=-{\displaystyle \frac{1}{{\gamma }_d{G}_d}}{\displaystyle \frac{\stackrel{\circ \circ }{a}}{a}}.$$

From here, one easily infers that certain assumptions on the instantaneous values of $\stackrel{\circ }{a}$, $\stackrel{\circ \circ }{a}$, and on k are sufficient to fulfill or violate the above energy conditions. For example, at any time ${\tau }_0$, we have the following implications:

$$k<0,\stackrel{\circ }{a}\left({\tau }_0\right)=0⟹\rho \left({\tau }_0\right)<0⟹\mathrm{WEC} \mathrm{fails} \mathrm{at} {\tau }_0;$$

(55)

$$k⩽0,\stackrel{\circ }{a}\left({\tau }_0\right)=0,\stackrel{\circ \circ }{a}\left({\tau }_0\right)>0⟹(\rho +p)\left({\tau }_0\right)<0⟹\mathrm{WEC} \mathrm{and} \mathrm{SEC} \mathrm{both} \mathrm{fail} \mathrm{at} {\tau }_0;$$

(56)

$$\stackrel{\circ \circ }{a}\left({\tau }_0\right)>0⟹\left((d-2)\rho +dp\right)\left({\tau }_0\right)<0⟹\mathrm{SEC} \mathrm{fails} \mathrm{at} {\tau }_0.$$

(57)

Note that the above three violations refer, respectively, to a generic stationary point for $a\left(\right)$, a local minimum point for $a\left(\right)$ (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 $\mathcal{T}\ni \tau \mapsto a\left(\tau \right)$, depending on an assigned constant $H_{\ast }\in {\left({\mathbb{T}}^{-1}\right)}^{+}={\left({\mathbb{L}}^{-1}\right)}^{+}$ and also involving the Hubble parameter H of Equation (28):

$$\mathrm{There} \mathrm{is} \mathrm{time} {\tau }_{\ast } \mathrm{such} \mathrm{that} a \left({\tau }_{\ast }\right)=1 \mathrm{and} \stackrel{\circ }{a}\left({\tau }_{\ast }\right)=H_{\ast }(\mathrm{i}.\mathrm{e}.,H\left({\tau }_{\ast }\right)=H_{\ast }).$$

(58)

In cosmological models which are presumed to be anyhow realistic, the condition $a\left({\tau }_{\ast }\right)=1$ (and $\stackrel{\circ }{a}\left({\tau }_{\ast }\right)>0$) typically defines the present time; if so, the equality $H\left({\tau }_{\ast }\right)=H_{\ast }$ indicates that $H_{\ast }$ 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”):

$$H_{\ast }\simeq 2.18\times {10}^{-18}{s}^{-1}.$$

(59)

3.15. Dimensionless Formalism

In the present Section 3.15, we fix a constant

$$H_{\ast }\in {\left({\mathbb{T}}^{-1}\right)}^{+}={\left({\mathbb{L}}^{-1}\right)}^{+};$$

(60)

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 $H_{\ast }$ as the present-time Hubble parameter is not required for the moment (even though being adopted in many subsequent applications).

Hereafter, we use $H_{\ast }$ 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 $\tau$, we will employ the dimensionless time

$$t:=H_{\ast }\tau ;$$

(61)

if $\tau$ ranges in an open interval $\mathcal{T}\subset \mathbb{T}$, t ranges in the open interval

$$\mathcal{J}:=\left\{H_{\ast }\tau \right|\tau \in \mathcal{T}\}\subset \mathbb{R}.$$

(62)

Given any function $\tau \mapsto u\left(\tau \right)$ of cosmic time, we can associate to it a function $t\mapsto u\left(t\right)$ of dimensionless time, so that $u\left(\tau \right)={\left.u\left(t\right)\right|}_{t=H_{\ast }\tau }$ (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 $t\mapsto a\left(t\right)$, the scalar field $t\mapsto \Phi \left(t\right)$, the density $t\mapsto {\rho }_n\left(t\right)$, and so on. We will intend

$$\dot{X}:={\displaystyle \frac{d}{dt}};$$

(63)

with the previous notation $\stackrel{\circ }{X}$ for the derivative with respect to $\tau$, and for each smooth function u of time, we have of course $\stackrel{\circ }{u}=H_{\ast }\dot{u}$, $\stackrel{\circ \circ }{u}=H_{\ast }^2\ddot{u}$, 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

$$\mathcal{J}=(t_{-},t_{+}),-\infty ⩽t_{-}<t_{+}⩽+\infty ,$$

(64)

and let us choose any point $t_0\in \mathcal{J}$. The FLRW spacetime under consideration is:

• Past timelike complete, if and only if

  $$t_{-}=-\infty ,{\int }_{-\infty }^{t_0}\mathrm{d}t{\displaystyle \frac{a\left(t\right)}{\sqrt{1+a^2\left(t\right)}}}=+\infty ;$$

  (65)
• Past lightlike complete, if and only if

  $${\int }_{t_{-}}^{t_0}\mathrm{d}ta\left(t\right)=+\infty$$

  (66)

  (without requiring $t_{-}=-\infty$);
• Future timelike complete, if and only if

  $$t_{+}=+\infty ,{\int }_{t_0}^{+\infty }\mathrm{d}t{\displaystyle \frac{a\left(t\right)}{\sqrt{1+a^2\left(t\right)}}}=+\infty ;$$

  (67)
• Future lightlike complete, if and only if

  $${\int }_{t_0}^{t_{+}}\mathrm{d}ta\left(t\right)=+\infty$$

  (68)

  (without requiring $t_{+}=+\infty$).

All the above completeness conditions are fulfilled, e.g., if15

$$\mathcal{J}=(-\infty ,+\infty ),\underset{t\in \mathcal{J}}{inf}a\left(t\right)=a_0>0.$$

(69)

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

$$H=H_{\ast }h,h:={\displaystyle \frac{\dot{a}}{a}};$$

(70)

we refer to h as the dimensionless Hubble parameter.

Spatial curvature. The constant sectional curvature $k\in {\mathbb{L}}^{-2}$ of the spatial metric will be represented as

$$k=-H_{\ast }^2{\Omega }_{k\ast },{\Omega }_{k\ast }\in \mathbb{R}.$$

(71)

Note that $\mathrm{sign}k=-\mathrm{sign}{\Omega }_{k\ast }$; this is a standard convention, whose convenience will be clear in the sequel.

Curvature scalar. Equations (36) and (71) imply

$$R=H_{\ast }^2\left(2d{\displaystyle \frac{\ddot{a}}{a}}+d(d-1){\displaystyle \frac{{\dot{a}}^2}{a^2}}-d(d-1){\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}\right).$$

(72)

Scalar field. In Section 2.3, we indicated that the scalar field $\Phi$ takes values in a space $\mathbb{F}$ such that ${\mathbb{F}}^2=\mathbb{M}{\mathbb{L}}^{2-d}=\mathbb{D}{\mathbb{L}}^2$, and its potential is a function $\mathcal{V}:\mathbb{F}\to \mathbb{D}=\mathbb{M}{\mathbb{L}}^{-d}$. We now represent each $\Phi \in \mathbb{F}$ as

$$\Phi ={\displaystyle \frac{1}{\sqrt{2{\gamma }_d{G}_d}}}\varphi ,\varphi \in \mathbb{R},$$

(73)

and introduce a dimensionless field potential $V:\mathbb{R}\to \mathbb{R}$ such that

$$\mathcal{V}(\Phi )={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}V\left(\varphi \right)\mathrm{for} \mathrm{all}\Phi \in \mathbb{F}, \mathrm{with} \varphi \in \mathbb{R}\mathrm{as} \mathrm{in}\left(73\right).$$

(74)

Densities and pressures. For $n\in \{1,\dots ,N\}$, let us consider a pair ${\rho }_n\in {\mathbb{D}}^{+}$, $p_n\in \mathbb{D}$, representing possible values for the density and pressure of the n-th fluid. The corresponding, dimensionless density and pressure ${\mathfrak{r}}_n\in {\mathbb{R}}^{+}$, ${\mathfrak{p}}_n\in \mathbb{R}$ are defined by

$${\rho }_n={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}{\mathfrak{r}}_n,p_n={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}{\mathfrak{p}}_n.$$

(75)

Let us recall the equation of state (5) $p_n={\mathcal{P}}_n\left({\rho }_n\right)$, with ${\mathcal{P}}_n:{\mathbb{D}}^{+}\to \mathbb{D}$ smooth. It is readily seen that there is a unique smooth function $P_n:{\mathbb{R}}^{+}\to \mathbb{R}$ such that

$$p_n={\mathcal{P}}_n\left({\rho }_n\right)⟺{\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right)$$

(76)

for all ${\rho }_n\in {\mathbb{D}}^{+}$, ${\mathfrak{r}}_n\in {\mathbb{R}}^{+}$ and $p_n\in \mathbb{D}$, ${\mathfrak{p}}_n\in \mathbb{R}$ related as in (75).

To continue, let us consider a possible time dependence for the scalar field $\Phi$ and its dimensionless equivalent $\varphi$. Due to (32), (73), and (74) (and also due to the remark after Equation (63)), we have

$${\rho }_{\Phi }={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}{\mathfrak{r}}_{\varphi },p_{\Phi }={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}{\mathfrak{p}}_{\varphi },$$

(77)

where we have introduced the dimensionless density and pressure

$${\mathfrak{r}}_{\varphi }:={\displaystyle \frac{\sigma }{2}}{\dot{\varphi }}^2+V\left(\varphi \right),{\mathfrak{p}}_{\varphi }:={\displaystyle \frac{\sigma }{2}}{\dot{\varphi }}^2-V\left(\varphi \right).$$

(78)

Of course, the total density and pressure $\rho ,p$ of Equations (33) and (34) have dimensionless analogs $\mathfrak{r},\mathfrak{p}$; indeed,

$$\rho ={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}\mathfrak{r},p={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}\mathfrak{p},$$

(79)

$$\mathfrak{r}:=\sum _{n=1}^N{\mathfrak{r}}_n+{\mathfrak{r}}_{\varphi },\mathfrak{p}:=\sum _{n=1}^N{\mathfrak{p}}_n+{\mathfrak{p}}_{\varphi }.$$

(80)

On the use of ϕ and Φ as labels. In the sequel, for convenience, we will use the symbol $\varphi$ as a label equivalent to $\Phi$ for quantities related to the scalar field. In particular, the densities ${\rho }_{\Phi },{\Omega }_{\Phi }$ and the pressure $p_{\Phi }$ of Equations (32) and (47) will also be indicated with ${\rho }_{\varphi },{\Omega }_{\varphi }$ and $p_{\varphi }$. 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 ${\rho }_A=(H_{\ast }^2/2{\gamma }_d{G}_d){\mathfrak{r}}_A$, $p_A=(H_{\ast }^2/2{\gamma }_d{G}_d){\mathfrak{p}}_A$, holding true both for $A=n\in \{1,\dots ,N\}$ and for $A=\varphi$.

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:

$${\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}=\mathfrak{r},$$

(81)

$$-{\displaystyle \frac{2}{d}}\left({\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}\right)=\mathfrak{p},$$

(82)

$$\sigma \left(\ddot{\varphi }+d{\displaystyle \frac{\dot{a}}{a}}\dot{\varphi }\right)+V^{\prime }\left(\varphi \right)=0,$$

(83)

$${\dot{\mathfrak{r}}}_n+d{\displaystyle \frac{\dot{a}}{a}}({\mathfrak{p}}_n+{\mathfrak{r}}_n)=0,$$

(84)

with $\mathfrak{r}={\sum }_{n=1}^N{\mathfrak{r}}_n+{\mathfrak{r}}_{\varphi }$, $\mathfrak{p}={\sum }_{n=1}^N{\mathfrak{p}}_n+{\mathfrak{p}}_{\varphi }$ as in (80), ${\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right)$ (see (76)), and ${\mathfrak{r}}_{\varphi },{\mathfrak{p}}_{\varphi }$ as in (78). In the dimensionless setting, we can regard this as a system of ODEs for the unknown smooth functions $\mathcal{J}\ni t$$\mapsto a\left(t\right)\in {\mathbb{R}}^{+}$, $\varphi \left(t\right)\in \mathbb{R}$, ${\mathfrak{r}}_n\left(t\right)\in {\mathbb{R}}^{+}$ ($n=1,\dots ,N$), with $\mathcal{J}\subset \mathbb{R}$ an open interval.

Total density, including the curvature contribution; normalized densities. Due to (44) and (71), the curvature density can be represented as

$${\rho }_k={\displaystyle \frac{H_{\ast }^2}{2{\gamma }_d{G}_d}}{\mathfrak{r}}_k,$$

(85)

where we have introduced the dimensionless curvature density as

$${\mathfrak{r}}_k:={\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}.$$

(86)

The first Einstein Equation (81) means

$$h^2=\tilde{\mathfrak{r}},$$

(87)

where $h:=\dot{a}/a$ is the dimensionless Hubble parameter (see Equation (70)), and we have introduced the total dimensionless density including the curvature contribution, which is by definition

$$\tilde{\mathfrak{r}}:=\mathfrak{r}+{\mathfrak{r}}_k=\sum _{n=1}^N{\mathfrak{r}}_n+{\mathfrak{r}}_k+{\mathfrak{r}}_{\varphi }.$$

(88)

To continue, we note that the definition (47) of the normalized densities is equivalent to

$${\Omega }_n={\displaystyle \frac{{\mathfrak{r}}_n}{\tilde{\mathfrak{r}}}}(n=1,\dots ,N),{\Omega }_k={\displaystyle \frac{{\mathfrak{r}}_k}{\tilde{\mathfrak{r}}}},{\Omega }_{\varphi }={\displaystyle \frac{{\mathfrak{r}}_{\varphi }}{\tilde{\mathfrak{r}}}}$$

(89)

(intending ${\Omega }_{\varphi }$ as an equivalent notation for ${\Omega }_{\Phi }$; see the paragraph after Equation (80)).

If Equation (87) (i.e., Equation (81)) is satisfied, we can also write

$${\Omega }_n={\displaystyle \frac{{\mathfrak{r}}_n}{h^2}}(n=1,\dots ,N),{\Omega }_k={\displaystyle \frac{{\mathfrak{r}}_k}{h^2}}={\displaystyle \frac{{\Omega }_{k\ast }}{{\dot{a}}^2}},{\Omega }_{\varphi }={\displaystyle \frac{{\mathfrak{r}}_{\varphi }}{h^2}}={\displaystyle \frac{1}{h^2}}\left[{\displaystyle \frac{\sigma }{2}}{\dot{\varphi }}^2+V\left(\varphi \right)\right].$$

(90)

Again on stationary points of the scale factor. The dimensionless analogs of statements (50) and (51) are as follows:

$$\dot{a}\left(t_0\right)=0⟺0=\tilde{\mathfrak{r}}\left(t_0\right)=\left(\sum _{n=1}^N{\mathfrak{r}}_n+{\mathfrak{r}}_k+{\mathfrak{r}}_{\varphi }\right)\left(t_0\right);$$

(91)

$${\Omega }_{k\ast }⩾0(\mathrm{i}.\mathrm{e}.,k⩽0)\mathrm{and}\dot{a}\left(t_0\right)=0⟹{\mathfrak{r}}_{\varphi }\left(t_0\right)<0$$

(92)

(with ${\mathfrak{r}}_{\varphi }=(\sigma /2){\dot{\varphi }}^2+V\left(\varphi \right)$, which is more likely to become negative in the phantom case $\sigma =-1$).

Again on the energy conditions. The dimensionless reformulation of the contents of Section 3.13 is straightforward. First of all,

$$\mathrm{WEC} \mathrm{holds}⟺\mathfrak{r}⩾0,\mathfrak{r}+\mathfrak{p}⩾0;$$

(93)

$$\mathrm{SEC} \mathrm{holds}⟺(d-2)\mathfrak{r}+d\mathfrak{p}⩾0,\mathfrak{r}+\mathfrak{p}⩾0.$$

(94)

Whenever the Einstein equations (81) and (82) hold, we have

$$\mathfrak{r}={\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}},\mathfrak{r}+\mathfrak{p}={\displaystyle \frac{2}{d}}\left(-{\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}\right),(d-2)\mathfrak{r}+d\mathfrak{p}=-2{\displaystyle \frac{\ddot{a}}{a}}.$$

(95)

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 $t_0$,

$${\Omega }_{k\ast }>0,\dot{a}\left(t_0\right)=0⟹\mathfrak{r}\left(t_0\right)<0⟹\mathrm{WEC} \mathrm{fails} at t_0;$$

(96)

$${\Omega }_{k\ast }⩾0,\dot{a}\left(t_0\right)=0,\ddot{a}\left(t_0\right)>0⟹(\mathfrak{r}+\mathfrak{p})\left(t_0\right)<0⟹\mathrm{WEC} \mathrm{and} \mathrm{SEC} \mathrm{both} \mathrm{fail} \mathrm{at} t_0;$$

(97)

$$\ddot{a}\left(t_0\right)>0⟹\left((d-2)\mathfrak{r}+d\mathfrak{p}\right)\left(t_0\right)<0⟹\mathrm{SEC} \mathrm{fails} \mathrm{at} t_0.$$

(98)

Again on the scale factor at a special time. Given $H_{\ast }$ as in (60), let us reconsider condition (58) on the scale factor. Clearly, the dimensionless analog of (58) is the following:

$$\mathrm{There} \mathrm{is} a \mathrm{time} t_{\ast } \mathrm{such} \mathrm{that} a \left(t_{\ast }\right)=1 \mathrm{and} \dot{a}\left(t_{\ast }\right)=1(\mathrm{i}.\mathrm{e}.,h\left(t_{\ast }\right)=1).$$

(99)

In agreement with the considerations after Equation (58), in realistic models, one often regards $t_{\ast }$ 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 $H_{\ast }$ as in (60)). The present Section 4 presents some basic results about Equations (81)–(84).

4.1. Determining the Fluids’ Densities

Let us choose $n\in \{1,\dots ,N\}$ and consider Equation (84) ${\dot{\mathfrak{r}}}_n+d{\displaystyle \frac{\dot{a}}{a}}({\mathfrak{p}}_n+{\mathfrak{r}}_n)=0$, with ${\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right)$. We make the ansatz ${\mathfrak{r}}_n\left(t\right)=F_n\left(a^{-d}\left(t\right)\right)$, with an unknown smooth function $F_n$ of a positive variable $\alpha$. It is readily seen that ${\mathfrak{r}}_n$ fulfills Equation (84) if $F_n^{\prime }\left(\alpha \right)=[P_n\left(F_n\left(\alpha \right)\right)+F_n\left(\alpha \right)]/\alpha$.

To formalize the above considerations, we introduce the function space

$${\mathbf{F}}_n:=\{F_n\in C^{\infty }({\mathbb{R}}^{+},{\mathbb{R}}^{+})|F_n^{\prime }\left(\alpha \right)={\displaystyle \frac{P_n\left(F_n\left(\alpha \right)\right)+F_n\left(\alpha \right)}{\alpha }}\mathrm{for} \mathrm{all} \alpha \in {\mathbb{R}}^{+}\}.$$

(100)

Moreover, we consider the following condition, discussed in the next paragraph and often assumed to hold in the sequel:

$$\mathrm{For} \mathrm{each} {\alpha }_0\in {\mathbb{R}}^{+} \mathrm{and} C_{n0}\in {\mathbb{R}}^{+},\mathrm{there} \mathrm{exists} F_n\in {\mathbf{F}}_n\mathrm{such} \mathrm{that} F_n\left({\alpha }_0\right)=C_{n0}$$

(101)

(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 $a:\mathcal{J}\to {\mathbb{R}}^{+}$, $t\mapsto a\left(t\right)$ (with $\mathcal{J}\subset \mathbb{R}$ an open interval). In Appendix G, we prove rigorously that, if (101) holds, for each smooth function ${\mathfrak{r}}_n:\mathcal{J}\to {\mathbb{R}}^{+}$, $t\mapsto {\mathfrak{r}}_n\left(t\right)$, we have the following equivalence:

$${\mathfrak{r}}_n \mathrm{fulfills}\left(84\right)(\mathrm{with} {\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right))⟺{\mathfrak{r}}_n=F_n\left(a^{-d}\right)\mathrm{for} \mathrm{some} F_n\in {\mathbf{F}}_n.$$

(102)

To continue, assume the first Einstein Equation (81) (or (87)) to be fulfilled by certain functions $\mathcal{J}\ni t\mapsto a\left(t\right),\varphi \left(t\right),{\mathfrak{r}}_n\left(t\right)$ ($n=1,\dots ,N$), and Equations (84) and (101) to hold at least for some n; then, from the relation ${\mathfrak{r}}_n=F_n\left(a^{-d}\right)$ ($F_n\in {\mathbf{F}}_n$) and from (90), we infer that the n-th normalized density is

$${\Omega }_n={\displaystyle \frac{F_n\left(a^{-d}\right)}{h^2}},$$

(103)

with h the dimensionless Hubble parameter; see (70).

On the condition (101). Let us again choose $n\in \{1,\dots ,N\}$. By the standard theory of the Cauchy problem, for any ${\alpha }_0,C_{n0}\in {\mathbb{R}}^{+}$, there is certainly a function $F_n\in C^{\infty }(\mathcal{A},{\mathbb{R}}^{+}$) (with $\mathcal{A}\subset {\mathbb{R}}^{+}$ an open interval) such that $F_n^{\prime }\left(\alpha \right)=(P_n\left(F_n\left(\alpha \right)\right)+F_n\left(\alpha \right))/\alpha$ for all $\alpha \in \mathcal{A}$ and ${\alpha }_0\in \mathcal{A}$, $F_n\left({\alpha }_0\right)=C_{n0}$. Equation (101) requires this to happen with $\mathcal{A}={\mathbb{R}}^{+}$ (the domain of the functions in the space ${\mathbf{F}}_n$) 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

$$P_n\left(r\right)+r\ne 0\mathrm{for} \mathrm{all} r\in {\mathbb{R}}^{+},$$

(104)

$${\int }_0^C{\displaystyle \frac{\mathrm{d}r}{|P_n\left(r\right)+r|}}=+\infty \mathrm{and}{\int }_C^{+\infty }{\displaystyle \frac{\mathrm{d}r}{|P_n\left(r\right)+r|}}=+\infty \mathrm{for} \mathrm{all} C\in {\mathbb{R}}^{+}.$$

In this case, for each ${\alpha }_0,C_{n0}\in {\mathbb{R}}^{+}$, the function $F_n$ of (101) is defined implicitly by the following quadrature formula:

$${\int }_{C_{n0}}^{F_n\left(\alpha \right)}{\displaystyle \frac{\mathrm{d}r}{P_n\left(r\right)+r}}=ln{\displaystyle \frac{\alpha }{{\alpha }_0}}\mathrm{for} \mathrm{all} \alpha \in {\mathbb{R}}^{+}.$$

(105)

(This formula actually individuates a unique smooth function $F_n:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$; see again Appendix G).

(ii)

It is

$$P_n\left(r\right)+r=0\mathrm{for} \mathrm{all} r\in {\mathbb{R}}^{+}.$$

(106)

In this trivial case, Equation (100) gives

$${\mathbf{F}}_n=\{F_n:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}|F_n=\mathrm{constant}\}$$

(107)

and the function $F_n$ in (101) is given by

$$F_n=\mathrm{constant}=C_{n0}.$$

(108)

Again on the fluids’ densities. Let us assume (101) for some n. By the uniqueness theorem for the Cauchy problem, for any fixed ${\alpha }_0\in {\mathbb{R}}^{+}$, the mapping $F_n\mapsto C_{n0}:=F_n\left({\alpha }_0\right)$ is a bijection between the function space ${\mathbf{F}}_n$ of Equation (100) and ${\mathbb{R}}^{+}$. In particular, let us choose ${\alpha }_0=1$ and consider the bijection

$${\mathbf{F}}_n\to {\mathbb{R}}^{+},F_n\mapsto {\Omega }_{n\ast }:=F_n\left(1\right)$$

(109)

(here we write ${\Omega }_{n\ast }$ instead of $C_{n0}$, for reasons that will become clear shortly afterward). In case (i) of the previous paragraph, the inverse of the map (109) sends ${\Omega }_{n\ast }\in {\mathbb{R}}^{+}$ to the function $F_n\in {\mathbf{F}}_n$ described by Equation (105) with ${\alpha }_0=1$ and $C_{n0}={\Omega }_{n\ast }$, i.e.,

$${\int }_{{\Omega }_{n\ast }}^{F_n\left(\alpha \right)}{\displaystyle \frac{\mathrm{d}r}{P_n\left(r\right)+r}}=ln\alpha \mathrm{for} \mathrm{all} \alpha \in {\mathbb{R}}^{+}.$$

(110)

To continue, assume the first Einstein equation (81) (or (87)) to be fulfilled by certain functions $\mathcal{J}\ni t\mapsto a\left(t\right),\varphi \left(t\right),{\mathfrak{r}}_n\left(t\right)$ ($n=1,\dots ,N$), 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

$${\Omega }_n\left(t_{\ast }\right)=F_n\left(1\right)\equiv {\Omega }_{n\ast }\mathrm{if} a \left(t_{\ast }\right)=1, h\left(t_{\ast }\right)=1 \mathrm{for} \mathrm{some} t_{\ast }$$

(111)

(here, we consider condition (99) about $t_{\ast }$, 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:

$$p_n=w_n{\rho }_n,\mathrm{i}.\mathrm{e}.,{\mathcal{P}}_n\left(\rho \right)=w_n\rho \mathrm{for} \mathrm{all} \rho \in {\mathbb{D}}^{+},$$

(112)

with $w_n\in \mathbb{R}$ 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:

$${\mathfrak{p}}_n=w_n{\mathfrak{r}}_n,\mathrm{i}.\mathrm{e}.,P_n\left(\mathfrak{r}\right)=w_n\mathfrak{r}\mathrm{for} \mathrm{all} \mathfrak{r}\in {\mathbb{R}}^{+}.$$

(113)

It should be noted that the linear case (112) and (113) fits items (i) and (ii) of the penultimate paragraph for $w_n\ne -1$ and $w_n=-1$, respectively. For any $w_n\in \mathbb{R}$, the space of Equation (100) is given by

$${\mathbf{F}}_n=\{F_n:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}|F_n\left(\alpha \right)={\Omega }_{n\ast }{\alpha }^{1+w_n}\mathrm{for} \mathrm{all}\alpha \in {\mathbb{R}}^{+}, \mathrm{with}{\Omega }_{n\ast }\in {\mathbb{R}}^{+}\};$$

(114)

note that the above representation is consistent with the relation (109) $F_n\left(1\right)={\Omega }_{n\ast }$. 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

$${\mathfrak{r}}_n={\displaystyle \frac{{\Omega }_{n\ast }}{a^{(1+w_n)d}}},{\mathfrak{p}}_n={\displaystyle \frac{w_n{\Omega }_{n\ast }}{a^{(1+w_n)d}}}({\Omega }_{n\ast }\in {\mathbb{R}}^{+}).$$

(115)

If the Einstein equation (81) is fulfilled, Equations (90) and (115) give for the n-th normalized density the following expression:

$${\Omega }_n={\displaystyle \frac{1}{h^2}}{\displaystyle \frac{{\Omega }_{n\ast }}{a^{(1+w_n)d}}}.$$

(116)

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 $n=1,\dots ,N$; 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 ${\mathfrak{r}}_n$ and ${\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right)$ coming from (102), which involve a function $F_n$ in the space (100); moreover, we use the explicit expressions (78) for ${\mathfrak{r}}_{\varphi },{\mathfrak{p}}_{\varphi }$. In this way, Equations (81) and (82) are converted to

$$\mathfrak{E}=0,\mathfrak{E}:=\left({\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}\right)-\left(\sum _{n=1}^N{F}_n\left(a^{-d}\right)+{\displaystyle \frac{\sigma }{2}}{\dot{\varphi }}^2+V\left(\varphi \right)\right),$$

(117)

$$\mathfrak{A}=0,$$

(118)

$$\mathfrak{A}:=-{\displaystyle \frac{2}{d}}\left({\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\dot{a}}^2}{a^2}}-{\displaystyle \frac{d-2}{2}}{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}\right)-\left(\sum _{n=1}^N(P_n\circ F_n)\left(a^{-d}\right)+{\displaystyle \frac{\sigma }{2}}{\dot{\varphi }}^2-V\left(\varphi \right)\right).$$

For future reference, we also report the Klein–Gordon Equation (83) that we rephrase as

$$\mathfrak{F}=0,\mathfrak{F}:=\sigma \left(\ddot{\varphi }+d{\displaystyle \frac{\dot{a}}{a}}\dot{\varphi }\right)+V^{\prime }\left(\varphi \right).$$

(119)

Equations (117)–(119) form a system in two unknown smooth functions $\mathcal{J}\ni t\mapsto a\left(t\right)\in {\mathbb{R}}^{+},\varphi \left(t\right)\in \mathbb{R}$; here and in the sequel, $\mathcal{J}$ will always indicate an unspecified, open real interval.

The equations in this system are not fully independent. In fact, if $\mathfrak{A}=0$ and $\mathfrak{F}=0$ at all times, for $\mathfrak{E}=0$ to be fulfilled at all times, it is sufficient that $\mathfrak{E}=0$ 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 $\mathcal{S}$ of Equation (6). In the present framework, based on the spacetime (15) with the metric (17), one has $\mathrm{d}v=a^{d}d\tau d\omega$, where $\mathrm{d}\omega$ is the volume element of the Riemannian metric h; in terms of the dimensionless time (61), ranging in a real interval $\mathcal{J}$, we have $\mathrm{d}\tau =\mathrm{d}t/H_{\ast }$. We also insert in Equation (6) the expression  (72) for R, the expression for ${\rho }_n$ arising from Equations (75) and (102) and the expressions (73) and (74) for $\Phi$ and $\mathcal{V}(\Phi )$. In this way, we obtain the following:

$$\mathcal{S}={\displaystyle \frac{H_{\ast }}{2{\gamma }_{\mathrm{d}}{G}_d}}\left({\int }_{{\mathfrak{S}}^d}d\omega \right)\times$$

(120)

$$\times {\int }_{\mathcal{J}}\mathrm{d}t\left[{\displaystyle \frac{2}{(d-1)}}{a}^{d-1}\ddot{a}+a^{d-2}{\dot{a}}^2-{\Omega }_{k\ast }{a}^{d-2}-\sum _{n=1}^N{a}^d{F}_n\left(a^{-d}\right)+{\displaystyle \frac{\sigma }{2}}{a}^d{\dot{\varphi }}^2-a^{d}V\left(\varphi \right)\right]=$$

$$=-{\displaystyle \frac{H_{\ast }}{2{\gamma }_d{G}_d}}\left({\int }_{{\mathfrak{S}}^d}\mathrm{d}\omega \right)\times {\int }_{\mathcal{J}}\mathrm{d}t\left[L(a,\varphi ,\dot{a},\dot{\varphi })-{\displaystyle \frac{2}{(d-1)}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(a^{d-1}\dot{a}\right)\right],$$

where

$$L:{\mathbb{R}}^{+}\times {\mathbb{R}}^3\to \mathbb{R},(a,\varphi ,\dot{a},\dot{\varphi })\mapsto L(a,\varphi ,\dot{a},\dot{\varphi }):=a^{d-2}{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^d{\dot{\varphi }}^2+U\left(a\right)+a^{d}V\left(\varphi \right),$$

(121)

$$U:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto U\left(a\right):=\sum _{n=1}^N{a}^d{F}_n\left(a^{-d}\right)+{\Omega }_{k\ast }{a}^{d-2}.$$

(122)

The above manipulations are to some extent formal, since the volume $\left({\int }_{{\mathfrak{S}}^d}d\omega \right)$ can be infinite (this certainly happens if ${\mathfrak{S}}^d$ is simply connected and it has curvature $k⩽0$). Forgetting this problem (and recalling that total derivatives like $(\mathrm{d}/\mathrm{d}t)\left(a^{d-1}\dot{a}\right)$ 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:

$${\displaystyle \frac{\delta L}{\delta a}}=0,{\displaystyle \frac{\delta L}{\delta \varphi }}=0,{\displaystyle \frac{\delta L}{\delta q}}:=-{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{q}}}\right)+{\displaystyle \frac{\partial L}{\partial q}}\mathrm{for} q=a,\varphi .$$

(123)

It is readily found that

$${\displaystyle \frac{\delta L}{\delta \varphi }}=a^d\mathfrak{F},$$

(124)

with $\mathfrak{F}$ as in Equation (119). The calculation of $\delta L/\delta a$ is a bit more engaging and involves amongst else the derivative $(\partial /\partial a)\left[F_n\left(a^{-d}\right)\right]$, to be computed recalling the expression for $F_n^{\prime }$ in (100); one ultimately obtains

$${\displaystyle \frac{\delta L}{\delta a}}=d{a}^{d-1}\mathfrak{A},$$

(125)

with $\mathfrak{A}$ as in Equation (118). The energy function $E:{\mathbb{R}}^{+}\times {\mathbb{R}}^3\to \mathbb{R}$ associated with the Lagrangian (121) is

$$E(a,\varphi ,\dot{a},\dot{\varphi }):=\left({\displaystyle \frac{\partial L}{\partial \dot{a}}}\dot{a}+{\displaystyle \frac{\partial L}{\partial \dot{\varphi }}}\dot{\varphi }-L\right)(a,\varphi ,\dot{a},\dot{\varphi })=a^{d-2}{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^d{\dot{\varphi }}^2-U\left(a\right)-a^{d}V\left(\varphi \right).$$

(126)

Of course, along any solution $t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the Lagrange equations, we have $E\left(t\right)=$ constant; in particular, $E\left(t\right)=0$ at all times if and only if $E\left(t_0\right)=0$ at some particular time $t_0$. On the other hand, it is evident that

$$E=a^d\mathfrak{E},$$

(127)

with $\mathfrak{E}$ an in Equation (117). To summarize, we should highlight the following:

(i)

The Einstein equation $\mathfrak{A}=0$ and the Klein–Gordon equation $\mathfrak{F}=0$ are equivalent to the Lagrange equations induced by L.

(ii)

The Einstein equation $\mathfrak{E}=0$ 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 ${\mathfrak{p}}_n=P_n\left({\mathfrak{r}}_n\right)$ 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 $w_n\in \mathbb{R}$, functions $F_n\in {\mathbf{F}}_n$ have the form (114) depending on a constant ${\Omega }_{n\ast }\in {\mathbb{R}}^{+}$; so, in Equation (122), for U, one must put

$$F_n\left(a^{-d}\right)={\displaystyle \frac{{\Omega }_{n\ast }}{a^{(1+w_n)d}}}.$$

(128)

4.5. Again on the Scale Factor at a Special Time

We have already considered for the scale factor $t\mapsto a\left(t\right)$ the condition $a\left(t_{\ast }\right)=1$, $\dot{a}\left(t_{\ast }\right)=1$ (i.e., $h\left(t_{\ast }\right)=1$), to be fulfilled at some special time $t_{\ast }$: 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 $\mathcal{J}\ni t\mapsto (a\left(t\right),\varphi \left(t\right))\in {\mathbb{R}}^{+}\times \mathbb{R}$ of the Lagrange equations induced by L, such that $a\left(t_{\ast }\right)=1$, $\dot{a}\left(t_{\ast }\right)=1$ at some time $t_{\ast }\in \mathcal{J}$.

(ii)

There are ${\varphi }_{\ast },{\dot{\varphi }}_{\ast }\in \mathbb{R}$ such that

$${\displaystyle \sum _{n=1}^N{\Omega }_{n\ast }+{\Omega }_{k\ast }+\frac{\sigma }{2}{\dot{\varphi }}_{\ast }^2+V\left({\varphi }_{\ast }\right)=1,\mathrm{where}{\Omega }_{n\ast }:=F_n\left(1\right).}$$

(129)

(Concerning the position ${\Omega }_{n\ast }:=F_n\left(1\right)$, recall Equation (109) and the subsequent discussion).

Let us first prove that (i) implies (ii). In fact, assume there is a zero-energy solution $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the Lagrange equations as in (i), and put ${\varphi }_{\ast }:=\varphi \left(t_{\ast }\right)$, ${\dot{\varphi }}_{\ast }:=\dot{\varphi }\left(t_{\ast }\right)$. Then, recalling Equations (126) and (122), we can write $0=E(a\left(t_{\ast }\right),\varphi \left(t_{\ast }\right),\dot{a}\left(t_{\ast }\right),\dot{\varphi }\left(t_{\ast }\right))$$=E(1,{\varphi }_{\ast },1,{\dot{\varphi }}_{\ast })$ $=1-(\sigma /2){\dot{\varphi }}_{\ast }^2-U\left(1\right)-V\left({\varphi }_{\ast }\right)$ $=1-(\sigma /2){\dot{\varphi }}_{\ast }^2-{\sum }_{n=1}^N{F}_n\left(1\right)-{\Omega }_{k\ast }-V\left({\varphi }_{\ast }\right)$, whence Equation (129) of (ii).

Conversely, let us assume (ii) and infer (i). For this purpose, we arbitrarily choose $t_{\ast }\in \mathbb{R}$; let $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ be the solution17 of the Lagrange equations with initial data $a\left(t_{\ast }\right)=1$, $\varphi \left(t_{\ast }\right)={\varphi }_{\ast }$, $\dot{a}\left(t_{\ast }\right)=1$, $\dot{\varphi }\left(t_{\ast }\right)={\dot{\varphi }}_{\ast }$. Clearly, (i) is true if we prove this to be a zero-energy solution. Indeed, from the assumption (129), we readily infer $E(a\left(t_{\ast }\right),\varphi \left(t_{\ast }\right),\dot{a}\left(t_{\ast }\right),\dot{\varphi }\left(t_{\ast }\right))=0$, whence $E(a\left(t\right),\varphi \left(t\right),\dot{a}\left(t\right),\dot{\varphi }\left(t\right))=0$ at each time $t\in \mathcal{J}$.

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 ${\mathbb{R}}^m$, 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 $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$$\in {\mathbb{R}}^{+}\times \mathbb{R}$ of the Lagrange equations in Section 4.4. Another typical application will concern the maximal, zero-energy solutions $\mathcal{J}\ni t\mapsto a\left(t\right)\in {\mathbb{R}}^{+}$ of the Lagrange equations induced by a certain “reduced” Lagrangian $\mathcal{L}(a,\dot{a})$, which is introduced in Section 5.3.

5. The Case with a Constant Potential for the Scalar Field: General Results on the Evolution of $a\left(t\right)$ and $\varphi \left(t\right)$

5.1. Basic Setting

Throughout the present Section 5, we assume the following for the dimensionless field potential:

$$V\left(\varphi \right)=\mathrm{constant}\equiv {{\rm Y}}_{\ast }\in \mathbb{R}.$$

(130)

Due to Equation (74), the corresponding dimensioned potential is

$$\mathcal{V}(\Phi )=\mathrm{constant}={\displaystyle \frac{H_{\ast }^2{{\rm Y}}_{\ast }}{2{\gamma }_d{G}_d}}.$$

(131)

According to Equations (12)–(14), this setting is equivalent to considering the Einstein equations with a cosmological constant as follows:

$$\Lambda ={\displaystyle \frac{d(d-1)H_{\ast }^2{{\rm Y}}_{\ast }}{2}}$$

(132)

(having the sign of ${{\rm Y}}_{\ast }$), and a free, massless scalar field.

Due to Equation (130), the Lagrangian (121) and its energy function (126) take the following form:

$$L(a,\varphi ,\dot{a},\dot{\varphi }):=a^{d-2}{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^d{\dot{\varphi }}^2+U\left(a\right)+{{\rm Y}}_{\ast }{a}^d,$$

(133)

$$E(a,\varphi ,\dot{a},\dot{\varphi }):=a^{d-2}{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^d{\dot{\varphi }}^2-U\left(a\right)-{{\rm Y}}_{\ast }{a}^d,$$

(134)

with $U\left(a\right):={\sum }_{n=1}^N{a}^d{F}_n\left(a^{-d}\right)+{\Omega }_{k\ast }{a}^{d-2}$, as in (122).

Let us recall again that a linear equation of state $p_n=w_n{\rho }_n$ for the n-th fluid ($w_n\in \mathbb{R}$) implies $F_n\left(\alpha \right)={\Omega }_{n\ast }{\alpha }^{1+w_n}$ with ${\Omega }_{n\ast }\in {\mathbb{R}}^{+}$, 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 $\Pi _{\ast }$

Clearly, the coordinate $\varphi$ is cyclic for the Lagrangian (133): $\partial L/\partial \varphi =0$. Due to this, the canonically conjugate momentum

$$\Pi _{\ast }:={\displaystyle \frac{\partial L}{\partial \dot{\varphi }}}=-\sigma a^d\dot{\varphi }$$

(135)

is constant along any solution of the Lagrange equations. The relation between $\dot{\varphi }$ and $\Pi _{\ast }$ in Equation (135) can be inverted, thus obtaining

$$\dot{\varphi }=-\sigma \Pi _{\ast }{a}^{-d}.$$

(136)

5.3. Reduced Lagrangian

Let us consider a solution $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the Lagrange equations with an assigned value of $\Pi _{\ast }$ (recalling that $\mathcal{J}$ indicates any open real interval). By the standard theory of Lagrangian systems with cyclic coordinates, the function $\mathcal{J}\ni t\mapsto a\left(t\right)$ can be characterized as a solution of the Lagrange equation induced by the reduced Lagrangian

$$\mathtt{L}:{\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R},(a,\dot{a})\mapsto \mathtt{L}(a,\dot{a}):={\left[L(a,\dot{a},\dot{\varphi })-\Pi _{\ast }\dot{\varphi }\right]}_{\dot{\varphi }=-\sigma \Pi _{\ast }{a}^{-d}}=a^{d-2}{\dot{a}}^2-\mathtt{V}\left(a\right),$$

(137)

$$\mathtt{V}\left(a\right):=-{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}{a}^{-d}-U\left(a\right)-{{\rm Y}}_{\ast }{a}^d=-{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}{a}^{-d}-\sum _{n=1}^N{a}^d{F}_n\left(a^{-d}\right)-{\Omega }_{k\ast }{a}^{d-2}-{{\rm Y}}_{\ast }{a}^d$$

(in the last passage, we have expressed U via Equation (122)). The energy function associated with $\mathtt{L}$ is

$$\mathtt{E}(a,\dot{a}):=\left({\displaystyle \frac{\partial \mathtt{L}}{\partial \dot{a}}}\dot{a}-\mathtt{L}\right)(a,\dot{a})=a^{d-2}{\dot{a}}^2+\mathtt{V}\left(a\right).$$

(138)

By comparison with the energy function E of Equation (134), we see that

$$\mathtt{E}(a,\dot{a})={\left[E(a,\varphi ,\dot{a},\dot{\varphi })\right]}_{\dot{\varphi }=-\sigma \Pi _{\ast }{a}^{-d}};$$

(139)

so, the energy constraint $E=0$ of Section 4.4 is equivalent to an energy constraint $\mathtt{E}=0$ on the motion $t\mapsto a\left(t\right)$.

To continue, let us note that we can write

$$\mathtt{L}(a,\dot{a})=a^{d-2}\mathcal{L}(a,\dot{a}),\mathtt{E}(a,\dot{a})=a^{d-2}\mathcal{E}(a,\dot{a})$$

(140)

where we have put

$$\mathcal{L}(a,\dot{a}):={\dot{a}}^2-\mathcal{V}\left(a\right),\mathcal{E}(a,\dot{a}):={\dot{a}}^2+\mathcal{V}\left(a\right),$$

(141)

$$\mathcal{V}\left(a\right):=a^{2-d}\mathtt{V}\left(a\right)=-{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}{a}^{2-2d}-\sum _{n=1}^N{a}^2{F}_n\left(a^{-d}\right)-{\Omega }_{k\ast }-{{\rm Y}}_{\ast }{a}^2.$$

(142)

Of course, $\mathcal{L}$ is a Lagrangian with energy function $\mathcal{E}$; the corresponding Lagrange equation reads as follows:

$$2\ddot{a}\left(t\right)=-{\mathcal{V}}^{\prime }\left(a\left(t\right)\right).$$

(143)

It turns out that the zero-energy solutions of the Lagrange equations for $\mathtt{L}$ and $\mathcal{L}$ coincide (this reflects a general result; see Appendix H); so, in the sequel, we will refer to the simplest Lagrangian $\mathcal{L}$.

5.4. Zero-Energy Solutions of the Reduced System

Clearly, $\mathcal{L}$ and $\mathcal{E}$ in Equation (141) are the Lagrangian and the energy function of a fictitious one-dimensional, conservative mechanical system with kinetic energy ${\dot{a}}^2$ and potential energy $\mathcal{V}\left(a\right)$. 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 $\mathcal{J}\ni t\mapsto a\left(t\right)$ of the Lagrange equations fulfilling at all times the constraint

$$0=\mathcal{E}\left(t\right)={\dot{a}}^2\left(t\right)+\mathcal{V}\left(a\left(t\right)\right),$$

(144)

which leads us to the following statements:

(i)

For all t in the (open, real) interval $\mathcal{J}$, one has $\mathcal{V}\left(a\left(t\right)\right)=-{\dot{a}}^2\left(t\right)⩽0$. So, the image of the function $t\mapsto a\left(t\right)$ is an interval contained in the following set:

$${\mathcal{V}}_{⩽0}:=\{a\in {\mathbb{R}}^{+}|\mathcal{V}\left(a\right)⩽0\}.$$

(145)

Of course, this is the union of the subsets

$${\mathcal{V}}_{=0}:=\{a\in {\mathbb{R}}^{+}|\mathcal{V}\left(a\right)=0\},{\mathcal{V}}_{<0}:=\{a\in {\mathbb{R}}^{+}|\mathcal{V}\left(a\right)<0\},$$

(146)

which must be distinguished for a qualitative analysis of the solution.

(ii)

The equations $\mathcal{V}\left(a\left(t\right)\right)=-{\dot{a}}^2\left(t\right)$ and $2\ddot{a}\left(t\right)=-{\mathcal{V}}^{\prime }\left(a\left(t\right)\right)$ have well known implications. In particular, if $t_1\in \mathcal{J}$ is an instant, one has $\dot{a}\left(t_1\right)=0$ if and only if $\mathcal{V}\left(a\left(t_1\right)\right)=0$; if, in addition, ${\mathcal{V}}^{\prime }\left(a\left(t_1\right)\right)\ne 0$, then $t_1$ is an inversion time, i.e., $\dot{a}\left(t\right)$ is nonzero with different signs when t ranges in two suitable intervals $(t_1-\delta ,t_1)$ and $(t_1,t_1+\delta )$ (this is readily inferred from the relations $\dot{a}\left(t_1\right)=0$ and $\ddot{a}\left(t_1\right)=-(1/2){\mathcal{V}}^{\prime }\left(a\left(t_1\right)\right)\ne 0$).

(iii)

If $(t^{\prime },t^{″})\subset \mathcal{J}$ is an interval such that $\mathrm{sign}\dot{a}\left(t\right)$ = constant $\equiv s_a\in \{\pm 1\}$ for all $t\in (t^{\prime },t^{″})$, everywhere in this interval, we have

$$\dot{a}\left(t\right)=s_a\sqrt{-\mathcal{V}\left(a\right(t\left)\right)}$$

(147)

so that, by separation of variables,

$${\int }_{a\left(t^{\prime }\right)}^{a\left(t^{″}\right)}{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}}=s_a(t^{″}-t^{\prime }).$$

(148)

(If $t^{\prime }$ or $t^{″}$ is an endpoint of $\mathcal{J}$, in the above formula, $a\left(t^{\prime }\right)$ or $a\left(t^{″}\right)$ should be intended as the limit of $a\left(t\right)$ for $t\to t^{\prime }$ or $t\to t^{″}$).

(iv)

The considerations in (iii) bring to our attention integrals of the following form:

$${\int }_{a^{\prime }}^{a^{″}}{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}};$$

(149)

let us assume, e.g., $0<a^{\prime }<a^{″}<+\infty$ and $\mathcal{V}\left(a\right)<0$ for all $a\in (a^{\prime },a^{″})$. If $\mathcal{V}\left(a^{\prime }\right)<0$ and $\mathcal{V}\left(a^{″}\right)<0$, the integral (149) is certainly convergent. Convergence is also ensured if $\mathcal{V}$ vanishes but ${\mathcal{V}}^{\prime }$ is nonvanishing, at one or both endpoints $a^{\prime },a^{″}$. For example, let $\mathcal{V}\left(a^{\prime }\right)=0$ and ${\mathcal{V}}^{\prime }\left(a^{\prime }\right)<0$; then, by Taylor’s formula, for $a\to {\left(a^{\prime }\right)}^{+}$, we have $1/\sqrt{-\mathcal{V}\left(a\right)}\sim$$1/\sqrt{-{\mathcal{V}}^{\prime }\left(a^{\prime }\right)}\times 1/\sqrt{a-a^{\prime }}$, so that the integrand in (149) diverges in an integrable way near $a^{\prime }$.

(v)

If $a_1\in {\mathbb{R}}^{+}$ is such that $\mathcal{V}\left(a_1\right)=0$, ${\mathcal{V}}^{\prime }\left(a_1\right)=0$, the constant function $a\left(t\right):=a_1$ for all $t\in \mathbb{R}$ 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 $a_1\in {\mathbb{R}}^{+}$ such that $\mathcal{V}\left(a_1\right)=0$, ${\mathcal{V}}^{\prime }\left(a_1\right)=0$. To explain this statement, let us consider, e.g., the case of a zero-energy solution $t\mapsto a\left(t\right)$ such that $\dot{a}\left(t_0\right)>0$ at some time $t_0$ and assume, with $a_0:=a\left(t_0\right)$, that $a_0<a_1$ and $\mathcal{V}\left(a\right)<0$ for $a\in [a_0,a_1)$. 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 $\dot{a}\left(t\right)>0$, until reaching point $a_1$. According to item (iii), $a_1$ will be reached at the time $t_1$ such that

$$t_1-t_0={\int }_{a_0}^{a_1}{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}};$$

(150)

on the other hand, the above integral diverges, i.e.,

$$t_1=+\infty .$$

(151)

In fact, the assumptions $\mathcal{V}\left(a_1\right)=0$, ${\mathcal{V}}^{\prime }\left(a_1\right)=0$ and Taylor’s formula grant the existence of $\delta ,C\in {\mathbb{R}}^{+}$ such that $-\mathcal{V}\left(a\right)⩽C{(a_1-a)}^2$ for $a\in (a_1-\delta ,a_1]$; this implies $1/\sqrt{-\mathcal{V}\left(a\right)}⩾(1/\sqrt{C})\times 1/(a_1-a)$, which ensures divergence of the integral in  (150).

The above argument has obvious variants. For example, again with $\mathcal{V}\left(a_1\right)=0$ and ${\mathcal{V}}^{\prime }\left(a_1\right)=0$, let us consider a maximal zero-energy solution such that $\dot{a}\left(t_0\right)>0$ at some time $t_0$ and assume, with $a_0:=a\left(t_0\right)$, that $a_1<a_0$ and $\mathcal{V}\left(a\right)<0$ for $a\in (a_1,a_0]$. Then, $a\left(t\right)$ exists for $t<t_0$ until reaching $a_1$ in the past of $t_0$; this occurs at time $-\infty$.

(vii)

Under specific assumptions, one can also discuss the time required for a maximal solution to diverge to $+\infty$. In this discussion, one essentially uses Equation (148), sending to $+\infty$ 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) $2\ddot{a}\left(t\right)=-{\mathcal{V}}^{\prime }\left(a\left(t\right)\right)$, we have

$$sign\ddot{a}\left(t\right)=- \mathrm{sign}{\mathcal{V}}^{\prime }\left(a\left(t\right)\right);$$

(152)

so, the concavity and the inflexion points in the graph of a solution $t\mapsto a\left(t\right)$ can be determined by studying the sign of ${\mathcal{V}}^{\prime }$ along the solution.

5.5. Time Evolution of the Scalar Field

After analyzing the zero-energy solutions $t\mapsto a\left(t\right)$ of the reduced system, let us proceed to discuss the zero-energy solutions $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the complete system. The following statements hold:

(i)

Due to (136), for all $t\in \mathcal{J}$, we have

$$\mathrm{sign}\dot{\varphi }\left(t\right)=\mathrm{constant}=-\sigma \mathrm{sign}\Pi _{\ast };$$

(153)

the function $t\mapsto \varphi \left(t\right)$ is constant if $\Pi _{\ast }=0$, and strictly monotonic if $\Pi _{\ast }\ne 0$.

(ii)

Consider an interval $(t^{\prime },t^{″})\subset \mathcal{J}$ on which $\dot{a}$ has constants sign $s_a\in \{\pm 1\}$, as in item (iii) of Section 5.4. Then, the function $t\mapsto a\left(t\right)$ is a diffeomorphism between $(t^{\prime },t^{″})$ and the interval $(a_{min},a_{max})$ where $a_{min}:=min(a\left(t^{\prime }\right),a\left(t^{″}\right))$ and $a_{max}:=max(a\left(t^{\prime }\right),a\left(t^{″}\right))$. We now consider the inverse function $(a_{min},a_{max})\ni a$$\mapsto t\left(a\right)$, and the composition $(a_{min},a_{max})\ni a$$\mapsto \varphi \left(a\right)$$:=\varphi \left(t\right(a\left)\right)$. Recalling Equations (136) and (147), we find

$${\displaystyle \frac{d\varphi }{da}}\left(a\right)={\displaystyle \frac{\dot{\varphi }\left(t\left(a\right)\right)}{\dot{a}\left(t\left(a\right)\right)}}=-{\displaystyle \frac{\sigma \Pi _{\ast }{a}^{-d}}{s_a\sqrt{-\mathcal{V}\left(a\right)}}}=-{\displaystyle \frac{\sigma s_a\Pi _{\ast }}{\sqrt{-a^{2d}\mathcal{V}\left(a\right)}}}$$

(154)

so that, integrating between $a\left(t^{\prime }\right)$ and $a\left(t^{″}\right)$,

$$\varphi \left(t^{″}\right)-\varphi \left(t^{\prime }\right)=-\sigma s_a\Pi _{\ast }{\int }_{a\left(t^{\prime }\right)}^{a\left(t^{″}\right)}{\displaystyle \frac{da}{\sqrt{-a^{2d}\mathcal{V}\left(a\right)}}}.$$

(155)

(as in the comment after Equation (148), here, $a\left(t^{\prime }\right)$ or $a\left(t^{″}\right)$ must be intended as a limit if $t^{\prime }$ or $t^{″}$ is an endpoint of $\mathcal{J}$; the same can be said of $\varphi \left(t^{\prime }\right)$ or $\varphi \left(t^{″}\right)$).

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 $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the Lagrange equations induced by the Lagrangian (133), with zero energy and with an assigned value of $\Pi _{\ast }$. The function $t\mapsto \varphi \left(t\right)$ is essentially determined by the function $t\mapsto a\left(t\right)$, as indicated in Section 5.5; from here, one infers that $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ is maximal if and only if $\mathcal{J}\ni t\mapsto a\left(t\right)$ 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 $\mathcal{J}\ni t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of the Lagrange equations with zero energy and an assigned value of $\Pi _{\ast }$. For the dimensionless Hubble parameter $h:=\dot{a}/a$, we infer from (144) and (147) that

$$h^2={\displaystyle \frac{-\mathcal{V}\left(a\right)}{a^2}},h=s_a{\displaystyle \frac{\sqrt{-\mathcal{V}\left(a\right)}}{a}};$$

(156)

the second equality holds on any time interval $(t^{\prime },t^{\prime \prime })\subset \mathcal{J}$ on which $\dot{a}$ has a constant sign $s_a\in \{\pm 1\}$.

For the densities and pressures of the fluids and for the curvature density in dimensionless form, we have the usual expressions ${\mathfrak{r}}_n=F_n\left(a^{-d}\right)$, ${\mathfrak{p}}_n=(P_n\circ F_n)\left(a^{-d}\right)$ ($n=1,\dots ,N$), ${\mathfrak{r}}_k={\Omega }_{k\ast }/a^2$ (see (102) and (86)).

Equation (78) on the dimensionless density and pressure of the scalar field and Equations (130) and (136) on V, $\dot{\varphi }$ give

$${\mathfrak{r}}_{\varphi }={\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}{a}^{-2d}+{{\rm Y}}_{\ast },{\mathfrak{p}}_{\varphi }={\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}{a}^{-2d}-{{\rm Y}}_{\ast }.$$

(157)

Let us also recall that, according to Equations (87) and (88), $h^2=-\mathcal{V}\left(a\right)/a^2$ also equals the total dimensionless density $\tilde{\mathfrak{r}}:={\sum }_{n=1}^N{\mathfrak{r}}_n+{\mathfrak{r}}_k+{\mathfrak{r}}_{\varphi }$.

Finally, according to (90), the normalized densities have the following expressions:

$${\Omega }_A={\displaystyle \frac{{\mathfrak{r}}_A}{h^2}}\mathrm{for} A=n,k,\varphi ,h^2 \mathrm{as} \mathrm{in}\left(156\right).$$

(158)

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 $\mathcal{L}$ for the reduced system (see Equations (141) and (142)). The conclusion is the equivalence of the forthcoming statements (i)(i’)(ii), for any $\Pi _{\ast }\in \mathbb{R}$:

(i)

There is a zero-energy solution $\mathcal{J}\ni t\mapsto (a\left(t\right),\varphi \left(t\right))\in {\mathbb{R}}^{+}\times \mathbb{R}$ of the Lagrange equations induced by L such that the canonically conjugate momentum of the scalar field has the assigned value $\Pi _{\ast }$, and $a\left(t_{\ast }\right)=1$, $\dot{a}\left(t_{\ast }\right)=1$ at some time $t_{\ast }\in \mathcal{J}$.

(i’)

There is a zero-energy solution $\mathcal{J}\ni t\mapsto a\left(t\right)\in {\mathbb{R}}^{+}$ of the Lagrange equations induced by $\mathcal{L}$, with the assigned value $\Pi _{\ast }$, such that $a\left(t_{\ast }\right)=1$, $\dot{a}\left(t_{\ast }\right)=1$ at some time $t_{\ast }\in \mathcal{J}$.

(ii)

It is

$$\sum _{n=1}^N{\Omega }_{n\ast }+{\Omega }_{k\ast }+{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}+{{\rm Y}}_{\ast }=1,\mathrm{where}{\Omega }_{n\ast }:=F_n\left(1\right).$$

(159)

Let us remark that Equation (159) is just the relation (129), where we have expressed V as in (130) and we have written ${\dot{\varphi }}_{\ast }^2=\Pi _{\ast }^2$, as prescribed by (136) when $a=a\left(t_{\ast }\right)=1$. By comparison with the definition (142) of $\mathcal{V}$, we readily see that Equation (159) is equivalent to

$$\mathcal{V}\left(1\right)=-1.$$

(160)

Assuming the above conditions to hold, let us consider a zero-energy solution $t\mapsto a\left(t\right)$ of the Lagrange equations induced by $\mathcal{L}$; if $t_{\ast }$ is such that $a\left(t_{\ast }\right)=1$ we have $0=\mathcal{E}(a\left(t_{\ast }\right),\dot{a}\left(t_{\ast }\right))={\dot{a}}^2\left(t_{\ast }\right)+\mathcal{V}\left(1\right)={\dot{a}}^2\left(t_{\ast }\right)-1$, whence $\dot{a}\left(t_{\ast }\right)=\pm 1$. To summarize, given a zero-energy solution $\mathcal{J}\ni t\mapsto a\left(t\right)$ of the reduced system and any time $t_{\ast }\in \mathcal{J}$, Equation (159) ensures the following equivalence:

$$a\left(t_{\ast }\right)=1\mathrm{and}\dot{a}\left(t_{\ast }\right)=1⟺a\left(t_{\ast }\right)=1\mathrm{and}\dot{a}\left(t_{\ast }\right)>0.$$

(161)

Let us recall the comment after Equation (99) on the possible interpretation of $t_{\ast }$ 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 $d=3$, ${{\rm Y}}_{\ast }=0$ 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 $V\left(\varphi \right)$ = 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 $H_{\ast }$ 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

$$d=3;$$

(162)

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:

$$p_n=w_n{\rho }_n(n=r,m),w_r:={\displaystyle \frac{1}{3}},w_m:=0.$$

(163)

Due to (b), all equations of the previous sections involving the evolution laws for the densities must be applied with

$$F_r\left(\alpha \right)={\Omega }_{r\ast }{\alpha }^{4/3},F_m\left(\alpha \right)={\Omega }_{m\ast }\alpha \mathrm{for} \mathrm{all} \alpha \in {\mathbb{R}}^{+},$$

(164)

$${\Omega }_{r\ast },{\Omega }_{m\ast }\in {\mathbb{R}}^{+}$$

(165)

(recall Equation (114)). In particular, the Lagrangian L and the energy E of Equations (133) and (134) take the following forms:

$$L(a,\varphi ,\dot{a},\dot{\varphi }):=a{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^3{\dot{\varphi }}^2+{\displaystyle \frac{{\Omega }_{r\ast }}{a}}+{\Omega }_{m\ast }+{\Omega }_{k\ast }a+{{\rm Y}}_{\ast }{a}^3,$$

(166)

$$E(a,\varphi ,\dot{a},\dot{\varphi }):=a{\dot{a}}^2-{\displaystyle \frac{\sigma }{2}}{a}^3{\dot{\varphi }}^2-{\displaystyle \frac{{\Omega }_{r\ast }}{a}}-{\Omega }_{m\ast }-{\Omega }_{k\ast }a-{{\rm Y}}_{\ast }{a}^3.$$

(167)

Let us recall that we have the constant of motion $\Pi _{\ast }$; see Equations (135) and (136). As in Equation (141), we have a reduced Lagrangian and an energy function

$$\mathcal{L}(a,\dot{a}):={\dot{a}}^2-\mathcal{V}\left(a\right),\mathcal{E}(a,\dot{a}):={\dot{a}}^2+\mathcal{V}\left(a\right)$$

where, according to (142),

$$\mathcal{V}:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto \mathcal{V}\left(a\right)=-{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2a^4}}-{\displaystyle \frac{{\Omega }_{r\ast }}{a^2}}-{\displaystyle \frac{{\Omega }_{m\ast }}{a}}-{\Omega }_{k\ast }-{{\rm Y}}_{\ast }{a}^2.$$

(168)

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

  $${\rho }_A={\displaystyle \frac{3H_{\ast }^2}{8\pi G}}{\mathfrak{r}}_A(A=r,m,k,\varphi ),p_A={\displaystyle \frac{3H_{\ast }^2}{8\pi G}}{\mathfrak{p}}_A(A=r,m,\varphi );$$

  (169)

  $${\mathfrak{r}}_r={\displaystyle \frac{{\Omega }_{r\ast }}{a^4}},{\mathfrak{p}}_r={\displaystyle \frac{{\Omega }_{r\ast }}{3a^4}};{\mathfrak{r}}_m={\displaystyle \frac{{\Omega }_{m\ast }}{a^3}},{\mathfrak{p}}_m=0;{\mathfrak{r}}_k:={\displaystyle \frac{{\Omega }_{k\ast }}{a^2}};$$

  (170)

  $${\mathfrak{r}}_{\varphi }={\displaystyle \frac{\sigma \Pi _{\ast }^2}{2a^6}}+{{\rm Y}}_{\ast },{\mathfrak{p}}_{\varphi }={\displaystyle \frac{\sigma \Pi _{\ast }^2}{2a^6}}-{{\rm Y}}_{\ast }.$$

  (171)
• Equations (80) and (88) for the total dimensionless pressure and the total dimensionless densities, excluding or including the curvature contribution, read, in this case,

  $$\mathfrak{r}:={\mathfrak{r}}_r+{\mathfrak{r}}_m+{\mathfrak{r}}_{\varphi },\mathfrak{p}:={\mathfrak{p}}_r+{\mathfrak{p}}_{\varphi };$$

  (172)

  $$\tilde{\mathfrak{r}}:={\mathfrak{r}}_r+{\mathfrak{r}}_m+{\mathfrak{r}}_k+{\mathfrak{r}}_{\varphi };$$

  (173)

  let us recall Equation (87).
• Equation (158) for the normalized energy densities reads, in this case,

  $${\Omega }_A={\displaystyle \frac{{\mathfrak{r}}_A}{h^2}}(A=r,m,k,\varphi ),$$

  (174)

  with $h=\dot{a}/a$ the dimensionless Hubble parameter.
• According to Equation (132), the present setting with a field potential $V\left(\varphi \right)=$ const. $={{\rm Y}}_{\ast }$ can be rephrased in terms of the Einstein equations with a cosmological constant

  $$\Lambda =3H_{\ast }^2{{\rm Y}}_{\ast }.$$

  (175)

To continue, we add the following assumptions to items (a) and (b):

(c)

It is

$${{\rm Y}}_{\ast }>0$$

(176)

(positive field potential or positive cosmological constant).

(d)

The values considered for the constants ${\Omega }_{A\ast }$ ($A=r,m,k$), $\Pi _{\ast }$ and ${{\rm Y}}_{\ast }$ are in any case related as follows:

$${\Omega }_{r\ast }+{\Omega }_{m\ast }+{\Omega }_{k\ast }+{\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}+{{\rm Y}}_{\ast }=1.$$

(177)

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 ${\Omega }_{r\ast },{\Omega }_{m\ast }>0$, while we have left the $sign{\Omega }_{k\ast }=-signk$ 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 $\Pi _{\ast }$, fulfilling the condition (99)

$$\mathrm{There} \mathrm{is} a \mathrm{time} t_{\ast }\mathrm{such} \mathrm{that} a \left(t_{\ast }\right)=1 \mathrm{and} \dot{a}\left(t_{\ast }\right)=1(\mathrm{i}.\mathrm{e}., h\left(t_{\ast }\right)=1),$$

or its dimensioned analog (58)

$$\mathrm{There} \mathrm{is} a \mathrm{time} {\tau }_{\ast }\mathrm{such} \mathrm{that} a\left({\tau }_{\ast }\right)=1 \mathrm{and} \stackrel{\circ }{a}\left({\tau }_{\ast }\right)=H_{\ast }(\mathrm{i}.\mathrm{e}., H\left({\tau }_{\ast }\right)=H_{\ast }).$$

Since the cosmological models of the present Section 6 aim for minimal realism, we will interpret $t_{\ast },{\tau }_{\ast }$ as describing present time, and thus $H_{\ast }$ 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) $\mathcal{V}\left(1\right)=-1$, where $\mathcal{V}$ is now the potential (168); let us also repeat that, for each zero-energy solution $t\mapsto a\left(t\right)$ of the reduced system and for any $t_{\ast }$ in its time domain, Equation (177) ensures the equivalence (161)

$$a\left(t_{\ast }\right)=1\mathrm{and}\dot{a}\left(t_{\ast }\right)=1⟺a\left(t_{\ast }\right)=1\mathrm{and}\dot{a}\left(t_{\ast }\right)>0.$$

To continue, let us reconsider the normalized densities ${\Omega }_A$. We already know (and, in any case, we can recover from Equations (170) and (174)) that

$${\Omega }_A\left(t_{\ast }\right)={\Omega }_{A\ast }\mathrm{for} A=r,m,k;$$

(178)

moreover, we readily obtain from Equations (171) and (174) that

$${\Omega }_{\varphi }\left(t_{\ast }\right)={\displaystyle \frac{\sigma \Pi _{\ast }^2}{2}}+{{\rm Y}}_{\ast }.$$

(179)

6.2. The Case of $\Pi _{\ast }=0$ : Recovering the Standard Model of Cosmology

Recalling that we are assuming ${\Omega }_{r\ast },{\Omega }_{m\ast },{{\rm Y}}_{\ast }>0$, we momentarily consider the solutions $t\mapsto \left(a\right(t),\varphi (t\left)\right)$ of our model with

$$\Pi _{\ast }=0.$$

(180)

Equation (177) becomes

$${\Omega }_{r\ast }+{\Omega }_{m\ast }+{\Omega }_{k\ast }+{{\rm Y}}_{\ast }=1;$$

(181)

according to item (i) in Section 5.5, we have

$$\varphi \left(t\right)=\mathrm{constant},$$

(182)

and Equation (179) allows us to interpret ${{\rm Y}}_{\ast }$ as the present time value ${\Omega }_{\varphi }\left(t_{\ast }\right)$. Equation (168) becomes

$$\mathcal{V}\left(a\right)=-{\displaystyle \frac{{\Omega }_{r\ast }}{a^2}}-{\displaystyle \frac{{\Omega }_{m\ast }}{a}}-{\Omega }_{k\ast }-{{\rm Y}}_{\ast }{a}^2;$$

(183)

of course,

$$\mathcal{V}\left(a\right)\sim -{\displaystyle \frac{{\Omega }_{r\ast }}{a^2}}\to -\infty \mathrm{for}a\to 0^{+},\mathcal{V}\left(a\right)\sim -{{\rm Y}}_{\ast }{a}^2\to -\infty \mathrm{for}a\to +\infty .$$

(184)

It can be shown that there is a unique $a_2$ such that

$$a_2\in {\mathbb{R}}^{+},{\mathcal{V}}^{\prime }\left(a_2\right)=0;$$

(185)

moreover,

$${\mathcal{V}}^{\prime }\left(a\right)>0\mathrm{for} a\in (0,a_2),{\mathcal{V}}^{\prime }\left(a\right)<0\mathrm{for} a\in (a_2,+\infty )$$

(186)

(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 $\mathcal{V}$.

The zero-energy solutions $t\mapsto a\left(t\right)$ of the reduced system with Lagrangian $\mathcal{L}(a,\dot{a})={\dot{a}}^2-\mathcal{V}\left(a\right)$ are easily analyzed with the methods of Section 5.4; in particular, we have the quadrature formula (148), with $\mathcal{V}$ as in (183). This is just the evolution law of the standard (or benchmark, or $\Lambda$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 ${{\rm Y}}_{\ast }$ with the present time value of the normalized dark energy density.

To fix ideas, let us consider the subcase

$${\Omega }_{k\ast }⩾0$$

(187)

(i.e., $k⩽0$). Then, we see from (183) that

$$\mathcal{V}\left(a\right)<0\mathrm{for} \mathrm{all} a\in {\mathbb{R}}^{+};$$

(188)

Figure 1 gives a qualitative plot of the function $\mathcal{V}$ under the assumption (187).18

Let $t\mapsto a\left(t\right)$ 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 ${\mathbb{R}}^{+}$ and $\dot{a}\left(t\right)$ 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 $\dot{a}>0$, and unbounded from below if $\dot{a}<0$. After a time translation $t\mapsto t+\mathrm{constant}$, this interval can be reduced to the form $(0,+\infty )$ if $\dot{a}>0$, and to the form $(-\infty ,0)$ if $\dot{a}<0$; as usual for the standard model, hereafter, we consider the case $\dot{a}>0$.

In this case, for t ranging in $(0,+\infty )$, $a\left(t\right)$ increases monotonically from 0 to $+\infty$ according to the following law (from (148)):

$${\int }_0^{a\left(t\right)}{\displaystyle \frac{da}{\sqrt{-\mathcal{V}\left(a\right)}}}=t$$

(189)

(with $1/\sqrt{-\mathcal{V}\left(a\right)}=$$1/\sqrt{{\Omega }_{r\ast }/a^2+{\Omega }_{m\ast }/a+{\Omega }_{k\ast }+{{\rm Y}}_{\ast }{a}^2}$$=a/\sqrt{{\Omega }_{r\ast }+{\Omega }_{m\ast }a+{\Omega }_{k\ast }{a}^2+{{\rm Y}}_{\ast }{a}^4}$; this function vanishes for $a\to 0^{+}$, so the above integral is nonsingular). Equation (189) and the first relation (184) imply

$$a\left(t\right)\sim \sqrt{2\sqrt{{\Omega }_{r\ast }}t}\mathrm{for} t\to 0^{+};$$

(190)

of course, the vanishing of the scale factor in this limit indicates a Big Bang. For $t\to +\infty$, an exponential behavior can be predicted for $a\left(t\right)$ since (by (147) and the second relation (184)) $\dot{a}\left(t\right)=\sqrt{-\mathcal{V}\left(a\right(t\left)\right)}\sim \sqrt{{{\rm Y}}_{\ast }}a\left(t\right)$.

The concavity features of the function $t\mapsto a\left(t\right)$ are inferred from Equation (152) $\mathrm{sign}\ddot{a}\left(t\right)=-\mathrm{sign}{\mathcal{V}}^{\prime }\left(a\left(t\right)\right)$, from Equation (186) about ${\mathcal{V}}^{\prime }$, and from the previous description of this solution. Clearly, there is a unique

$$t_2\in (0,+\infty )\mathrm{s}.\mathrm{t}.a\left(t_2\right)=a_2,$$

(191)

and $a\left(t\right)\in (0,a_2)$ for $t\in (0,t_2)$, $a\left(t\right)\in (a_2,+\infty )$ for $t\in (t_2,+\infty )$. These facts imply

$$\ddot{a}\left(t\right)<0\mathrm{for} t\in (0,t_2),\ddot{a}\left(t_2\right)=0,\ddot{a}\left(t\right)>0\mathrm{for} t\in (t_2,+\infty );$$

(192)

so, $t_2$ is an inflexion point. Needless to say, the present time $t_{\ast }\in (0,+\infty )$ is uniquely determined by the condition $a\left(t_{\ast }\right)=1$ (recall the discussion in the final part of Section 6.1). According to Equation (189),

$$t_2={\int }_0^{a_2}{\displaystyle \frac{da}{\sqrt{-\mathcal{V}\left(a\right)}}},t_{\ast }={\int }_0^1{\displaystyle \frac{da}{\sqrt{-\mathcal{V}\left(a\right)}}}.$$

(193)

Figure 2 describes qualitatively the behavior of the scale factor $t\mapsto a\left(t\right)$. 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 $(t_{-},t_{+})=(0,+\infty )$), 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:

$${\Omega }_{r\ast }\simeq 9.2·{10}^{-5},{\Omega }_{m\ast }\simeq 0.31,{\Omega }_{k\ast }\simeq 0,{{\rm Y}}_{\ast }\simeq 0.69;$$

(194)

these reference values will be taken into account in the sequel.

To fix the ideas, let us assume exactly the above numerical values for ${\Omega }_{r\ast }$, ${{\rm Y}}_{\ast }$ and set ${\Omega }_{m\ast }=0.31-{\Omega }_{r\ast }$, ${\Omega }_{k\ast }=0$ (so that (181) holds exactly). Using these values and solving Equation (185) for $a_2$ (i.e., numerically), we obtain

$$a_2\simeq 0.608.$$

(195)

From these values and from Equation (193), we obtain (computing numerically the corresponding integrals)

$$t_2\simeq 0.528,t_{\ast }\simeq 0.955.$$

(196)

Using Equation (59) involving the present-time Hubble parameter, we infer that the cosmic times ${\tau }_i=t_i/H_{\ast }$ ($i=2,\ast$) have the values

$${\tau }_2\simeq 7.68\times {10}^9\mathrm{yr},{\tau }_{\ast }\simeq 13.9\times {10}^9\mathrm{yr};$$

(197)

of course, ${\tau }_{\ast }$ 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 ${\Omega }_{k\ast }<0$ and $|{\Omega }_{k\ast }|$ not too large, while this equation fails for ${\Omega }_{k\ast }<0$ and $|{\Omega }_{k\ast }|$ 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 $\Pi _{\ast }\ne 0$

Maintaining all the assumptions of Section 6.1, we now pass from the case $\Pi _{\ast }=0$ (the standard model of cosmology) to cases with $\Pi _{\ast }\ne 0$; these will be treated assuming frequently that $\Pi _{\ast }$ is small. The deviation of these cases from the standard model is controlled by the term $-\sigma \Pi _{\ast }^2/\left(2a^4\right)$ in Equation (168) for $\mathcal{V}$.

In the sequel, most of our attention is devoted to the case

(i)

$\sigma =-1$ (phantom scalar field), $\Pi _{\ast }\ne 0$ (typically small), ${\Omega }_{k\ast }⩾0$ (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 ${\Omega }_{k\ast }⩾0$, 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 $t\to +\infty$, in case (i), the scale factor presents an exponential growth similar to that of the standard model with ${\Omega }_{k\ast }⩾0$; the same exponential growth appears for $t\to -\infty$, i.e., very far in the past before the Big Bounce. As we shall see, specifying values close to those in (194) for ${\Omega }_{r\ast },{\Omega }_{m\ast },{\Omega }_{k\ast }$ and ${{\rm Y}}_{\ast }$ and an extremely small value for $\Pi _{\ast }$, 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)

$\sigma =-1$ (phantom scalar field), $\Pi _{\ast }\ne 0$ (typically small), ${\Omega }_{k\ast }<0$ (i.e., $k>0$);

this will be sketched in Section 6.8. If $-{\Omega }_{k\ast }$ (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 $t\to \pm \infty$. If $-{\Omega }_{k\ast }$ 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 $t\to \pm \infty$ (different behaviors occur if $-{\Omega }_{k\ast }$ has exactly the threshold value; see again Section 6.8). The physical plausibility of the model when $-{\Omega }_{k\ast }$ 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)

$\sigma =1$ (canonical scalar field), $\Pi _{\ast }\ne 0$;

this presents a Big Bang19 and can be considered, at least for small $\Pi _{\ast }$, as a perturbation of the standard model of cosmology.

6.4. Preparing the Analysis of the Phantom Case with $\Pi _{\ast }\ne 0$

From here to the end of the present Section 6, we assume the scalar field is a phantom:

$$\sigma =-1.$$

(198)

We consider solutions of the model with $\Pi _{\ast }$ nonvanishing; for future convenience, we represent this quantity as

$$\Pi _{\ast }=s\sqrt{2{\Omega }_{r\ast }}\mu ,s\in \{\pm 1\},\mu \in {\mathbb{R}}^{+}.$$

(199)

We confirm the general conditions (165), (176), and (177), thus requiring that

$${\Omega }_{r\ast },{\Omega }_{m\ast },{{\rm Y}}_{\ast }>0,{\Omega }_{r\ast }+{\Omega }_{m\ast }+{\Omega }_{k\ast }-{\Omega }_{r\ast }{\mu }^2+{{\rm Y}}_{\ast }=1$$

(200)

(note that ${\Omega }_{r\ast }{\mu }^2$ is just the expression for $\Pi _{\ast }/2$ following from (199); the sign of ${\Omega }_{k\ast }$ 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

$$\mathcal{V}:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto \mathcal{V}\left(a\right)={\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^4}}-{\displaystyle \frac{{\Omega }_{r\ast }}{a^2}}-{\displaystyle \frac{{\Omega }_{m\ast }}{a}}-{\Omega }_{k\ast }-{{\rm Y}}_{\ast }{a}^2.$$

(201)

Needless to say,

$$\mathcal{V}\left(a\right)\sim {\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^4}}\to +\infty \mathrm{for} a\to 0^{+};\mathcal{V}\left(a\right)\sim -{{\rm Y}}_{\ast }{a}^2\to -\infty \mathrm{for} a\to +\infty .$$

(202)

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

$${\mathfrak{r}}_{\varphi }=-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast },{\mathfrak{p}}_{\varphi }=-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}-{{\rm Y}}_{\ast }.$$

(203)

6.5. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }⩾0$: Analysis of the Potential $\mathcal{V}\left(a\right)$ and the Associated Function $\mathcal{H}\left(a\right):=-\mathcal{V}\left(a\right)/a^2$

Throughout the present Section 6.5, we make the assumptions (198)–(200) and also prescribe

$${\Omega }_{k\ast }⩾0$$

(204)

(i.e., $k⩽0$). In the sequel, we will often put conditions of smallness on $\mu$ (i.e., on $\Pi _{\ast }$); the most important of such conditions will be the forthcoming inequality (211).

The potential $\mathcal{V}$. In Appendix L (which makes reference to the preceding Appendix I and Appendix J), we prove the following statements (a)–(d) about $\mathcal{V}$:

(a)

There is a unique point $a_0$ such that

$$a_0\in {\mathbb{R}}^{+},\mathcal{V}\left(a_0\right)=0;$$

(205)

in addition,

$$\mathcal{V}\left(a\right)>0 \mathrm{for} a\in (0,a_0),\mathcal{V}\left(a\right)<0 \mathrm{for} a\in (a_0,+\infty )$$

(206)

and

$$a_0<1.$$

(207)

(b)

Let

$$N_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }+{\Omega }_{k\ast }\mu +{{\rm Y}}_{\ast }{\mu }^3}{{\Omega }_{r\ast }}},F_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }+{\Omega }_{k\ast }\mu +{{\rm Y}}_{\ast }{\mu }^3}{2{\Omega }_{r\ast }+3{\Omega }_{m\ast }\mu +4{\Omega }_{k\ast }{\mu }^2+6{{\rm Y}}_{\ast }{\mu }^4}};$$

(208)

then, $N_{\ast }>0,F_{\ast }>0$ and $F_{\ast }\mu <1$. Moreover,

$${\displaystyle \frac{\mu }{\sqrt{1+N_{\ast }\mu }}}<a_0<\mu (1-F_{\ast }\mu ).$$

(209)

(Note that the upper bound on $a_0$ in (209) implies the rougher bound $a_0<\mu$).

(c)

Let us consider the derivative

$${\mathcal{V}}^{\prime }:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto {\mathcal{V}}^{\prime }\left(a\right)=-{\displaystyle \frac{4{\Omega }_{r\ast }{\mu }^2}{a^5}}+{\displaystyle \frac{2{\Omega }_{r\ast }}{a^3}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a^2}}-2{{\rm Y}}_{\ast }a,$$

(210)

and assume

$${\left({\displaystyle \frac{4\sqrt{2}{{\rm Y}}_{\ast }}{{\Omega }_{m\ast }}}\right)}^{1/3}\mu <1.$$

(211)

Then, there are just two points $a_1,a_2$ such that

$$a_1,a_2\in {\mathbb{R}}^{+},a_1<a_2,{\mathcal{V}}^{\prime }\left(a_1\right)={\mathcal{V}}^{\prime }\left(a_2\right)=0;$$

(212)

moreover

$${\mathcal{V}}^{\prime }\left(a\right)<0\mathrm{for} a\in (0,a_1)\cup (a_2,+\infty ),{\mathcal{V}}^{\prime }\left(a\right)>0\mathrm{for}\in a\in (a_1,a_2),$$

(213)

so that $a_1$ is a local minimum point, and $a_2$ is a local maximum point for $\mathcal{V}$. Also, we have the following (somehow rough) inequalities:

$$a_0<a_1<\sqrt{2}\mu <a_2.$$

(214)

(d)

Maintaining the assumption (211), let

$$M_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }}{\sqrt{2}{\Omega }_{r\ast }}},G_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }-4\sqrt{2}{{\rm Y}}_{\ast }{\mu }^3}{2\sqrt{2}{\Omega }_{r\ast }+3{\Omega }_{m\ast }\mu }};$$

(215)

then, $M_{\ast }>0$, $G_{\ast }>0$ by (211) and $G_{\ast }\mu <1$. Moreover, we have the following bounds (refining (214)) for $a_1$:

$${\displaystyle \frac{\sqrt{2}\mu }{\sqrt{1+M_{\ast }\mu }}}<a_1<\sqrt{2}\mu \left(1-G_{\ast }\mu \right).$$

(216)

Remarks. (i) Figure 3 presents a qualitative plot of the potential $\mathcal{V}$ 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 $\mu$-dependent family of models in which

$$\Pi _{\ast }=s\sqrt{2{\Omega }_{r\ast }}\mu ,s\in \{\pm 1\},{{\rm Y}}_{\ast }={{\rm Y}}_{\ast \ast }+{\Omega }_{r\ast }{\mu }^2(\mu \in {\mathbb{R}}^{+})$$

(217)

$${\Omega }_{r\ast }>0,{\Omega }_{m\ast }>0,{\Omega }_{k\ast }⩾0,{{\rm Y}}_{\ast \ast }>0 \mathrm{constants} \mathrm{such} \mathrm{that} {\Omega }_{r\ast }+{\Omega }_{m\ast }+{\Omega }_{k\ast }+{{\rm Y}}_{\ast \ast }=1.$$

The above requirement of sum one for ${\Omega }_{r\ast },\dots ,{{\rm Y}}_{\ast \ast }$ 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 ${\Omega }_{\varphi }\left(t_{\ast }\right)={{\rm Y}}_{\ast \ast }$.

For such a family of models, the quantities $a_0,a_1$ are functions of $\mu$, and it makes sense to discuss their behavior in the limit of vanishing $\mu$. For $\mu \to 0^{+}$, we have

$$a_0=\mu \left(1-{\displaystyle \frac{{\Omega }_{m\ast }}{2{\Omega }_{r\ast }}}\mu +O\left({\mu }^2\right)\right),$$

(218)

$$a_1=\sqrt{2}\mu \left(1-{\displaystyle \frac{{\Omega }_{m\ast }}{2\sqrt{2}{\Omega }_{r\ast }}}\mu +O\left({\mu }^2\right)\right).$$

(219)

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 $\mathcal{H}$. Maintaining the assumptions (198)–(200) and (204), we now define

$$\mathcal{H}:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto \mathcal{H}\left(a\right):=-{\displaystyle \frac{\mathcal{V}\left(a\right)}{a^2}}=-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}+{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}+{{\rm Y}}_{\ast }.$$

(220)

This function is especially interesting since, according to Equations  (156) and (87), along any solution of the cosmological model, we have

$$\mathcal{H}\left(a\left(t\right)\right)=h^2\left(t\right)=\tilde{\mathfrak{r}}\left(t\right),$$

(221)

where h is the dimensionless Hubble parameter (see Equation (70)) and $\tilde{\mathfrak{r}}$ is the total dimensionless density, including the curvature contribution (see Equation (173)). Of course,

$$\mathcal{H}\left(a\right)\sim -{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}\to -\infty \mathrm{for}a\to 0^{+},\mathcal{H}\left(a\right)\to {{\rm Y}}_{\ast }\mathrm{for}a\to +\infty .$$

(222)

Moreover, $\mathcal{H}$ has the following features (e) (an obvious fact) and (f)–(i) (proved in Appendix M):

(e)

For all $a\in {\mathbb{R}}^{+}$, we have

$$\mathrm{sign}\mathcal{H}\left(a\right)=-\mathrm{sign}\mathcal{V}\left(a\right).$$

(223)

Thus, if $a_0\in {\mathbb{R}}^{+}$ is the point mentioned in Equations (205) and (206), we have

$$\mathcal{H}\left(a\right)<0 \mathrm{for} a\in (0,a_0),\mathcal{H}\left(a_0\right)=0,\mathcal{H}\left(a\right)>0 \mathrm{for} a\in (a_0,+\infty ).$$

(224)

(f)

Let us consider the derivative

$${\mathcal{H}}^{\prime }:{\mathbb{R}}^{+}\to \mathbb{R},a\mapsto {\mathcal{H}}^{\prime }\left(a\right)={\displaystyle \frac{6{\Omega }_{r\ast }{\mu }^2}{a^7}}-{\displaystyle \frac{4{\Omega }_{r\ast }}{a^5}}-{\displaystyle \frac{3{\Omega }_{m\ast }}{a^4}}-{\displaystyle \frac{2{\Omega }_{k\ast }}{a^3}}.$$

(225)

Then, there is a unique point $a_{1/2}$ such that

$$a_{1/2}\in {\mathbb{R}}^{+},{\mathcal{H}}^{\prime }\left(a_{1/2}\right)=0;$$

(226)

moreover,

$${\mathcal{H}}^{\prime }\left(a\right)>0 \mathrm{for} a\in (0,a_{1/2}),{\mathcal{H}}^{\prime }\left(a\right)<0 \mathrm{for} a\in (a_{1/2},+\infty ).$$

(227)

Due to these facts and to the asymptotics (222), $a_{1/2}$ is the absolute maximum point of $\mathcal{H}$ and

$$\mathcal{H}\left(a_{1/2}\right)>{{\rm Y}}_{\ast }.$$

(228)

The last inequality obviously implies $\mathcal{H}\left(a_{1/2}\right)>0$ so that, by comparison with (224), we also infer

$$a_0<a_{1/2}.$$

(229)

(g)

Let

$$S_{\ast }:={\displaystyle \frac{3(\sqrt{3/2}{\Omega }_{m\ast }+{\Omega }_{k\ast }\mu )}{4{\Omega }_{r\ast }}},T_{\ast }:={\displaystyle \frac{3(\sqrt{3/2}{\Omega }_{m\ast }+{\Omega }_{k\ast }\mu )}{8{\Omega }_{r\ast }+9\sqrt{3/2}{\Omega }_{m\ast }\mu +12{\Omega }_{k\ast }{\mu }^2}};$$

(230)

then, $S_{\ast }>0,T_{\ast }>0$ and $T_{\ast }\mu <1$. Moreover,

$${\displaystyle \frac{\sqrt{3/2}\mu }{\sqrt{1+S_{\ast }\mu }}}<a_{1/2}<\sqrt{3/2}\mu \left(1-T_{\ast }\mu \right).$$

(231)

(Note that the upper bound on $a_{1/2}$ in (231) implies the rougher bound $a_{1/2}<\sqrt{3/2}\mu$.)

(h)

The bounds (231) imply

$${\displaystyle \frac{4{\Omega }_{r\ast }}{27{\mu }^4}}·{\displaystyle \frac{1+X_{\ast }\mu }{{(1-T_{\ast }\mu )}^4}}<\mathcal{H}\left(a_{1/2}\right)<{\displaystyle \frac{4{\Omega }_{r\ast }}{27{\mu }^4}}·{(1+S_{\ast }\mu )}^2(1+Y_{\ast }\mu ),$$

(232)

with $S_{\ast },T_{\ast }$ as in (230), and

$$X_{\ast }:=-2S_{\ast }+{\displaystyle \frac{3\sqrt{3/2}{\Omega }_{m\ast }}{{\Omega }_{r\ast }}}·{\displaystyle \frac{1}{\sqrt{1+S_{\ast }\mu }}}+{\displaystyle \frac{9{\Omega }_{k\ast }}{2{\Omega }_{r\ast }}}·{\displaystyle \frac{\mu }{1+S_{\ast }\mu }}+{\displaystyle \frac{27{{\rm Y}}_{\ast }}{4{\Omega }_{r\ast }}}·{\displaystyle \frac{{\mu }^3}{{(1+S_{\ast }\mu )}^2}},$$

(233)

$$Y_{\ast }:=-{\displaystyle \frac{4T_{\ast }}{{(1-T_{\ast }\mu )}^2}}+{\displaystyle \frac{3\sqrt{3/2}{\Omega }_{m\ast }}{{\Omega }_{r\ast }}}(1-T_{\ast }\mu )+{\displaystyle \frac{2T_{\ast }^2\mu }{{(1-T_{\ast }\mu )}^2}}$$

$$+{\displaystyle \frac{9{\Omega }_{k\ast }}{2{\Omega }_{r\ast }}}\mu {(1-T_{\ast }\mu )}^2+{\displaystyle \frac{27{{\rm Y}}_{\ast }}{4{\Omega }_{r\ast }}}{\mu }^3{(1-T_{\ast }\mu )}^4.$$

(i)

Assume (211) holds so that we can speak of the point $a_1$ defined by (212); also, assume

$${\displaystyle M_{\ast }\mu <\frac{1}{3},\mathrm{with} M_{\ast } \mathrm{as} \mathrm{in}\left(215\right).}$$

(234)

Then,

$$a_{1/2}<a_1.$$

(235)

Remarks. (i) Figure 4 presents a qualitative plot of the function $\mathcal{H}$ under the conditions (198)–(200), (204), (211), and (234).

(ii) Let us consider a family of models as in (217). For $\mu \to 0^{+}$, Equation (231) implies

$$a_{1/2}=\sqrt{3/2}\mu \left(1-{\displaystyle \frac{3\sqrt{3/2}{\Omega }_{m\ast }}{8{\Omega }_{r\ast }}}\mu +O\left({\mu }^2\right)\right);$$

(236)

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 $\mu \to 0^{+}$, we have

$$\mathcal{H}\left(a_{1/2}\right)={\displaystyle \frac{4{\Omega }_{r\ast }}{27{\mu }^4}}\left(1+{\displaystyle \frac{3\sqrt{3/2}{\Omega }_{m\ast }}{{\Omega }_{r\ast }}}\mu +O\left({\mu }^2\right)\right).$$

(237)

The last result can be obtained by directly inserting the expansion (236) of $a_{1/2}$ into Equation (220) for $\mathcal{H}$, or using the estimates (232) for $\mathcal{H}\left(a_{1/2}\right)$; in fact, both the lower and upper bound on $\mathcal{H}\left(a_{1/2}\right)$ in Equation (232) behave like the left-hand side of Equation (237).

6.6. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }⩾0$ : 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 $\mathcal{J}\ni t\mapsto a\left(t\right)$ of the reduced system with Lagrangian (141) ($\mathcal{J}$ 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 $\mathcal{V}$ in Section 6.5, we are led to the description reported in the following paragraphs:

Basic features of the solution; Big Bounce. The set ${\mathcal{V}}_{⩽0}:=\{a\in {\mathbb{R}}^{+}|\mathcal{V}\left(a\right)⩽0\}$ coincides in this case with $[a_0,+\infty )$, with $a_0$ as in (205). The motion $\mathcal{J}\ni t\mapsto a\left(t\right)$ has range $[a_0,+\infty )$ and possesses a unique inversion time $t_0$, characterized by the condition $a\left(t_0\right)=a_0$; up to a time shift $t\mapsto t+$ constant, we can assume $t_0=0$. Thus,

$$a\left(0\right)=a_0;$$

(238)

moreover

$$\mathrm{sign}\dot{a}\left(t\right)= \mathrm{sign}t$$

(239)

(even at $t=0$, when $\dot{a}$ vanishes). From the quadrature formula (148) with $t^{\prime },t^{\prime \prime }$ replaced with $0,t$ (or with $t,0$), and from Equations (238) and (239), we obtain

$${\int }_{a_0}^{a\left(t\right)}{\displaystyle \frac{da}{\sqrt{-\mathcal{V}\left(a\right)}}}=\left(\mathrm{sign}t\right)t=\left|t\right|.$$

(240)

Since $\mathcal{V}\left(a_0\right)=0$ and ${\mathcal{V}}^{\prime }\left(a_0\right)<0$, the function $1/\sqrt{-\mathcal{V}\left(a\right)}$ diverges like $1/\sqrt{a-a_0}$ for $a\to {\left(a_0\right)}^{+}$, so the above integral converges; this remark is related to item (iv) in Section 5.4.

The motion continues until $a\left(t\right)$ diverges in the far future and in the far past; divergence of $a\left(t\right)$ occurs for $t\to t_{\pm }$, where $t_{\pm }:=\pm {\int }_{a_0}^{+\infty }da/\sqrt{-\mathcal{V}\left(a\right)}$. On the other hand, the last integral diverges since $1/\sqrt{-\mathcal{V}\left(a\right)}\sim 1/\left(\sqrt{{{\rm Y}}_{\ast }}a\right)$ for $a\to +\infty$ (recall Equation (184)); thus, $t_{\pm }=\pm \infty$, and we can say that

$$\mathcal{J}=(-\infty ,+\infty ),a\left(t\right)\to +\infty \mathrm{for}t\to \pm \infty$$

(241)

(a posteriori, this also ensures that (239) and (240) hold for all $t\in \mathbb{R}$). By construction, the function $\mathbb{R}\to {\mathbb{R}}^{+}$, $t\mapsto a\left(t\right)$ solves the Cauchy problem $2\ddot{a}\left(t\right)=-{\mathcal{V}}^{\prime }\left(a\left(t\right)\right)$ (recall (143)), $a\left(0\right)=a_0$, $\dot{a}\left(0\right)=0$. It is evident that the function $\mathbb{R}\to {\mathbb{R}}^{+}$, $t\mapsto a(-t)$ solves the same Cauchy problem, so by the standard uniqueness theorem, we have

$$a(-t)=a\left(t\right)\mathrm{for} \mathrm{all} t\in \mathbb{R}.$$

(242)

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 $(-\infty ,+\infty )$ and the function $t\mapsto a\left(t\right)$ 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 $a_i\in (a_0,+\infty )$. From the features of the function $t\mapsto a\left(t\right)$ described previously, it appears that there is a unique time

$$t_i\in {\mathbb{R}}^{+}\mathrm{such} \mathrm{that}a\left(t_i\right)=a_i;$$

(243)

according to Equation (240), this is given by

$$t_i={\int }_{a_0}^{a_i}{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}}.$$

(244)

In the sequel, we will often refer to the above statements.

Concavity features of the solution. Let us reconsider Equation (152) $\mathrm{sign}\ddot{a}\left(t\right)=-\mathrm{sign}{\mathcal{V}}^{\prime }\left(a\left(t\right)\right)$. The sign of ${\mathcal{V}}^{\prime }$ 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,

$$t_1<t_2;a\left(t\right)\in [a_0,a_1)\mathrm{for} t\in [0,t_1),$$

(245)

$$a\left(t\right)\in (a_1,a_2)\mathrm{for} t\in (t_1,t_2),a\left(t\right)\in (a_2,+\infty )\mathrm{for} t\in (t_2,+\infty ).$$

Equations (212) and (213) on the sign of ${\mathcal{V}}^{\prime }$ and Equations (152), (243) and (245) imply

$$\ddot{a}\left(t\right)>0\mathrm{for} t\in [0,t_1)\cup (t_2,+\infty ),\ddot{a}\left(t_1\right)=\ddot{a}\left(t_2\right)=0,\ddot{a}\left(t\right)<0\mathrm{for} t\in (t_1,t_2);$$

(246)

thus, $t_1$ and $t_2$ are inflection times for the function $a\left(\right)$. One can make similar statements on the behavior of $a\left(\right)$ at negative times, noting that (242) implies $\ddot{a}(-t)=\ddot{a}\left(t\right)$.

Let us recall that, according to Equation (98), the strong energy condition (SEC) is violated whenever $\ddot{a}\left(t\right)>0$; 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 $(-\infty ,-t_2)$, $(t_2,+\infty )$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 $t_{\ast }$ is individuated by the conditions

$$t_{\ast }\in {\mathbb{R}}^{+},a\left(t_{\ast }\right)=1,$$

(247)

that automatically imply

$$\dot{a}\left(t_{\ast }\right)=1(\mathrm{i}.\mathrm{e}.,h\left(t_{\ast }\right)=1).$$

(248)

In fact, $\dot{a}\left(t_{\ast }\right)>0$ since $t_{\ast }>0$, and we have the equivalence (161) reviewed in the discussion after Equation (177). The present time is described by Equation (244) with $i=\ast$ and $a_{\ast }=1$, i.e.,

$$t_{\ast }={\int }_{a_0}^1{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}}.$$

(249)

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 $t_1⋘t_2<t_{\ast }$.

The large t limit: exponential growth of the scale factor. For all $t\in {\mathbb{R}}^{+}$, we have $\dot{a}\left(t\right)=\sqrt{-\mathcal{V}\left(a\right(t\left)\right)}$; for $t\to +\infty$, we have $a\left(t\right)\to +\infty$ and this fact, with Equation  (202) about $\mathcal{V}$, gives $\dot{a}\left(t\right)\sim \sqrt{{{\rm Y}}_{\ast }}a\left(t\right)$ (note that a similar statement was made in Section 6.2). Due to this asymptotic feature, $a\left(t\right)$ is expected to behave like const.×$e^{{\displaystyle \sqrt{{{\rm Y}}_{\ast }}t}}$ for $t\to +\infty$. In Appendix N, we rigorously prove that

$$a\left(t\right)\sim K_{\ast }{e}^{{\displaystyle \sqrt{{{\rm Y}}_{\ast }}t}}\mathrm{for} t\to +\infty ,$$

(250)

$$K_{\ast }:=exp\left({\int }_1^{+\infty }\mathrm{d}a\left({\displaystyle \frac{1}{a}}-\sqrt{-{\displaystyle \frac{{{\rm Y}}_{\ast }}{\mathcal{V}\left(a\right)}}}\right)-\sqrt{{{\rm Y}}_{\ast }}{t}_{\ast }\right).$$

The integral from 1 to $+\infty$ in the above definition of $K_{\ast }$ converges, since $1/a-\sqrt{-{{\rm Y}}_{\ast }/\mathcal{V}\left(a\right)}=O(1/a^3)$ for $a\to +\infty$ (see again Appendix N); let us also recall the expression (249) for $t_{\ast }$.

Needless to say, the relation (250) and the symmetry property $a(-t)=a\left(t\right)$ also imply

$$a\left(t\right)\sim K_{\ast }{e}^{{\displaystyle -\sqrt{{{\rm Y}}_{\ast }}t}}\mathrm{for} t\to -\infty .$$

(251)

For $k=0$, let us reconsider the spatial and spacetime line elements (16) and (17) and, referring to the dimensioned time $\tau =t/H_{\ast }$, 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 $\tau \to +\infty$ (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 $\tau \to -\infty$.

The (dimensionless) Hubble parameter $h:=\dot{a}/a$. Let us note that Equation (242) implies that h is an odd function of time:

$$h(-t)=-h\left(t\right)\mathrm{for} \mathrm{all} t\in \mathbb{R}.$$

(252)

Moreover, due to (239), we have

$$\mathrm{sign}h\left(t\right)=\mathrm{sign}t(\mathrm{in} \mathrm{particular}, h(0)=0).$$

(253)

Let us further discuss the behavior of the function $t\mapsto h\left(t\right)$; it will suffice to see what happens for $t⩾0$. With $\mathcal{H}$ as in Equation (220), we see from Equation (221) that

$$h\left(t\right)=\sqrt{\mathcal{H}\left(a\right(t\left)\right)}\mathrm{for} t\in [0,+\infty );$$

(254)

let us repeat that, while t ranges in this interval, $a\left(t\right)$ increases monotonically from $a_0$ to $+\infty$. We can combine these remarks with the available information about $\mathcal{H}$ (see the paragraph starting from Equation (220), and Figure 4); let us recall, in particular, that $\mathcal{H}$ has a unique maximum point $a_{1/2}$ (see Equation (226) and subsequent statements), where $a_{1/2}\in (a_0,a_1)$ (see (229) and (235)), and $\mathcal{H}\left(a\right)$ approaches ${{\rm Y}}_{\ast }$ for $a\to +\infty$ (see (222)).

In conclusion, the time behavior of h for $t⩾0$ is as follows: First of all, there is a unique time $t_{1/2}\in {\mathbb{R}}^{+}$ such that $a\left(t_{1/2}\right)=a_{1/2}$, and this is given by Equation (244) with $i=1/2$. Moreover,

$$t_{1/2}<t_1;$$

(255)

$$h \mathrm{is} \mathrm{strictly} \mathrm{increasing} \mathrm{on} [0,t_{1/2}]\mathrm{and} \mathrm{strictly} \mathrm{decreasing} \mathrm{on} [t_{1/2},+\infty );$$

(256)

$$\underset{t\in [0,+\infty )}{max}h\left(t\right)=h\left(t_{1/2}\right)=\sqrt{\mathcal{H}\left(a_{1/2}\right)},h\left(t\right)\to \sqrt{{{\rm Y}}_{\ast }}\mathrm{for} t\to +\infty .$$

(257)

Let us also note that $\dot{h}=\ddot{a}/a-h^2$; in particular, (recalling the relation $h\left(0\right)=0$ and Equation (143))

$$\dot{h}\left(0\right)={\displaystyle \frac{\ddot{a}\left(0\right)}{a\left(0\right)}}=-{\displaystyle \frac{{\mathcal{V}}^{\prime }\left(a_0\right)}{2a_0}}>0.$$

(258)

Preliminaries on the densities. Let us consider for $A=r,m,k,\varphi$ the corresponding densities, e.g., in the dimensionless versions ${\mathfrak{r}}_A$; these are given by Equations (170) and (203), and can be regarded as functions of the scale factor or of the dimensionless time: ${\mathfrak{r}}_A={\mathfrak{r}}_A\left(a\right)$ or ${\mathfrak{r}}_A={\mathfrak{r}}_A\left(t\right)$. In the sequel, we also refer to the total dimensionless density $\tilde{\mathfrak{r}}$, including the curvature contribution, which is the sum of all terms ${\mathfrak{r}}_A$ (see Equation (173)). Let us remark that the relation $a(-t)=a\left(t\right)$ implies the following for all $t\in \mathbb{R}$:

$${\mathfrak{r}}_A(-t)={\mathfrak{r}}_A\left(t\right)(A=r,m,k,\varphi ),\tilde{\mathfrak{r}}(-t)=\tilde{\mathfrak{r}}\left(t\right).$$

(259)

Time evolution of the total density. Let us recall again Equation (221), ensuring that

$$\tilde{\mathfrak{r}}\left(t\right)=h^2\left(t\right)$$

for all $t\in \mathbb{R}$. The results of the previous paragraph about h yield the following conclusions on the behavior of the function $t\mapsto \tilde{\mathfrak{r}}\left(t\right)$, say for $t⩾0$:

$$\tilde{\mathfrak{r}} \mathrm{is} \mathrm{strictly} \mathrm{increasing} \mathrm{on} [0,t_{1/2}]\mathrm{and} \mathrm{strictly} \mathrm{decreasing} on [t_{1/2},+\infty );$$

(260)

$$\underset{t\in [0,+\infty )}{max}\tilde{\mathfrak{r}}\left(t\right)=\tilde{\mathfrak{r}}\left(t_{1/2}\right)=\mathcal{H}\left(a_{1/2}\right),\tilde{\mathfrak{r}}\left(t\right)\to {{\rm Y}}_{\ast }\mathrm{for} t\to +\infty$$

(261)

(with $t_{1/2}$ as in (255)). Let us also note that

$$\tilde{\mathfrak{r}}\left(0\right)=h^2\left(0\right)=0,\dot{\tilde{\mathfrak{r}}}\left(0\right)=2h\left(0\right)\dot{h}\left(0\right)=0.$$

(262)

Comparison with the Planck density; classicality condition. Let us recall that, in our setting with $c=1$, the Planck length, time and mass are

$${\ell }_P={\tau }_P=\sqrt{\hslash G}\simeq 1.62\times {10}^{-33}\mathrm{cm}\simeq 5.39\times {10}^{-44}s,m_P=\sqrt{{\displaystyle \frac{\hslash }{G}}}\simeq 2.18\times {10}^{-5}\mathrm{gr}.$$

(263)

From these quantities, we define a Planck density

$${\rho }_P:={\displaystyle \frac{m_P}{{\ell }_P^3}}={\displaystyle \frac{1}{\hslash G^2}}\simeq 5.16\times {10}^{93}\mathrm{gr}/{\mathrm{cm}}^3.$$

(264)

Let us compare this quantity with the total (dimensioned) density, including the curvature contribution; this is

$$\tilde{\rho }:={\rho }_r+{\rho }_m+{\rho }_k+{\rho }_{\varphi }={\displaystyle \frac{3H_{\ast }^2}{8\pi G}}\tilde{\mathfrak{r}},$$

(265)

where $\tilde{\mathfrak{r}}$ 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 $\tilde{\rho }$ ($⩾0$) and the constant ${\rho }_P$; due to Equations (259), (261), and (265), we have

$${\displaystyle \frac{1}{{\rho }_P}}\underset{t\in \mathbb{R}}{max}\tilde{\rho }\left(t\right)={\displaystyle \frac{1}{Q}}\underset{t\in \mathbb{R}}{max}\tilde{\mathfrak{r}}\left(t\right)={\displaystyle \frac{\mathcal{H}\left(a_{1/2}\right)}{Q}},$$

(266)

where ${\displaystyle Q:=\frac{8\pi G}{3H_{\ast }^2}{\rho }_P}$. On the other hand, Equation (264) gives ${\displaystyle Q=\frac{8\pi }{3H_{\ast }^2\hslash G}}$, i.e., by Equation (263),

$$Q={\displaystyle \frac{8\pi }{3{\left(H_{\ast }{\tau }_P\right)}^2}}.$$

(267)

We note that the parameter Q is dimensionless; using for $H_{\ast },{\tau }_{\ast }$ the values in Equations (59) and (263), we obtain

$$Q\simeq 6.04\times {10}^{122}.$$

(268)

It is reasonable to prescribe the classicality condition for the cosmological model under analysis, that reads

$${\displaystyle \frac{1}{{\rho }_P}}\underset{t\in \mathbb{R}}{max}\tilde{\rho }\left(t\right)\ll 1,i.e.,{\displaystyle \frac{\mathcal{H}\left(a_{1/2}\right)}{Q}}\ll 1;$$

(269)

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 $\mu$ this condition is, essentially, a lower bound on $\mu$ (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 ${\mathfrak{r}}_A$ ($A=\varphi ,r,m,k$), viewing them as functions of the variable $a\in {\mathbb{R}}^{+}$ on the grounds of Equations (170) and (203)23; let us also remark that for all a, we have ${\mathfrak{r}}_r\left(a\right),{\mathfrak{r}}_m\left(a\right)>0$ and ${\mathfrak{r}}_k\left(a\right)⩾0$ (since we are assuming ${\Omega }_{k\ast }⩾0$).

That said, for any $a\in {\mathbb{R}}^{+}$, we have

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋0⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋0⟺a⪋a_{\varphi },a_{\varphi }:={\left({\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{{{\rm Y}}_{\ast }}}\right)}^{1/6}.$$

(270)

Again, for any $a\in {\mathbb{R}}^{+}$,

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋-{\mathfrak{r}}_r\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋-{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}⟺{{\rm Y}}_{\ast }{a}^6+{\Omega }_{r\ast }{a}^2-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi r-},$$

(271)

$$a_{\varphi r-}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in} {\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi r-}^6+{\Omega }_{r\ast }{a}_{\varphi r-}^2-{\Omega }_{r\ast }{\mu }^2=0;$$

for the proof that $a_{\varphi r-}$ exists and is unique, and for a justification of the conditions $a⪋a_{\varphi r-}$ in (271), see Appendix O. In the same appendix, we prove that

$$a_{\varphi r-}>a_0;$$

(272)

$${\displaystyle \frac{\mu }{\sqrt{1+{{\rm Y}}_{\ast }{\mu }^4/{\Omega }_{r\ast }}}}<a_{\varphi r-}<\mu \left(1-{\displaystyle \frac{{{\rm Y}}_{\ast }{\mu }^4}{2{\Omega }_{r\ast }+6{{\rm Y}}_{\ast }{\mu }^4}}\right).$$

(273)

It should be noted that, for small $\mu$, $a_{\varphi r-}$ is very close to $a_0$: Equations (209) and (273) give the bounds

$${\displaystyle \frac{1}{(1-F_{\ast }\mu )\sqrt{1+{{\rm Y}}_{\ast }{\mu }^4/{\Omega }_{r\ast }}}}<{\displaystyle \frac{a_{\varphi r-}}{a_0}}<\sqrt{1+N_{\ast }\mu }\left(1-{\displaystyle \frac{{{\rm Y}}_{\ast }{\mu }^4}{2{\Omega }_{r\ast }+6{{\rm Y}}_{\ast }{\mu }^4}}\right)$$

(274)

(with $N_{\ast },F_{\ast }$ as in Equation (208)). For a family of models as in (217), Equation (273) implies the following for $\mu \to 0^{+}$:

$$a_{\varphi r-}=\mu \left(1-{\displaystyle \frac{{{\rm Y}}_{\ast \ast }{\mu }^4}{2{\Omega }_{r\ast }}}+O\left({\mu }^6\right)\right)$$

(275)

(since both the lower and the upper bounds on $a_{\varphi r-}$ in (273) behave like the right-hand side of (275)); similarly, we infer from Equation (274) that

$${\displaystyle \frac{a_{\varphi r-}}{a_0}}=1+{\displaystyle \frac{{\Omega }_{m\ast }}{2{\Omega }_{r\ast }}}\mu +O\left({\mu }^2\right).$$

(276)

To continue, we note that, for any $a\in {\mathbb{R}}^{+}$, we have

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋{\mathfrak{r}}_r\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}⟺{{\rm Y}}_{\ast }{a}^6-{\Omega }_{r\ast }{a}^2-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi r+},$$

(277)

$$a_{\varphi r+}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in} {\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi r+}^6-{\Omega }_{r\ast }{a}_{\varphi r+}^2-{\Omega }_{r\ast }{\mu }^2=0;$$

the existence and uniqueness of $a_{\varphi r+}$, as well as the above conditions $a⪋a_{\varphi r+}$, are justified with arguments similar to those employed in connection with Equation (271) and $a_{\varphi r-}$.

Using again similar arguments, one proves the following for any $a\in {\mathbb{R}}^{+}$24:

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋-{\mathfrak{r}}_m\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋-{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}⟺{{\rm Y}}_{\ast }{a}^6+{\Omega }_{m\ast }{a}^3-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi m-},$$

$$a_{\varphi m-}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in} {\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi m-}^6+{\Omega }_{m\ast }{a}_{\varphi m-}^3-{\Omega }_{r\ast }{\mu }^2=0,$$

(278)

$$\mathrm{i}.\mathrm{e}.,a_{\varphi m-}={\left({\displaystyle \frac{-{\Omega }_{m\ast }+\sqrt{{\Omega }_{m\ast }^2+4{{\rm Y}}_{\ast }{\Omega }_{r\ast }{\mu }^2}}{2{{\rm Y}}_{\ast }}}\right)}^{1/3};$$

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋{\mathfrak{r}}_m\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}⟺{{\rm Y}}_{\ast }{a}^6-{\Omega }_{m\ast }{a}^3-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi m+},$$

$$a_{\varphi m+}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in} {\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi m+}^6-{\Omega }_{m\ast }{a}_{\varphi m+}^3-{\Omega }_{r\ast }{\mu }^2=0,$$

(279)

$$\mathrm{i}.\mathrm{e}.,a_{\varphi m+}={\left({\displaystyle \frac{{\Omega }_{m\ast }+\sqrt{{\Omega }_{m\ast }^2+4{{\rm Y}}_{\ast }{\Omega }_{r\ast }{\mu }^2}}{2{{\rm Y}}_{\ast }}}\right)}^{1/3};$$

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋-{\mathfrak{r}}_k\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}⟺{{\rm Y}}_{\ast }{a}^6+{\Omega }_{k\ast }{a}^4-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi k-},$$

(280)

$$a_{\varphi k-}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in}{\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi k-}^6+{\Omega }_{k\ast }{a}_{\varphi k-}^4-{\Omega }_{r\ast }{\mu }^2=0;$$

$${\mathfrak{r}}_{\varphi }\left(a\right)⪋{\mathfrak{r}}_k\left(a\right)⟺-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{{\rm Y}}_{\ast }⪋{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}⟺{{\rm Y}}_{\ast }{a}^6-{\Omega }_{k\ast }{a}^4-{\Omega }_{r\ast }{\mu }^2⪋0⟺a⪋a_{\varphi k+},$$

(281)

$$a_{\varphi k+}:= \mathrm{the} \mathrm{unique} \mathrm{point} \mathrm{in} {\mathbb{R}}^{+} \mathrm{such} \mathrm{that} {{\rm Y}}_{\ast }{a}_{\varphi k+}^6-{\Omega }_{k\ast }{a}_{\varphi k+}^4-{\Omega }_{r\ast }{\mu }^2=0$$

Finally, for any $a\in {\mathbb{R}}^{+}$, we have

$${\mathfrak{r}}_r\left(a\right)⪌{\mathfrak{r}}_m\left(a\right)⟺{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}⪌{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}⟺a⪋a_{rm},a_{rm}:={\displaystyle \frac{{\Omega }_{r\ast }}{{\Omega }_{m\ast }}};$$

(282)

$${\mathfrak{r}}_r\left(a\right)⪌{\mathfrak{r}}_k\left(a\right)⟺{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}⪌{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}⟺a⪋a_{rk},a_{rk}:=\sqrt{{\displaystyle \frac{{\Omega }_{r\ast }}{{\Omega }_{k\ast }}}};$$

(283)

$${\mathfrak{r}}_m\left(a\right)⪌{\mathfrak{r}}_k\left(a\right)⟺{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}⪌{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}}⟺a⪋a_{mk},a_{mk}:={\displaystyle \frac{{\Omega }_{m\ast }}{{\Omega }_{k\ast }}}$$

(284)

(intending ${\Omega }_{r\ast }/{\Omega }_{k\ast }:={\Omega }_{m\ast }/{\Omega }_{k\ast }:=+\infty$ if ${\Omega }_{k\ast }=0$).

Let i denote anyone of the above subscripts $\varphi$, $\varphi r-,\varphi r+,\dots ,rk,mk$. Of course, given the motion $t\in \mathbb{R}\mapsto a\left(t\right)$, if $a_i>a_0$, there is a unique time $t_i>0$ such that $a\left(t_i\right)=a_i$, and this is given by Equation (244); for any $t⩾0$, we have that $a\left(t\right)⪋a_i$⟺ $t⪋t_i$.

On the computation of certain times. Let us consider a time $t_i$ admitting an integral representation of the form (244) $t_i={\int }_{a_0}^{a_i}da/\sqrt{-\mathcal{V}\left(a\right)}$, with $a_i>a_0$.

We have already noted that $1/\sqrt{-\mathcal{V}\left(a\right)}$ diverges (like $1/\sqrt{a-a_0}$) for $a\to {\left(a_0\right)}^{+}$; 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 $\mu \to 0^{+}$ for the times $t_{1/2},t_1$, and $t_{\varphi r-}$, which are determined as in (244) with $a_i=a_{1/2},a_1$, and $a_{\varphi r-}$ (see Equations (212), (226), and (271)). In this limit, it is found that

$$t_{1/2}={\displaystyle \frac{K_{1/2}}{\sqrt{{\Omega }_{r\ast }}}}{\mu }^2(1+O\left(\mu \right)),K_{1/2}:={\displaystyle \frac{\sqrt{3}}{4}}+\mathrm{arcsinh}\sqrt{{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{1}{2}}}\simeq 0.7623;$$

(285)

$$t_1={\displaystyle \frac{K_1}{\sqrt{{\Omega }_{r\ast }}}}{\mu }^2(1+O\left(\mu \right)),K_1:={\displaystyle \frac{1}{\sqrt{2}}}+\mathrm{arcsinh}\sqrt{{\displaystyle \frac{1}{\sqrt{2}}}-{\displaystyle \frac{1}{2}}}\simeq 1.148;$$

(286)

$$t_{\varphi r-}={\displaystyle \frac{\sqrt{{\Omega }_{m\ast }}}{{\Omega }_{r\ast }}}{\mu }^{5/2}(1+O\left(\mu \right)).$$

(287)

Time evolution of the scalar field. Equation (136) with $\sigma =-1,d=3$ and Equation (199), ensure that everywhere on the time domain $(-\infty ,+\infty )$,

$$\dot{\varphi }={\displaystyle \frac{s\sqrt{2{\Omega }_{r\ast }}\mu }{a^3}},\mathrm{sign}\dot{\varphi }=\mathrm{constant}=s.$$

(288)

In particular,

$$\dot{\varphi }\left(0\right)={\displaystyle \frac{s\sqrt{2{\Omega }_{r\ast }}\mu }{a_0^3}};$$

(289)

the bounds (209) on $a_0$, involving $F_{\ast },N_{\ast }$ as in Equation (208), imply

$${\displaystyle \frac{\sqrt{2{\Omega }_{r\ast }}}{{\mu }^2}}·{\displaystyle \frac{1}{{(1-F_{\ast }\mu )}^3}}<\left|\dot{\varphi }\left(0\right)\right|<{\displaystyle \frac{\sqrt{2{\Omega }_{r\ast }}}{{\mu }^2}}·{(1+N_{\ast }\mu )}^{3/2},$$

(290)

showing that $\dot{\varphi }\left(0\right)$ is very large for small $\mu$. From (288) and Equations (250) and (251) on the large t behavior of $a\left(t\right)$, we also infer that

$$\dot{\varphi }\left(t\right)\sim {\displaystyle \frac{s\sqrt{2{\Omega }_{r\ast }}\mu }{K_{\ast }^3{e}^{{\displaystyle \mp 3\sqrt{{{\rm Y}}_{\ast }}t}}}}\to 0\mathrm{for} t\to \mp \infty .$$

(291)

Again from (288), we see that the function $t\in \mathbb{R}\mapsto \varphi \left(t\right)$ is strictly increasing or decreasing in the two cases $s=\pm 1$; 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 $\sigma =-1,d=3$ and $t^{\prime }=-\infty ,t^{\prime \prime }=0$ or $t^{\prime }=0,t^{\prime \prime }=+\infty$, and recalling again Equation (199), we conclude that

$$\varphi \left(t\right)\to \varphi \left(0\right)\mp s\Delta \varphi \mathrm{for} t\to \mp \infty ,\Delta \varphi :=\sqrt{2{\Omega }_{r\ast }}\mu {\int }_{a_0}^{+\infty }{\displaystyle \frac{da}{\sqrt{-a^6\mathcal{V}\left(a\right)}}}.$$

(292)

According to (202), for $a\to +\infty$, we have $1/\sqrt{-a^6\mathcal{V}\left(a\right)}\sim 1/\left(\sqrt{{{\rm Y}}_{\ast }}{a}^4\right)$ (while, for $a\to {\left(a_0\right)}^{+}$, we have $1/\sqrt{-a^6\mathcal{V}\left(a\right)}\sim 1/\left(a_0^3\sqrt{-{\mathcal{V}}^{\prime }\left(a_0\right)}\sqrt{a-a_0}\right)$ ); so, the integral defining $\Delta \varphi$ converges.

In Appendix S, we derive precise lower and upper bounds on $\Delta \varphi$. For a family of models as in (217), these bounds ensure that, in the limit $\mu \to 0^{+}$,

$$\Delta \varphi ={\displaystyle \frac{\pi }{\sqrt{2}}}(1+O\left({\mu }^{\vartheta }\right))\mathrm{for} \mathrm{any} \vartheta \in \left({\displaystyle 0,\frac{1}{4}}\right).$$

(293)

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:

$$\mathrm{WEC} \mathrm{holds}⟺\mathfrak{r}⩾0,\mathfrak{r}+\mathfrak{p}⩾0;$$

(294)

$$\mathrm{SEC} \mathrm{holds}⟺\mathfrak{r}+3\mathfrak{p}⩾0,\mathfrak{r}+\mathfrak{p}⩾0.$$

(295)

Again, in the case under analysis, the total dimensionless density and pressure $\mathfrak{r},\mathfrak{p}$ are determined by Equations (170), (203), and (172), which imply

$$\mathfrak{r}=-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{a^6}}+{\displaystyle \frac{{\Omega }_{r\ast }}{a^4}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}+{{\rm Y}}_{\ast },\mathfrak{r}+\mathfrak{p}=-{\displaystyle \frac{2{\Omega }_{r\ast }{\mu }^2}{a^6}}+{\displaystyle \frac{4{\Omega }_{r\ast }}{3a^4}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}},$$

(296)

$$\mathfrak{r}+3\mathfrak{p}=-{\displaystyle \frac{4{\Omega }_{r\ast }{\mu }^2}{a^6}}+{\displaystyle \frac{2{\Omega }_{r\ast }}{3a^4}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a^3}}-2{{\rm Y}}_{\ast }.$$

Comparing the above results with the expressions (201), (210), and (225) for the function $\mathcal{V}$ and for the derivatives ${\mathcal{V}}^{\prime },{\mathcal{H}}^{\prime }$ we see that25

$$\mathfrak{r}=-{\displaystyle \frac{\mathcal{V}\left(a\right)}{a^2}}-{\displaystyle \frac{{\Omega }_{k\ast }}{a^2}},\mathfrak{r}+\mathfrak{p}=-{\displaystyle \frac{1}{3}}a{\mathcal{H}}^{\prime }\left(a\right)-{\displaystyle \frac{2{\Omega }_{k\ast }}{3a^2}},\mathfrak{r}+3\mathfrak{p}={\displaystyle \frac{{\mathcal{V}}^{\prime }\left(a\right)}{a}}.$$

(297)

In the sequel of this paragraph, just for simplicity, we consider the zero curvature case

$${\Omega }_{k\ast }=0.$$

(298)

Due to this assumption, the signs of the quantities in Equation (297) are determined by the signs of $\mathcal{V}$, ${\mathcal{V}}^{\prime }$, and ${\mathcal{H}}^{\prime }$, 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

$$\mathfrak{r}\left\{\begin{array}{cc}=0\hfill & \mathrm{for} a=a_0,\hfill \\ >0\hfill & \mathrm{for} a\in (a_0,+\infty ),\hfill \end{array}\right.\mathfrak{r}+\mathfrak{p}\left\{\begin{array}{cc}<0\hfill & \mathrm{for} a\in [a_0,a_{1/2}),\hfill \\ =0\hfill & \mathrm{for} a=a_{1/2},\hfill \\ >0\hfill & \mathrm{for} a\in (a_{1/2},+\infty ),\hfill \end{array}\right.$$

(299)

$$\mathfrak{r}+3\mathfrak{p}\left\{\begin{array}{cc}<0\hfill & \mathrm{for} a\in [a_0,a_1)\cup (a_2,+\infty ),\hfill \\ =0\hfill & \mathrm{for} a=a_1,a_2,\hfill \\ >0\hfill & \mathrm{for} a\in (a_1,a_2).\hfill \end{array}\right.$$

These facts and (294) and (295) imply

$$\mathrm{WEC} \mathrm{fails} \mathrm{for} a\in [a_0,a_{1/2}), \mathrm{holds} \mathrm{for} a\in [a_{1/2},+\infty );$$

(300)

$$\mathrm{SEC} \mathrm{fails} \mathrm{for} a\in [a_0,a_1)\cup (a_2,+\infty ), \mathrm{holds} \mathrm{for} a\in [a_1,a_2].$$

(301)

The equivalent statements in terms of the dimensionless time are obvious; for example, SEC fails for $t\in (-\infty ,-t_2)\cup (-t_1,t_1)\cup (t_2,+\infty )$ and holds for $t\in [-t_2,-t_1]\cup [t_1,t_2]$.

For extending the above results to the case ${\Omega }_{k\ast }>0$, 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 $\mathcal{V},{\mathcal{V}}^{\prime }$, and ${\mathcal{H}}^{\prime }$.

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 $\Pi _{\ast }\ne 0$  (Small), ${\Omega }_{k\ast }=0$ 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

$${\Omega }_{r\ast }=9.2\times {10}^{-5},{\Omega }_{m\ast }=0.31-{\Omega }_{r\ast },{\Omega }_{k\ast }=0(\mathrm{i}.\mathrm{e}.,k=0),{{\rm Y}}_{\ast }=0.69+{\Omega }_{r\ast }{\mu }^2$$

(302)

and

$$0<\mu ⋘1.$$

The value of the parameter $\mu$ 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 $\mu ⋘1$ is intended appropriately. We also remark that Equation (179) for the present-time normalized energy density of the scalar field gives in this case

$${\Omega }_{\varphi }\left(t_{\ast }\right)=0.69.$$

(303)

The above values of ${\Omega }_{r\ast }$, ${\Omega }_{m\ast }$, and ${\Omega }_{\varphi }\left(t_{\ast }\right)$ 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 $a\mapsto \mathcal{V}\left(a\right),\mathcal{H}\left(a\right)$ 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

$$a_0\simeq \mu ,a_{1/2}\simeq \sqrt{3/2}\mu ,a_1\simeq \sqrt{2}\mu ,a_1<a_2;$$

(304)

in particular, $a_0$ is the minimum of the zero-energy solution $t\mapsto a\left(t\right)$ analyzed in Section 6.6, to which we systematically refer from now on. We now fix the value of $\mu$ on the grounds of the classicality condition (269); this involves the ratio

$${\displaystyle \frac{1}{{\rho }_P}}\underset{t\in \mathbb{R}}{max}\tilde{\rho }\left(t\right)={\displaystyle \frac{\mathcal{H}\left(a_{1/2}\right)}{Q}},$$

(305)

with ${\rho }_P$ the Planck density (see Equation (264)) and Q as in Equations (267) and (268). However, due to Equations (232) and the smallness of $\mu$,

$$\mathcal{H}\left(a_{1/2}\right)\simeq {\displaystyle \frac{4{\Omega }_{r\ast }}{27{\mu }^4}};$$

(306)

therefore,

$${\displaystyle \frac{1}{{\rho }_P}}\underset{t\in \mathbb{R}}{max}\tilde{\rho }\left(t\right)\simeq {\displaystyle \frac{4{\Omega }_{r\ast }}{27Q{\mu }^4}}\simeq {\displaystyle \frac{2.26\times {10}^{-128}}{{\mu }^4}},$$

(307)

where, in the last step, we used the numerical values in Equations (268) and (302) for Q and ${\Omega }_{r\ast }$.

The classicality condition (269) requires the ratio (307) to be very small; we now choose an extremely small value for $\mu$, which, however, is sufficiently large to fulfill (269). We will adopt the value

$$\mu :={10}^{-31},$$

(308)

implying

$${\displaystyle \frac{1}{{\rho }_P}}\underset{t\in \mathbb{R}}{max}\tilde{\rho }\left(t\right)\simeq 2.26\times {10}^{-4}.$$

(309)

From here to the end of the present Section 6.7, we stick to the choices (302) and (308) for ${\Omega }_{r\ast },{\Omega }_{m\ast },{\Omega }_{k\ast },{{\rm Y}}_{\ast }$, and $\mu$, confirming that all conditions (200) (204) and (211) (234) are fulfilled; whenever necessary, we use the value (59) for $H_{\ast }$.

In the sequel, we report many distinguished values $a_i$ of the scale factor and the corresponding times $t_i$ (or ${\tau }_i$), 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 $a_0,\dots ,a_2$ 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 $a\left(t_i\right)=a_i$ and its equivalent ${\tau }_i=t_i/H_{\ast }$ in terms of cosmic time. Here are the values:

$$\begin{array}{ccccccccccc}\hfill a_0& \simeq {10}^{-31}\hfill & \hfill ,& \hfill & \hfill t_0& =0\hfill & \hfill ,& \hfill & \hfill {\tau }_0& =0s\hfill & ;\\ \hfill a_{1/2}& \simeq 1.22\times {10}^{-31}\hfill & \hfill ,& \hfill & \hfill t_{1/2}& \simeq 7.95\times {10}^{-61}\hfill & \hfill ,& \hfill & \hfill {\tau }_{1/2}& \simeq 3.65\times {10}^{-43}s\hfill & ;\\ \hfill a_1& \simeq 1.41\times {10}^{-31}\hfill & \hfill ,& \hfill & \hfill t_1& \simeq 1.20\times {10}^{-60}\hfill & \hfill ,& \hfill & \hfill {\tau }_1& \simeq 5.49\times {10}^{-43}s\hfill & ;\\ \hfill a_2& \simeq 0.608\hfill & \hfill ,& \hfill & \hfill t_2& \simeq 0.528\hfill & \hfill ,& \hfill & \hfill {\tau }_2& \simeq 7.68\times {10}^{9}yr\hfill & .\end{array}$$

(310)

Let us recall that the function $t\mapsto a\left(t\right)$ is even and, for $0⩽t<+\infty$, increases from the minimum value $a_0$ to $+\infty$. 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 $[0,+\infty )$.

The present time. The present time $t_{\ast }$, individuated by the conditions $t_{\ast }>0$ and $a\left(t_{\ast }\right)=1$, is determined by Equation (249). It is found that $t_{\ast }$ and its cosmic time equivalent ${\tau }_{\ast }=t_{\ast }/H_{\ast }$ have the values

$$t_{\ast }\simeq 0.955,{\tau }_{\ast }\simeq 13.9\times {10}^{9}yr.$$

(311)

${\tau }_{\ast }$ 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 ($i=\varphi ,\varphi r\mp ,\varphi m\mp ,rm$), 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 ${\tau }_i=t_i/H_{\ast }$. These special values, arranged in increasing order, are as follows:

$$\begin{array}{ccccccccccc}\hfill a_{\varphi r-}& \simeq (1+1.68\times {10}^{-28})a_0\hfill & \hfill ,& \hfill & \hfill t_{\varphi r-}& =1.91\times {10}^{-74}\hfill & \hfill ,& \hfill & \hfill {\tau }_{\varphi r-}& =8.78\times {10}^{-57}s\hfill & ;\\ \hfill a_{\varphi m-}& \simeq 1.44\times {10}^{-22}\hfill & \hfill ,& \hfill & \hfill t_{\varphi m-}& \simeq 1.08\times {10}^{-42}\hfill & \hfill ,& \hfill & \hfill {\tau }_{\varphi m-}& \simeq 4.94\times {10}^{-25}s\hfill & ;\\ \hfill a_{\varphi }& \simeq 1.05\times {10}^{-11}\hfill & \hfill ,& \hfill & \hfill t_{\varphi }& \simeq 5.74\times {10}^{-21}\hfill & \hfill ,& \hfill & \hfill {\tau }_{\varphi }& \simeq 2.63\times {10}^{-3}s\hfill & ;\\ \hfill a_{rm}& \simeq 2.97\times {10}^{-4}\hfill & \hfill ,& \hfill & \hfill t_{rm}& \simeq 3.59\times {10}^{-6}\hfill & \hfill ,& \hfill & \hfill {\tau }_{rm}& \simeq 52.2\times {10}^{3}yr\hfill & ;\\ \hfill a_{\varphi r+}& \simeq 0.107\hfill & \hfill ,& \hfill & \hfill t_{\varphi r+}& \simeq 4.20\times {10}^{-2}\hfill & \hfill ,& \hfill & \hfill {\tau }_{\varphi r+}& \simeq 0.611\times {10}^{9}yr\hfill & ;\\ \hfill a_{\varphi m+}& \simeq 0.766\hfill & \hfill ,& \hfill & \hfill t_{\varphi m+}& \simeq 0.707\hfill & \hfill ,& \hfill & \hfill {\tau }_{\varphi m+}& \simeq 10.3\times {10}^{9}yr.\hfill \end{array}$$

(312)

Let us consider the evolution of the cosmological model for $\tau ⩾0$, recalling that the behavior for $\tau ⩽0$ is specular due to $a(-\tau )=a\left(\tau \right)$; using the inequalities (270), (271), (277)–(279) and (282), we obtain the following description:

0.

$sign\left({\mathfrak{r}}_{\varphi }\right)= \mathrm{sign} (a-a_{\varphi })= \mathrm{sign} (\tau -{\tau }_{\varphi })$. We repeat that $a_{\varphi }\simeq 1.05\times {10}^{-11}$ and ${\tau }_{\varphi }\simeq 2.63\times {10}^{-3}s$.

1.

$a_0⩽a⩽a_{\varphi r-}$, i.e., $0⩽\tau ⩽{\tau }_{\varphi r-}$: a field dominated era, with ${\mathfrak{r}}_{\varphi }<0$ and $|{\mathfrak{r}}_{\varphi }|⩾{\mathfrak{r}}_r,{\mathfrak{r}}_m$;

we repeat that $a_0\simeq {10}^{-31}$, $a_{\varphi r-}\simeq (1+1.68\times {10}^{-28})a_0$ and ${\tau }_{\varphi r-}\simeq 8.78\times {10}^{-57}s$.

2.

$a_{\varphi r-}⩽a⩽a_{rm}$, i.e., ${\tau }_{\varphi r-}⩽\tau ⩽{\tau }_{rm}$: a radiation dominated era, with ${\mathfrak{r}}_r⩾\left|{\mathfrak{r}}_{\varphi }\right|,{\mathfrak{r}}_m$.

We repeat that $a_{rm}\simeq 2.97\times {10}^{-4}$ and ${\tau }_{rm}\simeq 52.2\times {10}^{3}yr$; according to 0., ${\mathfrak{r}}_{\varphi }$ changes sign shortly after the beginning of this era.

3.

$a_{rm}⩽a⩽a_{\varphi m+}$, i.e., ${\tau }_{rm}⩽\tau ⩽{\tau }_{\varphi m+}$: a matter dominated era, with ${\mathfrak{r}}_{\varphi }>0$ and ${\mathfrak{r}}_m⩾{\mathfrak{r}}_{\varphi },{\mathfrak{r}}_r$; we repeat that $a_{\varphi m+}\simeq 0.766$ and ${\tau }_{\varphi m+}\simeq 10.3\times {10}^9\mathrm{yr}$.

4.

$a_{\varphi m+}⩽a<+\infty$, i.e., ${\tau }_{\varphi m+}⩽\tau <+\infty$: a field dominated era, with ${\mathfrak{r}}_{\varphi }>0$ and ${\mathfrak{r}}_{\varphi }⩾{\mathfrak{r}}_m,{\mathfrak{r}}_r$. This era contains the present time with $a=1$ and $\tau ={\tau }_{\ast }\simeq 13.9\times {10}^9\mathrm{yr}$.

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 $\mu$, 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 $a_{rm}$ and ${\tau }_{rm}$ 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 $\tau \to +\infty$, 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 $[0,{\tau }_{\varphi r-}]$; 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 $a_{\varphi r-}$ 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 $a_{\varphi r-}/a_0$ 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 $\mu$, 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 $O\left({\mu }^2\right)$, are practically indistinguishable.

By definition, $a_2$ is the second positive solution of the equation ${\mathcal{V}}^{\prime }\left(a\right)=0$ (or of the equivalent algebraic equation $\mathcal{Q}\left(a\right)=0$, where $\mathcal{Q}$ is the polynomial in Equation (A123) of Appendix L); the value of $a_2$ was estimated by solving numerically this equation. The value of $a_{\varphi r+}$ was estimated by solving numerically the algebraic equation in (277).

The values of $a_i$ for $i=\varphi ,\varphi m\mp ,rm$ were obtained from the analytic expressions in (270), (278), (279), and (282).

For all choices of i in (310)–(312) we have $t_i={\int }_{a_0}^{a_i}da/\sqrt{-\mathcal{V}\left(a\right)}$, as in Equation (244) (note that the present time $t_{\ast }$ corresponds to $a_{\ast }:=1$). 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 $i=1/2,1,\varphi r-$ (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 $i=2,\ast ,\varphi r+,\varphi m\mp ,rm$ (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 $i=1/2,1,\varphi r-$, these values are also indistinguishable from those provided by the asymptotic expressions (285)–(287) neglecting the reminders $O\left({\mu }^2\right)$.

Other topics. Of course, all statements in the last two paragraphs of Section 6.6 about the field history $t\mapsto \varphi \left(t\right)$ and the energy conditions apply to the present framework.

6.8. The Phantom Case with $\Pi _{\ast }\ne 0$ (Small) and ${\Omega }_{k\ast }<0$: Sketching the Basic Results

Throughout the present Section 6.8, we keep the assumptions (198)–(200), but we prescribe

$${\Omega }_{k\ast }<0$$

(313)

(i.e., $k>0$). We also put three smallness conditions on $\mu$; the first one is Equation (211), the other two are

$$2\sqrt{-{\displaystyle \frac{{\Omega }_{k\ast }}{{\Omega }_{r\ast }}}}\mu <1,$$

(314)

$$-{\displaystyle \frac{{\Omega }_{k\ast }}{{\Omega }_{m\ast }}}\mu <1.$$

(315)

The potential $\mathcal{V}$. Again, this is given by Equation (201) and behaves as in Equation (202) for $a\to 0^{+}$ and $a\to +\infty$. The following facts about $\mathcal{V}$ are proved in Appendix U.

($\alpha$)

There are just two points $a_1,a_2$ as in Equation (212), reproduced here:

$$a_1,a_2\in {\mathbb{R}}^{+},a_1<a_2,{\mathcal{V}}^{\prime }\left(a_1\right)={\mathcal{V}}^{\prime }\left(a_2\right)=0.$$

Moreover, we have again the inequalities (213)

$${\mathcal{V}}^{\prime }\left(a\right)<0\mathrm{for} a\in (0,a_1)\cup (a_2,+\infty ),{\mathcal{V}}^{\prime }\left(a\right)>0\mathrm{for} a\in (a_1,a_2),$$

so that $a_1$ is a local minimum point, and $a_2$ is a local maximum point for $\mathcal{V}$. Also, we have the (rough) inequalities

$$a_1<\sqrt{2}\mu <a_2.$$

(316)

($\beta$)

Defining $M_{\ast }$, $G_{\ast }$ as in Equation (215), we have again the inequalities $M_{\ast }>0,G_{\ast }>0$, $G_{\ast }\mu <1$ and the bounds (216) $\sqrt{2}\mu /\sqrt{1+M_{\ast }\mu }<a_1<\sqrt{2}\mu \left(1-G_{\ast }\mu \right)$.

($\gamma$)

We have

$$\mathcal{V}\left(a_1\right)<0.$$

(317)

($\delta$)

Let

$$W_{\ast }:=-{\displaystyle \frac{{\Omega }_{r\ast }{\mu }^2}{{\left(a_2\right)}^4}}+{\displaystyle \frac{{\Omega }_{r\ast }}{{\left(a_2\right)}^2}}+{\displaystyle \frac{{\Omega }_{m\ast }}{a_2}}+{{\rm Y}}_{\ast }{\left(a_2\right)}^2;$$

(318)

then,

$$W_{\ast }>0.$$

(319)

The sign of $\mathcal{V}\left(a_2\right)$ is determined by a comparison between $W_{\ast }$ and ${\Omega }_{k\ast }$, and it determines the sign of $\mathcal{V}$ on ${\mathbb{R}}^{+}$ as in the forthcoming items (${\delta }_1$) (${\delta }_2$) (${\delta }_3$).

(${\delta }_1$)

We have the equivalence

$$-W_{\ast }<{\Omega }_{k\ast }<0⟺\mathcal{V}\left(a_2\right)<0.$$

(320)

In the case (320), there is a unique point $a_0$ such that

$$a_0\in {\mathbb{R}}^{+},\mathcal{V}\left(a_0\right)=0;$$

(321)

moreover,

$$a_0<a_1,$$

(322)

$$\mathcal{V}\left(a\right)>0 \mathrm{for} a\in (0,a_0),\mathcal{V}\left(a\right)<0 \mathrm{for} a\in (a_0,+\infty )$$

(323)

(so, we again have the situation described by Figure 3).

(${\delta }_2$)

We have the equivalence

$${\Omega }_{k\ast }=-W_{\ast }⟺\mathcal{V}\left(a_2\right)=0.$$

(324)

In the case (324), there is a unique point $a_0$ such that

$$a_0\in {\mathbb{R}}^{+},a_0\ne a_2,\mathcal{V}\left(a_0\right)=0;$$

(325)

(i.e., $\mathcal{V}$ has just two zeroes $a_0,a_2$ in ${\mathbb{R}}^{+}$). We have again the inequality (322) $a_0<a_1$. Moreover,

$$\mathcal{V}\left(a\right)>0 \mathrm{for} a\in (0,a_0),\mathcal{V}\left(a\right)<0 \mathrm{for} a\in (a_0,a_2)\cup (a_2,+\infty )$$

(326)

(see Figure 6).

(${\delta }_3$)

We have the equivalence

$${\Omega }_{k\ast }<-W_{\ast }⟺\mathcal{V}\left(a_2\right)>0.$$

(327)

In the case (327), there is a unique triplet of points $a_0,a_{3/2},a_{5/2}$ such that

$$a_0<a_{3/2}<a_{5/2}\in {\mathbb{R}}^{+},\mathcal{V}\left(a_0\right)=\mathcal{V}\left(a_{3/2}\right)=\mathcal{V}\left(a_{5/2}\right)=0;$$

(328)

moreover,

$$a_0<a_1<\sqrt{2}\mu <a_{3/2}<a_2<a_{5/2},$$

(329)

$$\mathcal{V}\left(a\right)>0 \mathrm{for} a\in (0,a_0)\cup (a_{3/2},a_{5/2}),\mathcal{V}\left(a\right)<0 \mathrm{for} a\in (a_0,a_{3/2})\cup (a_{5/2},+\infty )$$

(330)

(see Figure 7).

($\epsilon$)

In any one of the cases (${\delta }_1$)–(${\delta }_3$), we have again the inequality (207) $a_0<1$.

($\zeta$)

Let

$${\mathcal{N}}_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }+{{\rm Y}}_{\ast }{\mu }^3}{{\Omega }_{r\ast }}},{\mathcal{F}}_{\ast }:={\displaystyle \frac{{\Omega }_{m\ast }+{\Omega }_{k\ast }\mu +{{\rm Y}}_{\ast }{\mu }^3}{2{\Omega }_{r\ast }+3{\Omega }_{m\ast }\mu +6{{\rm Y}}_{\ast }{\mu }^4}};$$

(331)

then, ${\mathcal{N}}_{\ast }>0,{\mathcal{F}}_{\ast }>0$ and ${\mathcal{F}}_{\ast }\mu <1$. Moreover, in any one of the cases (${\delta }_1$)–(${\delta }_3$), we have

$${\displaystyle \frac{\mu }{\sqrt{1+{\mathcal{N}}_{\ast }\mu }}}<a_0<\mu (1-{\mathcal{F}}_{\ast }\mu ).$$

(332)

Behavior of the scale factor. Let us refer to the cases (${\delta }_1$)–(${\delta }_3$) in the previous paragraph, which correspond, respectively, to the assumptions (320), (324) and (327). Recalling the definitions (146) of ${\mathcal{V}}_{=0},{\mathcal{V}}_{<0}$, we have the following:

• Under the assumption (320), we have ${\mathcal{V}}_{=0}=\left\{a_0\right\}$ and ${\mathcal{V}}_{<0}=(a_0,+\infty )$. The behavior of a (maximal) zero-energy solution $a\left(\right)$$:t\mapsto a\left(t\right)$ is similar to that described in the previous Section 6.6. So, $a\left(\right)$ is defined on the whole real axis $(-\infty ,+\infty )$ with image $[a_0,+\infty )$. There is a Big Bounce at a time that we take conventionally to be $t=0$, with $a\left(0\right)=a_0$; moreover, $a(-t)=a\left(t\right)$ and $a\left(t\right)\to +\infty$ (exponentially) for $t\to \mp \infty$.
• Under the assumption (324), we have ${\mathcal{V}}_{=0}=\{a_0,a_2\}$ and ${\mathcal{V}}_{<0}=(a_0,a_2)\cup (a_2,+\infty )$, with ${\mathcal{V}}^{\prime }\left(a_0\right)<0$ and ${\mathcal{V}}^{\prime }\left(a_2\right)=0$; 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., $a\left(0\right)=a_0$; then, $a(-t)=a\left(t\right)$ for all t. The function $a\left(\right)$ 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 $\mp \infty$ (see item (vi) in Section 5.4). In conclusion, $a\left(\right)$ has domain $(-\infty ,+\infty )$, range $[a_0,a_2)$ and $a_2={lim}_{t\to \mp \infty }a\left(t\right)$. See Figure 8, where we also indicated the time $t_1>0$ such that $a\left(t_1\right)=a_1$; recalling item (viii) in Section 5.4, we see that $t_1$ is an inflection point for the scale factor since ${\mathcal{V}}^{\prime }\left(a_1\right)=0$ and ${\mathcal{V}}^{\prime }$ has different signs on the left and on the right of $a_1$.
  Second type: $a\left(\right)$ has domain $(-\infty ,+\infty )$ and range $(a_2,+\infty )$; $\dot{a}\left(t\right)>0$ for all real t, or $\dot{a}\left(t\right)<0$ for all real t. If $\dot{a}>0$, we have $a\left(t\right)\to a_2$ for $t\to -\infty$ and $a\left(t\right)\to +\infty$ for $t\to +\infty$; if $\dot{a}<0$, the values of the limits for $t\to \mp \infty$ 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 $a\left(\right)$ to diverge is related to item (vii) in the same section). In Figure 8, we illustrate the case $\dot{a}>0$.
  Third type: This corresponds to the equilibrium solution $a\left(t\right)=a_2$ for all $t\in (-\infty ,+\infty )$ (see item (v) in Section 5.4).
• Under the assumption (327), we have ${\mathcal{V}}_{=0}=\{a_0,a_{3/2},a_{5/2}\}$ and ${\mathcal{V}}_{<0}=(a_0,a_{3/2})\cup (a_{5/2},+\infty )$, with ${\mathcal{V}}^{\prime }\left(a_0\right)<0$, ${\mathcal{V}}^{\prime }\left(a_{3/2}\right)>0$ and ${\mathcal{V}}^{\prime }\left(a_{5/2}\right)<0$; there are two types of zero-energy solutions, described hereafter.
  First type: The function $a\left(\right)$ has domain $(-\infty ,+\infty )$ and range $[a_0,a_{3/2}]$; it oscillates periodically with period

  $$T:=2{\int }_{a_0}^{a_{3/2}}{\displaystyle \frac{\mathrm{d}a}{\sqrt{-\mathcal{V}\left(a\right)}}}.$$

  (333)
  (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 $a\left(\right)$ has inflection points at the infinitely many times t such that $a\left(t\right)=a_1$; these times are not indicated in the figure).
  Second type: The function $a\left(\right)$ has domain $(-\infty ,+\infty )$ and range $[a_{5/2},+\infty )$. There is a Big Bounce at the time when $a\left(\right)$ equals $a_{5/2}$, which we can assume to be $t=0$; we have $a(-t)=a\left(t\right)$ for all real t, $a\left(\right)$ is strictly increasing (resp., decreasing) on $[0,+\infty )$ (resp., on $(-\infty ,0]$), and $a\left(t\right)\to +\infty$ (exponentially) for $t\to \mp \infty$. See Figure 9 again.
• In all the cases considered above, we have a cosmology with the time domain $(-\infty ,+\infty )$, 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 $t\mapsto a\left(t\right)$ of the reduced system fulfilling at some time $t_{\ast }$ the condition $a\left(t_{\ast }\right)=1,\dot{a}\left(t_{\ast }\right)=1$, which is indeed equivalent to $a\left(t_{\ast }\right)=1,\dot{a}\left(t_{\ast }\right)>0$: 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 $a\left(t_{\ast }\right)=1,\dot{a}\left(t_{\ast }\right)>0$ at some time $t_{\ast }$ 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

$$\sigma =-1.$$

(334)

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

Let us consider the Lagrangian L of Equations (121) and (122) with $\sigma =-1$, and pass from the coordinates $a>0$ and $\varphi$ to the new coordinates $r>0$ and $\theta$, defined by

$$a={\left(C_{d}r\right)}^{2/d},\varphi ={\displaystyle \frac{\theta }{C_d}},C_d:={\displaystyle \frac{d}{2\sqrt{2}}}.$$

(335)

The Lagrangian as a function of the new coordinates, again indicated with L, is

$$L(r,\theta ,\dot{r},\dot{\theta })={\displaystyle \frac{1}{2}}{\dot{r}}^2+{\displaystyle \frac{1}{2}}{r}^2{\dot{\theta }}^2+\mathcal{U}\left(r\right)+r^2\mathcal{W}\left(\theta \right),$$

(336)

$$\mathcal{U}\left(r\right):=\sum _{n=1}^N{\left(C_{d}r\right)}^2{F}_n\left({\left(C_{d}r\right)}^{-2}\right)+{\Omega }_{k\ast }{\left(C_{d}r\right)}^{{\displaystyle \frac{2(d-2)}{d}}},\mathcal{W}\left(\theta \right):=C_d^{2}V(\theta /C_d)$$

(337)

(the $F_n$’s are as in Section 4.1 and Section 4.2). Let us note that $(1/2){\dot{r}}^2+(1/2)r^2{\dot{\theta }}^2$ 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 $\theta$.

As a matter of fact, for a genuine interpretation of $(r,\theta )$ as polar coordinates it is required that29

$$\theta \in \mathbb{R}/\left(2\pi \mathbb{Z}\right);$$

(338)

due to the relation between $\theta$ and $\varphi$ in (335), (338) holds if and only if

$$\varphi \in \mathbb{R}/\left(P_d\mathbb{Z}\right),P_d:={\displaystyle \frac{2\pi }{C_d}}={\displaystyle \frac{4\sqrt{2}\pi }{d}}.$$

(339)

This makes some difference with respect to the general setting employed up to now, where $\varphi \in \mathbb{R}$ is the dimensionless form of the scalar field $\Phi \in \mathbb{F}$, and these two objects are connected by the relation (73) $\Phi =\varphi /\sqrt{2{\gamma }_d{G}_d}$. 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:

$$\Phi \in \mathbb{F}/\left(F_d\mathbb{Z}\right),F_d:={\displaystyle \frac{P_d}{\sqrt{2{\gamma }_d{G}_d}}}={\displaystyle \frac{4\pi }{d\sqrt{{\gamma }_d{G}_d}}}\in \mathbb{F}.$$

(340)

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

$$\mathcal{V}:\mathbb{F}/\left(F_d\mathbb{Z}\right)\to \mathbb{D},V:\mathbb{R}/\left(P_d\mathbb{Z}\right)\to \mathbb{R},$$

(341)

still connected through the relation (74) $\mathcal{V}(\Phi )=(H_{\ast }^2/2{\gamma }_d{G}_d)V\left(\varphi \right)$. In the same vein, Equation (337) defines a function

$$\mathcal{W}:\mathbb{R}/\left(2\pi \mathbb{Z}\right)\to \mathbb{R}.$$

(342)

It should be noted that a function $\mathcal{V}$ as in Equation (341) can be identified with a periodic function $\mathbb{F}\to \mathbb{D}$ of period $F_d$; similarly, the functions $V,\mathcal{W}$ in Equations (341) and (342) can be identified with periodic functions $\mathbb{R}\to \mathbb{R}$ of periods $P_d$ and $2\pi$, respectively.

In the present Section 7 and in the subsequent Section 8, we will stick to the setting (338)–(342). The transformation $(r,\theta )\mapsto (a,\varphi )$ defined by Equation (335) is a smooth diffeomorphism between ${\mathbb{R}}^{+}\times \mathbb{R}/\left(2\pi \mathbb{Z}\right)$ and ${\mathbb{R}}^{+}\times \mathbb{R}/\left(P_d\mathbb{Z}\right)$. The Lagrangian (336) is a function of the variables $(r,\theta ,\dot{r},\dot{\theta })\in {\mathbb{R}}^{+}\times \mathbb{R}/\left(2\pi \mathbb{Z}\right)\times \mathbb{R}\times \mathbb{R}$, and depends on the smooth functions $\mathcal{U}:{\mathbb{R}}^{+}\to \mathbb{R}$, $\mathcal{W}:\mathbb{R}/\left(2\pi \mathbb{Z}\right)\to \mathbb{R}$ defined by Equation (337).

7.2. Cartesian Coordinates

Maintaining the setup of the previous Section 7.1, we can pass from the polar coordinates $(r,\theta )$ to the Cartesian coordinates

$$x:=rcos\theta ,y:=rsin\theta .$$

(343)

Obviously enough, the above transformation $(r,\theta )\to (x,y)$ is a smooth diffeomorphism between ${\mathbb{R}}^{+}\times \mathbb{R}/\left(2\pi \mathbb{Z}\right)$ and ${\mathbb{R}}^2\setminus \left\{(0,0)\right\}$, whose inverse map can be described via the following equations:

$$r=\sqrt{x^2+y^2},cos\theta ={\displaystyle \frac{x}{\sqrt{x^2+y^2}}},sin\theta ={\displaystyle \frac{y}{\sqrt{x^2+y^2}}}.$$

(344)

The Lagrangian (336) (337) becomes, in Cartesian coordinates,

$$L(x,y,\dot{x},\dot{y})={\displaystyle \frac{1}{2}}{\dot{x}}^2+{\displaystyle \frac{1}{2}}{\dot{y}}^2+\mathcal{U}\left(\sqrt{x^2+y^2}\right)+(x^2+y^2)\mathcal{W}\left(\theta \right){|}_{cos\theta =x/\sqrt{x^2+y^2},sin\theta =y/\sqrt{x^2+y^2}}.$$

(345)

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,

  $$N=1,F_1\left(\alpha \right)={\Omega }_{m\ast }\alpha forall\alpha \in {\mathbb{R}}^{+},({\Omega }_{m\ast }\in {\mathbb{R}}^{+})$$

  (346)

  (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

  $${\Omega }_{k\ast }=0.$$

  (347)
  Due to (346) and (347), the functions U, $\mathcal{U}$ of Equations (122) and (337) are constant: $U\left(a\right)={\Omega }_{m\ast }$, $\mathcal{U}\left(r\right)={\Omega }_{m\ast }$ for all $a,r\in {\mathbb{R}}^{+}$. Thus, the Lagrangian in $(a,\varphi )$ coordinates and in polar coordinates $(r,\theta )$ (Equation (121) with $\sigma =-1$ and Equation (336)) reads, respectively, as follows:

  $$L(a,\varphi ,\dot{a},\dot{\varphi }):=a^{d-2}{\dot{a}}^2+a^d{\dot{\varphi }}^2+a^{d}V\left(\varphi \right)+{\Omega }_{m\ast },$$

  (348)

  $$L(r,\theta ,\dot{r},\dot{\theta })={\displaystyle \frac{1}{2}}{\dot{r}}^2+{\displaystyle \frac{1}{2}}{r}^2{\dot{\theta }}^2+r^2\mathcal{W}\left(\theta \right)+{\Omega }_{m\ast }.$$

  (349)
• The function $\mathcal{W}$ of Equation (337), giving the potential of the scalar field in terms of the coordinate $\theta$, is assumed to have the form

  $$\mathcal{W}\left(\theta \right)={\displaystyle \frac{1}{2}}{W}_1{cos}^2\theta +{\displaystyle \frac{1}{2}}{W}_2{sin}^2\theta \mathrm{for} \mathrm{all} \theta \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)(W_1,W_2\in \mathbb{R})$$

  (350)
  (i.e., $\mathcal{W}\left(\theta \right)=(1/4)(W_1+W_2)+(1/4)(W_1-W_2)cos\left(2\theta \right)$ for all $\theta$). Due to the relation between $\mathcal{W}$ and V in Equation (337), the prescription (350) is equivalent to

  $$V\left(\varphi \right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{W_1}{C_d^2}}{cos}^2\left(C_d\varphi \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{W_2}{C_d^2}}{sin}^2\left(C_d\varphi \right);$$

  (351)

  a further equivalent, in terms of the dimensioned potential $\mathcal{V}$, is readily obtained from Equations (73) and (74).

Due to Equations (349) and (350), the Lagrangian (336) reads

$$L(r,\theta ,\dot{r},\dot{\theta })={\displaystyle \frac{1}{2}}{\dot{r}}^2+{\displaystyle \frac{1}{2}}{r}^2{\dot{\theta }}^2+{\displaystyle \frac{1}{2}}{W}_1{r}^2{cos}^2\theta +{\displaystyle \frac{1}{2}}{W}_2{r}^2{sin}^2\theta +{\Omega }_{m\ast };$$

(352)

passing to the Cartesian coordinates (343), we have

$$L(x,y,\dot{x},\dot{y})={\displaystyle \frac{1}{2}}{\dot{x}}^2+{\displaystyle \frac{1}{2}}{\dot{y}}^2+{\displaystyle \frac{1}{2}}{W}_1{x}^2+{\displaystyle \frac{1}{2}}{W}_2{y}^2+{\Omega }_{m\ast }.$$

(353)

The corresponding energy function reads

$$E(x,y,\dot{x},\dot{y})={\displaystyle \frac{1}{2}}{\dot{x}}^2+{\displaystyle \frac{1}{2}}{\dot{y}}^2-{\displaystyle \frac{1}{2}}{W}_1{x}^2-{\displaystyle \frac{1}{2}}{W}_2{y}^2-{\Omega }_{m\ast },$$

(354)

and the Lagrange equations are

$$\ddot{x}=W_{1}x,\ddot{y}=W_{2}y.$$

(355)

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

$$\mathbf{r}:=(x,y)\in {\mathbb{R}}^2,$$

(356)

so that the radius $r=\sqrt{x^2+y^2}$ coincides with usual norm $\left|\mathbf{r}\right|$; we will generically indicate with boldface symbols the elements of ${\mathbb{R}}^2$ and set $0:=(0,0)$.

Let us recall that we are interested in the solutions $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ of the Lagrange equations taking values in ${\mathbb{R}}^2\setminus \left\{0\right\}$ and fulfilling the energy constraint $E=0$ (see Section 4.4). For any such curve, the instantaneous distance between the point $\mathbf{r}\left(t\right)$ and the origin gives the radius $r\left(t\right)$ and, consequently, the scale factor $a\left(t\right)$ via Equation (335). The angle $\theta \left(t\right)$ and, consequently, the field $\varphi \left(t\right)$ 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

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}{\dot{x}}^2+{\displaystyle \frac{1}{2}}{\dot{y}}^2-{\displaystyle \frac{1}{2}}{W}_1{x}^2-{\displaystyle \frac{1}{2}}{W}_2{y}^2;$$

(357)

the right-hand side of (357) is automatically constant along any solution of the Lagrange equations (355), and we can use (357) to determine ${\Omega }_{m\ast }$ for any given solution. However, we must keep in mind our requirement ${\Omega }_{m\ast }>0$. 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 ${\mathbb{R}}^2\setminus \left\{0\right\}$. 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 $W_i={\lambda }_i^2>0$ ($i=1,2$)

If $W_1$ and $W_2$ are both positive we can represent them as indicated in the title, with ${\lambda }_i\in {\mathbb{R}}^{+}$; in this case, Equation (355) describes two harmonic repulsors.

In the forthcoming Section 8.4, we will treat the case ${\lambda }_1={\lambda }_2$; 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 ${\lambda }_1$ and ${\lambda }_2$ considered arbitrary, which gives a nonconstant field potential whenever ${\lambda }_1\ne {\lambda }_2$.

8.4. Cosmologies with ${\lambda }_1={\lambda }_2$

Let

$$W_1=W_2={\lambda }^2,\lambda \in {\mathbb{R}}^{+}.$$

(358)

This prescription corresponds to a constant and positive field potential since Equations (350), (351), and (358) give

$$\mathcal{W}\left(\theta \right)=\mathrm{const}.={\displaystyle \frac{1}{2}}{\lambda }^2,V\left(\varphi \right)=\mathrm{const}.={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\lambda }{C_d}}\right)}^2.$$

(359)

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

$$\Lambda ={\displaystyle \frac{d(d-1){\lambda }^2{H}_{\ast }^2}{4C_d^2}}={\displaystyle \frac{2(d-1){\lambda }^2{H}_{\ast }^2}{d}}.$$

(360)

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 $W_i={\lambda }^2$ ($i=1,2$), using the vector variable $\mathbf{r}:=(x,y)$ and the standard norm $\left|\right|$ of ${\mathbb{R}}^2$. In this way, we obtain

$$\ddot{\mathbf{r}}={\lambda }^2\mathbf{r},$$

(361)

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}|\dot{\mathbf{r}}{|}^2-{\displaystyle \frac{1}{2}}{\lambda }^2{\left|\mathbf{r}\right|}^2.$$

(362)

(Let us repeat that $\left|\mathbf{r}\right|$ coincides with the usual radius r.) As recalled in Appendix V, a function $t\mapsto \mathbf{r}\left(t\right)$ is a maximal solution of Equation (361) if and only if one of the following cases (i)–(iii) occurs:

(i)

We have

$$\mathbf{r}\left(t\right)=\mathbf{H}{e}^{\lambda t}\mathrm{for} \mathrm{all} t\in \mathbb{R}(\mathbf{H}\in {\mathbb{R}}^2);$$

(363)

(ii)

We have

$$\mathbf{r}\left(t\right)=\mathbf{K}{e}^{-\lambda t}\mathrm{for} \mathrm{all} t\in \mathbb{R}(\mathbf{K}\in {\mathbb{R}}^2);$$

(364)

(iii)

Up to a time translation $t\mapsto t+$ const., we have

$$\mathbf{r}\left(t\right)=Acosh\left(\lambda t\right)\mathbf{i}+Bsinh\left(\lambda t\right)\mathbf{j}\mathrm{for} \mathrm{all} t\in \mathbb{R},$$

(365)

where

$$A,B\in \mathbb{R},A,B⩾0;\mathbf{i},\mathbf{j}\in {\mathbb{R}}^2,\left|\mathbf{i}\right|=\left|\mathbf{j}\right|=1,\mathbf{i}•\mathbf{j}=0$$

(366)

($•$ is the standard inner product of ${\mathbb{R}}^2$; the previous conditions on $(\mathbf{i},\mathbf{j})$ indicate that this pair is an orthonormal basis of ${\mathbb{R}}^2$).

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

$$A,B>0.$$

(367)

Assuming (367), let us write Equation (365) as

$$\mathbf{r}\left(t\right)=\xi \left(t\right)\mathbf{i}+\eta \left(t\right)\mathbf{j}\mathrm{for} \mathrm{all} t\in \mathbb{R},\xi \left(t\right):=Acosh\left(\lambda t\right),\eta \left(t\right):=Bsinh\left(\lambda t\right));$$

(368)

for all $t\in \mathbb{R}$ we have

$${\displaystyle \frac{{\xi }^2\left(t\right)}{A^2}}-{\displaystyle \frac{{\eta }^2\left(t\right)}{B^2}}=1,\xi \left(t\right)>0,$$

(369)

and we recognize that the point $\mathbf{r}\left(t\right)$ describes a branch of a hyperbola (see Figure 10). Let us remark that $\mathbf{r}\left(t\right)\ne 0$ for all $t\in \mathbb{R}$.

To continue, let us note that Equations (362) and (365) imply

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}(B^2-A^2){\lambda }^2;$$

(370)

we want ${\Omega }_{m\ast }>0$, which is equivalent to

$$B>A.$$

(371)

Due to (365) and (366), the radius $r\left(t\right)=\left|\mathbf{r}\right(t\left)\right|$ is such that

$$r\left(t\right)=\sqrt{A^2{cosh}^2\left(\lambda t\right)+B^2{sinh}^2\left(\lambda t\right)}=\sqrt{A^2+(A^2+B^2){sinh}^2\left(\lambda t\right)}\mathrm{for} \mathrm{all} t\in \mathbb{R}.$$

(372)

From here, it is evident that $t=0$ is a point of absolute minimum for the function $t\mapsto r\left(t\right)$, with

$$r\left(0\right)=A;$$

(373)

it is also evident that $r(-t)=r\left(t\right)$ and that the function $r\left(\right)$ is strictly decreasing on $(-\infty ,0]$, strictly increasing on $[0,+\infty )$ (see Figure 11). Finally,

$$r\left(t\right)\sim {\displaystyle \frac{1}{2}}\sqrt{A^2+B^2}{e}^{\mp \lambda t}\mathrm{for} t\to \mp \infty .$$

(374)

The above features of the radius function are easily rephrased in terms of the scale factor $a\left(t\right)={\left(C_{d}r\left(t\right)\right)}^{2/d}$ (recall Equation (335)). Thus, the function $a\left(\right)$ has an absolute minimum at $t=0$, with $a\left(0\right)={\left(C_{d}A\right)}^{2/d}$; this function decreases strictly on $(-\infty ,0]$, increases strictly on $[0,+\infty )$, and grows exponentially for $t\to \mp \infty$.

To summarize, the present cosmological model presents a Big Bounce at $t=0$; note that the minimum value $a\left(0\right)$ (or $r\left(0\right)$) can be made arbitrarily small by choosing a sufficiently small A. Needless to say, the strict positivity of the minimum $a\left(0\right)$ 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 ${\lambda }_1$ and ${\lambda }_2$ Arbitrary

We now discuss the general case

$$W_i={\lambda }_i^2,{\lambda }_i\in {\mathbb{R}}^{+}(i=1,2)$$

(375)

(this will also give an alternative treatment of the special case ${\lambda }_1={\lambda }_2$, already discussed in the previous section by different means). Equation (355) reads

$$\ddot{x}={\lambda }_1^{2}x,\ddot{y}={\lambda }_2^{2}y;$$

(376)

a function $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ is a maximal solution of these equations if and only if

$$x\left(t\right)=A_1{e}^{{\lambda }_{1}t}+B_1{e}^{-{\lambda }_{1}t},y\left(t\right)=A_2{e}^{{\lambda }_{2}t}+B_2{e}^{-{\lambda }_{2}t}\mathrm{for} \mathrm{all} t\in \mathbb{R},$$

(377)

where

$$A_1,B_1,A_2,B_2\in \mathbb{R}.$$

(378)

For simplicity, from now on, we just consider the nondegenerate case

$$A_i\ne 0,B_i\ne 0(i=1,2).$$

(379)

Let us investigate conditions under which $x\left(t\right),y\left(t\right)$ or $\mathbf{r}\left(t\right)$ vanish. For any $t\in \mathbb{R}$, we readily find the following equivalences:

$$x\left(t\right)=0⟺{\displaystyle \frac{B_1}{A_1}}<0,t={\displaystyle \frac{1}{2{\lambda }_1}}ln\left(-{\displaystyle \frac{B_1}{A_1}}\right);$$

(380)

$$y\left(t\right)=0⟺{\displaystyle \frac{B_2}{A_2}}<0,t={\displaystyle \frac{1}{2{\lambda }_2}}ln\left(-{\displaystyle \frac{B_2}{A_2}}\right).$$

(381)

Clearly, the function $\mathbf{r}\left(\right)$ has a zero if and only if the conditions on the parameters $A_i,B_i$ indicated above hold, and the above zeroes of $x\left(\right)$ and $y\left(\right)$ coincide; the last condition means equality of the times $(1/2{\lambda }_i)ln\left(-B_i/A_i\right)=(1/2)ln\left({(-B_i/A_i)}^{1/{\lambda }_i}\right)$ ($i=1,2$). Thus,

$$\exists t\in \mathbb{R}\mathrm{such} \mathrm{that}\mathbf{r}\left(t\right)=0⟺{\displaystyle \frac{B_1}{A_1}}<0,{\displaystyle \frac{B_2}{A_2}}<0,{\left(-{\displaystyle \frac{B_1}{A_1}}\right)}^{1/{\lambda }_1}={\left(-{\displaystyle \frac{B_2}{A_2}}\right)}^{1/{\lambda }_2}.$$

(382)

From now on, we assume

$${\displaystyle \frac{B_1}{A_1}}>0,or{\displaystyle \frac{B_2}{A_2}}>0,or\left({\displaystyle \frac{B_1}{A_1}}<0,{\displaystyle \frac{B_2}{A_2}}<0,{\left(-{\displaystyle \frac{B_1}{A_1}}\right)}^{1/{\lambda }_1}\ne {\left(-{\displaystyle \frac{B_2}{A_2}}\right)}^{1/{\lambda }_2}\right);$$

(383)

this means that we are denying the conditions on $A_i,B_i$ in (382), so that

$$\mathbf{r}\left(t\right)\ne 0\mathrm{for} \mathrm{all} t\in \mathbb{R}.$$

(384)

To continue, let us note that Equation (357) with $W_i={\lambda }_i^2$ and Equation (377) give

$${\Omega }_{m\ast }=-2({\lambda }_1^2{A}_1{B}_1+{\lambda }_2^2{A}_2{B}_2);$$

(385)

we want ${\Omega }_{m\ast }>0$, so from now on we also assume

$${\lambda }_1^2{A}_1{B}_1+{\lambda }_2^2{A}_2{B}_2<0.$$

(386)

Let us proceed to the analysis of the (never vanishing) radius $r\left(t\right)=\left|\mathbf{r}\right(t\left)\right|$. Equation (377) gives

$$r\left(t\right)=\sqrt{A_1^2{e}^{2{\lambda }_{1}t}+A_2^2{e}^{2{\lambda }_{2}t}+B_1^2{e}^{-2{\lambda }_{1}t}+B_2^2{e}^{-2{\lambda }_{2}t}+2(A_1{B}_1+A_2{B}_2)}\mathrm{for} t\in \mathbb{R};$$

(387)

from here, we infer that

$$\dot{r}\left(t\right)={\displaystyle \frac{\mathcal{R}\left(t\right)}{r\left(t\right)}}\mathrm{for} t\in \mathbb{R},\mathcal{R}\left(t\right):={\lambda }_1{A}_1^2{e}^{2{\lambda }_{1}t}+{\lambda }_2{A}_2^2{e}^{2{\lambda }_{2}t}-{\lambda }_1{B}_1^2{e}^{-2{\lambda }_{1}t}-{\lambda }_2{B}_2^2{e}^{-2{\lambda }_{2}t}.$$

(388)

The function $\mathcal{R}:\mathbb{R}\to \mathbb{R}$ defined above has derivative $\dot{\mathcal{R}}\left(t\right)=2({\lambda }_1^2{A}_1^2{e}^{2{\lambda }_{1}t}+{\lambda }_2^2{A}_2^2{e}^{2{\lambda }_{2}t}+{\lambda }_1^2{B}_1^2{e}^{-2{\lambda }_{1}t}+{\lambda }_2^2{B}_2^2{e}^{-2{\lambda }_{2}t})>0$, and $\mathcal{R}\left(t\right)\to \mp \infty$ for $t\to \mp \infty$. Thus, $\mathcal{R}$ is strictly increasing with range $(-\infty ,+\infty )$; therefore, there is a unique

$$t_0\in \mathbb{R}\mathrm{such} \mathrm{that} \mathcal{R}\left(t_0\right)=0,$$

(389)

and $\mathcal{R}\left(t\right)<0$ for $t<t_0$, $\mathcal{R}\left(t\right)>0$ for $t>t_0$.

Since $\mathrm{sign}\dot{r}\left(t\right)=\mathrm{sign}\mathcal{R}\left(t\right)$ for all real t, we conclude the following: the radius function $t\mapsto r\left(t\right)$ admits $t_0$ as a point of absolute minimum, is strictly decreasing on $(-\infty ,t_0]$ and strictly increasing on $[t_0,+\infty )$ (we know that $r\left(t\right)>0$ for all t, this holds, in particular, for the minimum value $r\left(t_0\right)$). Finally, let us remark that Equation (387) implies

$$r\left(t\right)\sim B_{\infty }{e}^{-{\lambda }_{\infty }t}\mathrm{for} t\to -\infty ,r\left(t\right)\sim A_{\infty }{e}^{{\lambda }_{\infty }t}\mathrm{for} t\to +\infty ,$$

(390)

where ${\lambda }_{\infty },A_{\infty },B_{\infty }$ are defined as follows: If ${\lambda }_1>{\lambda }_2$, then ${\lambda }_{\infty }:={\lambda }_1$, $A_{\infty }:=\left|A_1\right|$, and $B_{\infty }=\left|B_1\right|$; if ${\lambda }_1<{\lambda }_2$, then ${\lambda }_{\infty }:={\lambda }_2$, $A_{\infty }:=\left|A_2\right|$, and $B_{\infty }=\left|B_2\right|$; if ${\lambda }_1={\lambda }_2$, then, ${\lambda }_{\infty }:={\lambda }_1={\lambda }_2$, $A_{\infty }:=\sqrt{A_1^2+A_2^2}$ and $B_{\infty }:=\sqrt{B_1^2+B_2^2}$.

The above results on the radius function $r\left(t\right)$ imply similar statements on the scale factor $a=a\left(t\right)$ given by (335). Thus, the function $a\left(\right)$ has a (strictly positive) absolute minimum at $t=t_0$, decreases strictly before $t_0$, increases strictly after $t_0$, and diverges exponentially for $t\to \mp \infty$. 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 $t\mapsto \mathbf{r}\left(t\right)$ and the graph of the radius $t\mapsto r\left(t\right)$ for a special choice of the parameters, compatible with (383) and (386).

8.6. The Case $W_i=-{\omega }_i^2<0$ ($i=1,2$): Lissajous Cosmologies

We now address the case where $W_1$ and $W_2$ are both negative and represent them as

$$W_i=-{\omega }_i^2,{\omega }_i\in {\mathbb{R}}^{+}(i=1,2).$$

(391)

Then, Equation (355) reads

$$\ddot{x}=-{\omega }_1^{2}x,\ddot{y}=-{\omega }_2^{2}y.$$

(392)

and describes two uncoupled harmonic oscillators with pulsations ${\omega }_i$. A function $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ is a maximal solution of (392) if and only if, up to a time translation $t\mapsto t+$ const., it can be expressed as follows:

$$x\left(t\right)=A_{1}sin\left({\omega }_{1}t\right),y\left(t\right)=A_{2}sin({\omega }_{2}t+\psi )\mathrm{for} \mathrm{all} t\in \mathbb{R},$$

(393)

where

$$A_1,A_2\in \mathbb{R},A_1,A_2⩾0,\psi \in \mathbb{R}/\left(2\pi \mathbb{Z}\right).$$

(394)

The curves in ${\mathbb{R}}^2$ 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 ${\omega }_2/{\omega }_1$ and on the phase $\psi$ (see the already cited references [63,64]). In the sequel, for particular values of ${\omega }_2/{\omega }_1$, 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 $\mathbf{r}\left(t\right)\ne 0$. Let us also remark that Equation (357) with $W_i=-{\omega }_i^2$ and Equation (393) imply the following expression for the density parameter:

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}{\omega }_1^2{A}_1^2+{\displaystyle \frac{1}{2}}{\omega }_2^2{A}_2^2.$$

(395)

In the forthcoming Section 8.7 and Section 8.8, we will discuss as examples the cases ${\omega }_1={\omega }_2$ and ${\omega }_2=2{\omega }_1$; 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 ${\omega }_1={\omega }_2$

Let us consider the case

$${\omega }_1={\omega }_2\equiv \omega \in {\mathbb{R}}^{+}.$$

(396)

This corresponds to a constant and negative field potential since Equations (350), (351), (391), and (396) give

$$\mathcal{W}\left(\theta \right)=\mathrm{const}.=-{\displaystyle \frac{1}{2}}{\omega }^2,V\left(\varphi \right)=\mathrm{const}.=-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\omega }{C_d}}\right)}^2.$$

(397)

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

$$\Lambda =-{\displaystyle \frac{d(d-1){\omega }^2{H}_{\ast }^2}{4C_d^2}}=-{\displaystyle \frac{2(d-1){\omega }^2{H}_{\ast }^2}{d}}.$$

(398)

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 $W_i=-{\omega }^2$ for $i=1,2$, using the vector variable $\mathbf{r}:=(x,y)$ (and the standard norm $\left|\right|$ of ${\mathbb{R}}^2$); in this way, the cited equations become

$$\ddot{\mathbf{r}}=-{\omega }^2\mathbf{r},$$

(399)

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}|\dot{\mathbf{r}}{|}^2+{\displaystyle \frac{1}{2}}{\omega }^2{\left|\mathbf{r}\right|}^2.$$

(400)

As recalled in Appendix W, a function $t\mapsto \mathbf{r}\left(t\right)$ is a maximal solution of Equation (399) if and only if, up to a time translation $t\mapsto t+$ const., it has the form

$$\mathbf{r}\left(t\right)=Acos\left(\omega t\right)\mathbf{i}+Bsin\left(\omega t\right)\mathbf{j}\mathrm{for} \mathrm{all} t\in \mathbb{R},$$

(401)

involving the parameters

$$A,B\in \mathbb{R}\mathrm{s}.\mathrm{t}.0⩽A⩽B;\mathbf{i},\mathbf{j}\in {\mathbb{R}}^2\mathrm{s}.\mathrm{t}.\left|\mathbf{i}\right|=\left|\mathbf{j}\right|=1,\mathbf{i}•\mathbf{j}=0$$

(402)

(again, $•$ denotes the standard inner product of ${\mathbb{R}}^2$; the above conditions on $(\mathbf{i},\mathbf{j})$ indicate that this pair is an orthonormal basis of ${\mathbb{R}}^2$). It is evident that the function $t\mapsto \mathbf{r}\left(t\right)$ in Equation (401) is periodic of period $2\pi /\omega$.

From now on, we consider solutions of the form (401) and (402) with the nondegeneracy property

$$0<A⩽B.$$

(403)

Let us rephrase Equation (401) as

$$\mathbf{r}\left(t\right)=\xi \left(t\right)\mathbf{i}+\eta \left(t\right)\mathbf{j}\mathrm{for} t\in \mathbb{R},\xi \left(t\right):=Acos\left(\omega t\right),\eta \left(t\right)=Bsin\left(\omega t\right).$$

(404)

It is evident that

$${\displaystyle \frac{{\xi }^2\left(t\right)}{A^2}}+{\displaystyle \frac{{\eta }^2\left(t\right)}{B^2}}=1,$$

(405)

so we have an ellipse: the minor semiaxis has direction $\mathbf{i}$ and length A, while the major semiaxis has direction $\mathbf{j}$ and length B (see Figure 14). Of course, $\mathbf{r}\left(t\right)\ne 0$ for all $t\in \mathbb{R}$.

The radius $r\left(t\right)=\left|\mathbf{r}\right(t\left)\right|$ is given by

$$r\left(t\right)=\sqrt{A^2{cos}^2\left(\omega t\right)+B^2{sin}^2\left(\omega t\right)}=\sqrt{A^2+(B^2-A^2){sin}^2\left(\omega t\right)}\mathrm{for} \mathrm{all} \in \mathbb{R};$$

(406)

this is a periodic function of time, with period $\pi /\omega$ (see Figure 15). Clearly, we have

$$\underset{t\in \mathbb{R}}{min}r\left(t\right)=A;ifA<B,r\left(t\right)=A⟺t=n{\displaystyle \frac{\pi }{\omega }}(n\in \mathbb{Z}).$$

(407)

$$\underset{t\in \mathbb{R}}{max}r\left(t\right)=B;ifA<B,r\left(t\right)=B⟺t=\left({\displaystyle \frac{1}{2}}+n\right){\displaystyle \frac{\pi }{\omega }}(n\in \mathbb{Z}).$$

(408)

Let us recall that the scale factor $a=a\left(t\right)$ is related to $r\left(t\right)$ through Equation (335); we have again a periodic function of period $\pi /\omega$, with

$$\underset{t\in \mathbb{R}}{min}a\left(t\right)={\left(C_{d}A\right)}^{2/d},\underset{t\in \mathbb{R}}{max}a\left(t\right)={\left(C_{d}B\right)}^{2/d}.$$

(409)

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 $a\left(\right)$ arbitrarily small (resp., the maximum of $a\left(\right)$ arbitrarily large); moreover, the period $\pi /\omega$ can be made arbitrarily long by choosing a sufficiently small $\omega$.

Concerning the mass density parameter ${\Omega }_{m\ast }$, from Equations (400)–(402), we readily infer

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}{\omega }^2(A^2+B^2).$$

(410)

The special case $A=B$. In this case, the ellipse described by Equation (405) becomes a circumference of radius A; thus, $r\left(t\right)=$ const. $=A$ and $a\left(t\right)=$ const. $={\left(C_{d}A\right)}^{2/d}$. Consequently, the spacetime line element in Equation (17) becomes $d{s}^2=-d{\tau }^2+{\left(C_{d}A\right)}^{4/d}d{\ell }^2$; let us recall that we are assuming zero spatial curvature, so $d{\ell }^2$ 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 ${\omega }_2=2{\omega }_1$

Let us consider the case

$${\omega }_1=\omega ,{\omega }_2=2\omega ,\omega \in {\mathbb{R}}^{+}.$$

(411)

The maximal solutions $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ 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 $A_1=2RA$, $A_2=A$; thus,

$$x\left(t\right)=2RAsin\left(\omega t\right),y\left(t\right)=Asin(2\omega t+\psi )\mathrm{for} t\in \mathbb{R},$$

(412)

where

$$A,R\in \mathbb{R},A,R>0,\psi \in \mathbb{R}/\left(2\pi \mathbb{Z}\right).$$

(413)

For future use, we also introduce the parameter

$$\vartheta :={\displaystyle \frac{2}{\pi }}arccos\left(R^2\right)\in (0,1)ifR\in (0,1),$$

(414)

whose relevance will appear in the sequel (with the above limitation on R). The function $t\mapsto \mathbf{r}\left(t\right)$ in Equation (412) is clearly periodic, of period $2\pi /\omega$. As is well known, the shape of the Lissajous curve described by $\mathbf{r}\left(t\right)$ is very sensitive to the phase $\psi$. Figure 16 describes the familiar curves corresponding to the choices $\psi =-\pi /2,-\pi /4,0,\pi /4$ ($mod.2\pi$); by a reflection $y\mapsto -y$ we also obtain the curves with $\psi =\pi /2,3\pi /4,\pi ,5\pi /4$ ($mod.2\pi$), since $y(t;\psi +\pi )=-y(t;\psi )$. For any $\psi$, Equation (395) for the mass density parameter gives

$${\Omega }_{m\ast }=2{\omega }^2(1+R^2)A^2.$$

(415)

Hereafter, we will consider the cases $\psi =-\pi /2,0$ ($mod.2\pi$) as examples, which will be treated in detail.

The case $\psi =-\pi /2$ ($mod.2\pi$). We have

$$x\left(t\right)=2RAsin\left(\omega t\right),y\left(t\right)=-Acos\left(2\omega t\right)\mathrm{for} t\in \mathbb{R}$$

(416)

(with $A,R>0$, as assumed before); the functions $t\mapsto x\left(t\right),y\left(t\right)$ never vanish simultaneously, so

$$\mathbf{r}\left(t\right)\ne \mathbf{0}\mathrm{for} \mathrm{all} \in \mathbb{R}.$$

(417)

To continue, let us remark that $y\left(t\right)/A=-cos\left(2\omega t\right)=2{sin}^2\left(\omega t\right)-1=$$2{(x\left(t\right)/2RA)}^2-1$; so,

$$y\left(t\right)={\displaystyle \frac{x^2\left(t\right)}{2R^{2}A}}-A,-2RA⩽x\left(t\right)⩽2RA.$$

(418)

Thus, the Lissajous curve we are considering is a parabolic segment with the y axis as a symmetry axis, vertex $\mathbf{v}$ and endpoints ${\mathbf{e}}_{\mp }$, where

$$\mathbf{v}:=(0,-A),{\mathbf{e}}_{-}:=(-2RA,A),{\mathbf{e}}_{+}:=(2RA,A)$$

(419)

(see Figure 17 and Figure 19). Let us allow t to range in the time interval $[-\pi /(2\omega ),3\pi /(2\omega \left)\right]$, whose length corresponds to one period of the function $\mathbf{r}\left(\right)$; then the point $\mathbf{r}\left(t\right)$ describes twice the parabolic segment, from ${\mathbf{e}}_{-}$ to ${\mathbf{e}}_{+}$ and back to ${\mathbf{e}}_{-}$. In particular,

$$\mathbf{r}\left(-{\displaystyle \frac{\pi }{2\omega }}\right)={\mathbf{e}}_{-},\mathbf{r}\left(0\right)=\mathbf{v},\mathbf{r}\left({\displaystyle \frac{\pi }{2\omega }}\right)={\mathbf{e}}_{+},\mathbf{r}\left({\displaystyle \frac{\pi }{\omega }}\right)=\mathbf{v},\mathbf{r}\left({\displaystyle \frac{3\pi }{2\omega }}\right)={\mathbf{e}}_{-}.$$

(420)

Of course, a similar description of the behavior of $\mathbf{r}\left(t\right)$ can be given for t ranging in any time interval $\left[\right(2n-1/2)\pi /\omega ,(2n+3/2)\pi /\omega ]$ ($n\in \mathbb{Z}$).

Due to (416), the (never vanishing) radius $r\left(t\right)$ is given by

$$r\left(t\right)=\sqrt{4R^2{sin}^2\left(\omega t\right)+{cos}^2\left(2\omega t\right)}A=\sqrt{2R^2-2R^{2}cos\left(2\omega t\right)+{cos}^2\left(2\omega t\right)}A;$$

(421)

this is a periodic function of time, of period $\pi /\omega$, and $r(-t)=r\left(t\right)$. It is readily found that, for all $t\in \mathbb{R}$,

$$\dot{r}\left(t\right)={\displaystyle \frac{2\omega sin\left(2\omega t\right)}{r\left(t\right)}}\left(R^2-cos\left(2\omega t\right)\right)A^2.$$

(422)

The sign of $\dot{r}\left(t\right)$ is readily studied, and one concludes the following on the behavior of $r\left(t\right)$ over one period, e.g., on the interval $[-\pi /(2\omega ),\pi /(2\omega \left)\right]$:

Case $R⩾1$: $r\left(t\right)$ behaves as follows on $[-\pi /(2\omega ),\pi /(2\omega \left)\right]$. The times $\mp \pi /\left(2\omega \right)$ are absolute maximum points, 0 is an absolute minimum point, and

$$r\left(\mp {\displaystyle \frac{\pi }{2\omega }}\right)=\sqrt{1+4R^2}A,r\left(0\right)=A.$$

(423)

The function $t\mapsto r\left(t\right)$ is strictly decreasing on $[-\pi /(2\omega ),0]$ and strictly increasing on $[0,\pi /(2\omega \left)\right]$. From a geometrical viewpoint, the above results indicate that the points of the parabolic segment maximizing the distance from the origin are $\mathbf{r}(\mp \pi /\left(2\omega \right))={\mathbf{e}}_{\mp }$; the point minimizing the distance from the origin is $\mathbf{r}\left(0\right)=\mathbf{v}$. See Figure 17 and Figure 18.

Case $0<R<1$: Let us consider the times

$$\mp {\displaystyle \frac{1}{2\omega }}arccos\left(R^2\right)=\mp {\displaystyle \frac{\vartheta \pi }{4\omega }},$$

(424)

with $\vartheta$ as in Equation (414); then,

$$-{\displaystyle \frac{\pi }{2\omega }}<-{\displaystyle \frac{\vartheta \pi }{4\omega }}<0<{\displaystyle \frac{\vartheta \pi }{4\omega }}<{\displaystyle \frac{\pi }{2\omega }},$$

(425)

and the function $t\mapsto r\left(t\right)$ behaves as follows over one period $[-\pi /(2\omega ),\pi /(2\omega \left)\right]$. The times $\mp \pi /\left(2\omega \right)$ are absolute maximum points and 0 is a relative maximum point, with $r(\mp \pi /(2\omega \left)\right)$, and $r\left(0\right)$ as in (423). The times $\mp \vartheta \pi /\left(4\omega \right)$ are absolute minimum points, with

$$r\left(\mp {\displaystyle \frac{\vartheta \pi }{4\omega }}\right)=R\sqrt{2-R^2}A.$$

(426)

The function $t\mapsto r\left(t\right)$ is strictly decreasing on $[-\pi /(2\omega ),-\vartheta \pi /(4\omega \left)\right]$ and on $[0,\vartheta \pi /(4\omega \left)\right]$, strictly increasing on $[-\vartheta \pi /(4\omega ),0]$ and on $[\vartheta \pi /(4\omega ),\pi /(2\omega \left)\right]$.

From a geometrical viewpoint, the above results indicate that the points of the parabolic segment maximizing the distance from the origin are ${\mathbf{e}}_{\mp }=\mathbf{r}(\mp \pi /\left(2\omega \right))$. The points minimizing the distance from the origin are

$${\mathbf{m}}_{\mp }:=\mathbf{r}\left(\mp {\displaystyle \frac{\vartheta \pi }{4\omega }}\right)=(\mp \sqrt{2}R\sqrt{1-R^2}A,-R^{2}A)$$

(427)

(the above explicit expressions for ${\mathbf{m}}_{\mp }$ follow from Equations (416) and (424) for $x\left(\right)$, $y\left(\right)$ 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 $t\mapsto r\left(t\right)$ on any interval $\left[\right(-1/2+n)\pi /\omega ,(1/2+n)\pi /\omega ]$ ($n\in \mathbb{Z}$) is of course similar to that described above for $n=0$. One can choose A and R so that the absolute minimum of $r\left(t\right)$ is arbitrarily small and the absolute maximum of $r\left(t\right)$ is arbitrarily large. Also, the periods involved can be made arbitrarily large by choosing $\omega$ sufficiently small.

Of course, analogous conclusions hold for the scale factor $a=a\left(t\right)$ given by (335): this is a periodic function of period $\pi /\omega$, with maxima and minima at the times indicated above for the function $r=r\left(t\right)$. 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 $\psi =0$ ($mod.2\pi$). Equation (412) gives

$$x\left(t\right)=2RAsin\left(\omega t\right),y\left(t\right)=Asin\left(2\omega t\right)\mathrm{for} t\in \mathbb{R}$$

(428)

(with $A,R>0$, as assumed before); the corresponding Lissajous curve is described in Figure 21 and Figure 24. For all $t\in \mathbb{R}$, we have

$$\mathbf{r}(-t)=-\mathbf{r}\left(t\right);$$

(429)

moreover,

$$\mathbf{r}\left(t\right)=\mathbf{0}⟺t=t_k(k\in \mathbb{Z}),t_k:=k{\displaystyle \frac{\pi }{\omega }}.$$

(430)

The existence of zeroes of $\mathbf{r}\left(\right)$ for $A,R>0$ is an exceptional feature of the present case $\psi =0$ ($mod.2\pi$), which also appears for $\psi =\pi$ ($mod.2\pi$): on the contrary, for any $A,R>0$ and for any $\psi \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)\setminus \{0,\pi (mod.2\pi \left)\right\}$, the solution described by Equation (412) is such that $\mathbf{r}\left(t\right)\ne \mathbf{0}$ for all $t\in \mathbb{R}$35.

Keeping in mind (430), we build the present cosmological model restricting the solution $\mathbf{r}\left(\right)$ to a maximal interval $\mathcal{J}$ such that $\mathbf{r}\left(t\right)\ne \mathbf{0}$ for all $t\in \mathcal{J}$; we can take $\mathcal{J}=(t_k,t_{k+1})$ for any $k\in \mathbb{Z}$. The choice of k is immaterial, and we will set $k=0$, i.e.,

$$\mathcal{J}:=\left(0,{\displaystyle \frac{\pi }{\omega }}\right);$$

(431)

this will be our time domain from now on.

Due to (428), the radius $r\left(t\right)$ is given by

$$r\left(t\right)=\sqrt{4R^2{sin}^2\left(\omega t\right)+{sin}^2\left(2\omega t\right)}A=\sqrt{1+2R^2-2R^{2}cos\left(2\omega t\right)-{cos}^2\left(2\omega t\right)}A;$$

(432)

we have

$$r\left(t\right)\to 0{\displaystyle fort\to 0,\frac{\pi }{\omega }}.$$

(433)

It is readily found that, for all $t\in \mathcal{J}$,

$$\dot{r}\left(t\right)={\displaystyle \frac{2\omega sin\left(2\omega t\right)}{r\left(t\right)}}\left(R^2+cos\left(2\omega t\right)\right)A^2.$$

(434)

The sign of $\dot{r}\left(t\right)$ is readily studied, and one concludes the following on the behavior of $r\left(t\right)$ over the interval $\mathcal{J}$:

Case $R⩾1$: $t=\pi /\left(2\omega \right)$ is an absolute maximum point and

$$r\left({\displaystyle \frac{\pi }{2\omega }}\right)=2RA.$$

(435)

The function $t\mapsto r\left(t\right)$ is strictly increasing on $(0,\pi /(2\omega \left)\right]$ and strictly decreasing on $[\pi /(2\omega ),\pi /\omega )$; see Figure 22. More geometrically, considering the curve $\left\{\mathbf{r}\right(t\left)\right|t\in \mathcal{J}\}$ (see again Figure 21), we can say that the point of this curve maximizing the distance from the origin is

$$\mathbf{r}\left({\displaystyle \frac{\pi }{2\omega }}\right)=(2RA,0).$$

(436)

With appropriate choices of A and R, the maximum $r(\pi /(2\omega \left)\right)$ can be made arbitrarily large.

Case $0<R<1$: Let us consider the times

$${\displaystyle \frac{\pi }{2\omega }}\mp {\displaystyle \frac{1}{2\omega }}arccos\left(R^2\right)=\left(1\mp {\displaystyle \frac{\vartheta }{2}}\right){\displaystyle \frac{\pi }{2\omega }},$$

(437)

with $\vartheta$ as in Equation (414). Then,

$$0<\left(1-{\displaystyle \frac{\vartheta }{2}}\right){\displaystyle \frac{\pi }{2\omega }}<{\displaystyle \frac{\pi }{2\omega }}<\left(1+{\displaystyle \frac{\vartheta }{2}}\right){\displaystyle \frac{\pi }{2\omega }}<{\displaystyle \frac{\pi }{\omega }},$$

(438)

and the function $t\mapsto r\left(t\right)$ behaves as follows on $\mathcal{J}$. The times $(1\mp \vartheta /2)\pi /\left(2\omega \right)$ are absolute maximum points and $\pi /\left(2\omega \right)$ is a relative minimum point, with $r(\pi /(2\omega \left)\right)$ as in (435) and

$$r\left(\left(1\mp {\displaystyle \frac{\vartheta }{2}}\right){\displaystyle \frac{\pi }{2\omega }}\right)=(1+R^2)A.$$

(439)

The function $t\mapsto r\left(t\right)$ is strictly increasing on $(0,(1-\vartheta /2)\pi /(2\omega \left)\right]$ and on $[\pi /(2\omega )$, $(1+\vartheta /2)\pi /\left(2\omega \right)]$, strictly decreasing on $\left[\right(1-\vartheta /2)\pi /(2\omega ),\pi /(2\omega \left)\right]$ and on $\left[\right(1+\vartheta /2)\pi /$$\left(2\omega \right),$$\pi /\omega )$; see Figure 23.

More geometrically, considering the curve $\left\{\mathbf{r}\right(t\left)\right|t\in \mathcal{J}\}$ (see again Figure 24), we can say that the points of this curve maximizing the distance from the origin are

$${\mathbf{p}}_{\pm }:=\mathbf{r}\left(\left(1\mp {\displaystyle \frac{\vartheta }{2}}\right){\displaystyle \frac{\pi }{2\omega }}\right)=\left(\sqrt{2}R\sqrt{1+R^2}A,\pm \sqrt{1-R^4}A\right);$$

(440)

the latter distance has a relative minimum point at $\mathbf{r}(\pi /(2\omega \left)\right)$ as in (436).

With appropriate choices of A and R, the maximum $r\left(\right(1\mp \vartheta /2)\pi /(2\omega \left)\right)$ can be made arbitrarily large, and the relative minimum $r(\pi /(2\omega \left)\right)$ arbitrarily small. Finally, the time interval $\mathcal{J}$ can be made arbitrarily long by choosing a sufficiently small $\omega$.

Our conclusions about the radius $r=r\left(t\right)$ have obvious analogs for the scale factor $a=a\left(t\right)$, given by (335). Both for $R⩾1$ and for $0<R<1$, $a\left(t\right)$ vanishes for $t\to 0,\pi /\omega$; so, the evolution of this model starts with a Big Bang at $t=0$, and ends with a Big Crunch at $t=\pi /\omega$. If $R⩾1$, $a\left(\right)$ has an absolute maximum point at time $\pi /\left(2\omega \right)$; if $0<R<1$, $a\left(\right)$ has two absolute maximum points at times $(1\mp \vartheta /2)\pi /\left(2\omega \right)$, and a relative minimum point at $\pi /\left(2\omega \right)$. The occurrence of a Big Bang and a Big Crunch is strictly related to the existence of zeroes of $\mathbf{r}\left(\right)$ and therefore reflects the exceptional feature of the case $\psi =0$ ($mod.2\pi$) 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 $t_{\mp }$ appearing in the cited equations are 0 and $\pi /\omega$ in the present case).

8.9. On General Lissajous Cosmologies

Let us spend a few words on the general case (391) $W_i=-{\omega }_i^2$ ($i=1,2$), where ${\omega }_1,{\omega }_2$ are chosen arbitrarily in ${\mathbb{R}}^{+}$. We know that the maximal solutions $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ of the evolution equations (392) have the form (393), depending on the constants $A_1,A_2,\psi$ as in (394). Here, we only consider the nondegerate case $A_1,A_2\ne 0$, so that

$$x\left(t\right)=A_{1}sin\left({\omega }_{1}t\right),y\left(t\right)=A_{2}sin({\omega }_{2}t+\psi )\mathrm{for} \mathrm{all} \in \mathbb{R},(A_1,A_2>0,\psi \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)).$$

(441)

Let us also recall that the zero-energy constraint gives the expression (395) for the mass density parameter.

Periodicity conditions for the function $\mathbf{r}\left(\right)$. From Equation (441), it follows that the function $t\mapsto \mathbf{r}\left(t\right)$ is periodic if and only if ${\omega }_2/{\omega }_1$ is rational, i.e., if and only if

$${\displaystyle \frac{{\omega }_2}{{\omega }_1}}={\displaystyle \frac{N_2}{N_1}}\left(N_1,N_2\in \{1,2,3,\dots .\}\right);$$

(442)

in this case,

$$\mathbf{r}(t+T)=\mathbf{r}\left(t\right)\mathrm{for} \mathrm{all} \in \mathbb{R},T:={\displaystyle \frac{2N_1\pi }{{\omega }_1}}={\displaystyle \frac{2N_2\pi }{{\omega }_2}}.$$

(443)

In the same case, we have a periodic cosmological model provided that $\mathbf{r}\left(t\right)\ne 0$ for all $t\in \mathbb{R}$; whether or not this happens depends on the phase $\psi$, 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

$${\omega }_1=3\omega ,{\omega }_2=4\omega ,(\omega \in {\mathbb{R}}^{+});$$

(444)

Equation (442) clearly holds with $N_1=3,N_2=4$, and Equation (443) ensures the function $\mathbb{R}\ni t\mapsto \mathbf{r}\left(t\right)$ to be periodic of period $T:=2\pi /\omega$. In Figure 25, we illustrate the range of this function for several choices of the phase $\psi$ (assuming for simplicity $A_1=A_2$); we will return to this example in the final paragraph of the present section.

If ${\omega }_2/{\omega }_1$ is irrational, it can be shown that the range of the (non-periodic) function $\mathbb{R}\ni t\mapsto \mathbf{r}\left(t\right)$ densely fills a rectangle centered at the origin in the $(x,y)$ plane [64]: see Figure 26. Again, the function $\mathbf{r}\left(\right)$ is never vanishing or possesses zeroes, depending on the phase $\psi$. If $\mathbf{r}\left(\right)$ never vanishes, the radius $r\left(t\right)=\left|\mathbf{r}\right(t\left)\right|$ (hence, the scale factor $a\left(t\right)$ 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 $\mathbb{R}\ni t\mapsto \mathbf{r}\left(t\right),r\left(t\right),a\left(t\right)$ are almost periodic, in the rigorous sense given usually to this expression (see [81], page 1).

When does $\mathbf{r}\left(t\right)$ vanish? We will discuss the problem for arbitrary ${\omega }_1$ and ${\omega }_2$, with ${\omega }_2/{\omega }_1$ either rational or irrational; for this purpose, we choose any real representative ${\psi }_{♭}$ of the phase $\psi$, so that

$${\psi }_{♭}\in \mathbb{R},\psi ={\psi }_{♭}\left(mod.2\pi \right).$$

(445)

Let us first individuate the times t at which $x\left(t\right)$ and $y\left(t\right)$ vanish separately. From Equation (441) (with $\psi$ replaced by ${\psi }_{♭}$), one easily infers the following, for any $t\in \mathbb{R}$:

$$x\left(t\right)=0⟺t={\tilde{t}}_h(h\in \mathbb{Z}),{\tilde{t}}_h:=h{\displaystyle \frac{\pi }{{\omega }_1}};$$

(446)

$$\phantom{x}y\left(t\right)=0⟺t={\widehat{t}}_{\ell }(\ell \in \mathbb{Z}),{\widehat{t}}_{\ell }:=\ell {\displaystyle \frac{\pi }{{\omega }_2}}-{\displaystyle \frac{{\psi }_{♭}}{{\omega }_2}}.$$

(447)

This implies the following statement about $\mathbf{r}\left(t\right)=\left(x\right(t),y(t)$):

$$\exists t\in \mathbb{R}s.t.\mathbf{r}\left(t\right)=0⟺\exists \ell ,h\in \mathbb{Z}s.t.{\widehat{t}}_{\ell }={\tilde{t}}_h⟺\exists \ell ,h\in \mathbb{Z}s.t.{\psi }_{♭}={\zeta }_{\ell h},$$

(448)

where

$${\zeta }_{\ell h}:=\left(\ell -h{\displaystyle \frac{{\omega }_2}{{\omega }_1}}\right)\pi \mathrm{for}\ell ,h\in \mathbb{Z}.$$

(449)

We can rephrase the result (448) in terms of the phase $\psi \in \mathbb{R}/\left(2\pi \mathbb{Z}\right)$, which is represented by ${\psi }_{♭}$; after noting that ${\zeta }_{\ell h}$ (mod. $2\pi$) equals ${\zeta }_{0h}$ (mod. $2\pi$) or ${\zeta }_{1h}$ (mod. $2\pi$) for $\ell \in \mathbb{Z}$ even or odd, respectively, we conclude that

$$\exists t\in \mathbb{R}s.t.\mathbf{r}\left(t\right)=\mathbf{0}⟺\psi \in Z,Z:=\{{\zeta }_{0h},{\zeta }_{1h}\left(mod.2\pi \right)|h\in \mathbb{Z}\}.$$

(450)

Let us note that, for any choice of ${\omega }_1,{\omega }_2$, we have $Z\ni 0\left(mod.2\pi \right)$ (if $\psi =0\left(mod.2\pi \right)$, it is evident from (441) that $\mathbf{r}\left(0\right)=\mathbf{0}$). To continue our discussion on the zeroes of $\mathbf{r}\left(\right)$, we must distinguish the cases with ${\omega }_2/{\omega }_1$ rational or irrational.

The case of rational ${\omega }_2/{\omega }_1$: In this case, which is described by Equation (442), Z is a finite subset of $\mathbb{R}/\left(2\pi \mathbb{Z}\right)$36. For each phase $\psi \in Z$, the periodicity of $\mathbf{r}\left(\right)$ (see Equation (443)) implies that there are infinitely many times $t\in \mathbb{R}$ such that $\mathbf{r}\left(t\right)=0$.

The case of irrational ${\omega }_2/{\omega }_1$: In this case, the subset Z is infinite and dense in $\mathbb{R}/\left(2\pi \mathbb{Z}\right)$; however, Z is countable, so its Lebesgue measure is zero. For each phase $\psi \in Z$, there is a unique time $t\in \mathbb{R}$ such that $\mathbf{r}\left(t\right)=0$37.

A final comment: We have just noted that, depending on the rational or irrational nature of ${\omega }_2/{\omega }_1$, the phases $\psi$ giving rise to zeroes for $\mathbf{r}\left(\right)$ form a finite or an infinite but countable, zero-measure subset of $\mathbb{R}/\left(2\pi \mathbb{Z}\right)$. In both cases, these phases must be regarded as exceptional.

Nonsingular Lissajous cosmologies. Given ${\omega }_1,{\omega }_2\in {\mathbb{R}}^{+}$, let us consider a phase $\psi$ outside the exceptional set Z of Equation (450) and any two amplitudes $A_1,A_2\in {\mathbb{R}}^{+}$. Then, $\mathbf{r}\left(t\right)\ne 0$ for all $t\in \mathbb{R}$, and we have a cosmological model with time domain $(-\infty ,+\infty )$, in which $a\left(t\right)$ 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 ${\omega }_2/{\omega }_1$ is rational, the function $\mathbf{r}\left(\right)$ on $\mathbb{R}$ is periodic (let us recall again Equation (443)), and this implies periodicity of the functions $r\left(\right),a\left(\right):\mathbb{R}\to {\mathbb{R}}^{+}$. Thus, ${inf}_{t\in \mathbb{R}}a\left(t\right)$$={min}_{t\in \mathbb{R}}a\left(t\right)\equiv 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 ${\omega }_2/{\omega }_1$ is irrational, the functions $\mathbf{r}\left(\right)$, $r\left(\right),a\left(\right)$ on $\mathbb{R}$ are not periodic, and ${inf}_{t\in \mathbb{R}}r\left(t\right)$$={inf}_{t\in \mathbb{R}}a\left(t\right)=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 ${\omega }_1,{\omega }_2\in {\mathbb{R}}^{+}$, let us consider a phase $\psi$ in the exceptional set Z of Equation (450), and any two amplitudes $A_1,A_2\in {\mathbb{R}}^{+}$. Then, according to the discussion after Equation (450), the function $\mathbb{R}\ni t\to \mathbf{r}\left(t\right)$ has infinitely many zeroes if ${\omega }_2/{\omega }_1$ is rational and unique zero if ${\omega }_2/{\omega }_1$ is irrational. To establish a cosmological model, we must confine the time variable to a maximal interval $\mathcal{J}$ not containing zeroes. In the rational case, we will have $\mathcal{J}=(t^{\prime },t^{\prime \prime })$, where $-\infty <t^{\prime }<t^{\prime \prime }<+\infty$ are two consecutive zeroes; in the irrational case, $\mathcal{J}=(t_{\times },+\infty )$ or $\mathcal{J}=(-\infty ,t_{\times })$, where $t_{\times }\in \mathbb{R}$ is the unique zero. The scale factor $a\left(\right):\mathcal{J}\to {\mathbb{R}}^{+}$, $t\mapsto a\left(t\right)$ determined by (335) vanishes for $t\to t_{\mp }$ or for $t\to t_{\times }$. So, in the rational case, the model has a Big Bang at $t^{\prime }$ and a Big Crunch at $t^{\prime \prime }$, whereas in the irrational case, there is either a Big Bang or a Big Crunch at $t_{\times }$.

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 $\mathcal{J}=(t^{\prime },t^{″})$, 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 $t_{\mp }$ in the cited equations coincide with $t^{\prime }$ and $t^{\prime \prime }$, respectively).
• In the irrational case with $\mathcal{J}=(t_{\times },+\infty )$, 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 $t_{-}=t_{\times }$ and $t_{+}=+\infty$)39.
• In the irrational case with $\mathcal{J}=(-\infty ,t_{\times })$, 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 $t_{-}=-\infty$ and $t_{+}=t_{\times }$).

Regarding an example mentioned previously. Let us return to the example (444) ${\omega }_1=3\omega$, ${\omega }_2=4\omega$ ($\omega \in {\mathbb{R}}^{+}$; 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 $\mathbb{R}\ni t\mapsto \mathbf{r}\left(t\right)$ is periodic of period $T:=2\pi /\omega$.

Using Equations (449) and (450), we see that the set of exceptional phases giving rise to zeroes for the function $t\mapsto \mathbf{r}\left(t\right)$ is $Z=\{-2\pi /3,-\pi /3,0,\pi /3,2\pi /3,\pi (mod.2\pi \left)\right\}$; for all the other phases $\psi$, we have nonsingular, periodic Lissajous cosmologies. We have already mentioned Figure 25, related to this case; the phases $\psi$ considered therein include the exceptional choice $\psi =0$ ($mod.2\pi$)40.

8.10. The Case $W_1>0,W_2<0$

We finally address the case

$$W_1={\lambda }^2,W_2=-{\omega }^2,(\lambda ,\omega \in {\mathbb{R}}^{+}).$$

(451)

Equation (355) reads

$$\ddot{x}={\lambda }^{2}x,\ddot{y}=-{\omega }^{2}y,$$

(452)

thus describing a repulsor and an oscillator. A function $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$ is a maximal solution of these equations if and only if it can be expressed as follows, up to a time translation $t\mapsto t+$ const.:

$$x\left(t\right)=A{e}^{\lambda t}+B{e}^{-\lambda t},y\left(t\right)=Csin\left(\omega t\right)\mathrm{for} \mathrm{all} \in \mathbb{R},$$

(453)

where

$$A,B,C\in \mathbb{R},C⩾0.$$

(454)

In the rest of the present Section 8.10, we direct our attention to the nondegenerate case

$$A,B\ne 0,C>0.$$

(455)

Let us investigate conditions under which $x\left(t\right),y\left(t\right)$ or $\mathbf{r}\left(t\right)$ vanish. For any $t\in \mathbb{R}$, we readily find the following equivalences:

$$x\left(t\right)=0⟺{\displaystyle \frac{B}{A}}<0,t={\displaystyle \frac{1}{2\lambda }}ln\left(-{\displaystyle \frac{B}{A}}\right);$$

(456)

$$y\left(t\right)=0⟺t=n{\displaystyle \frac{\pi }{\omega }}\mathrm{for} \mathrm{some} n\in \mathbb{Z}.$$

(457)

Clearly, the function $\mathbf{r}\left(\right)$ has a zero if and only $B/A<0$, and the zero of $x\left(\right)$ indicated in (456) coincides with one of the zeroes of $y\left(\right)$ described by (457); the last condition holds if an only if $(\omega /2\pi \lambda )ln\left(-B/A\right)=n$ for some $n\in \mathbb{Z}$. Thus,

$$\exists t\in \mathbb{R}suchthat\mathbf{r}\left(t\right)=\mathbf{0}⟺{\displaystyle \frac{B}{A}}<0,{\displaystyle \frac{\omega }{2\pi \lambda }}ln\left(-{\displaystyle \frac{B}{A}}\right)\in \mathbb{Z}.$$

(458)

From here to the end of the present Section 8.10, we assume

$${\displaystyle \frac{B}{A}}>0,or\left({\displaystyle \frac{B}{A}}<0,{\displaystyle \frac{\omega }{2\pi \lambda }}ln\left(-{\displaystyle \frac{B}{A}}\right)\notin \mathbb{Z}\right);$$

(459)

since we are denying the conditions on $A,B$ in (458), we have

$$\mathbf{r}\left(t\right)\ne \mathbf{0}\mathrm{for} \mathrm{all} \in \mathbb{R}.$$

(460)

To continue, let us note that Equation (357) with $W_1={\lambda }^2$, $W_2=-{\omega }^2$ and Equation (453) give

$${\Omega }_{m\ast }=-2{\lambda }^{2}AB+{\displaystyle \frac{1}{2}}{\omega }^2{C}^2;$$

(461)

we want ${\Omega }_{m\ast }>0$, so, from now on, we also assume

$$C^2>{\displaystyle \frac{4{\lambda }^2}{{\omega }^2}}AB.$$

(462)

Let us proceed to the analysis of the radius $r\left(t\right)=\left|\mathbf{r}\right(t\left)\right|$; Equation (453) gives

$$r\left(t\right)=\sqrt{A^2{e}^{2\lambda t}+B^2{e}^{-2\lambda t}+2AB+C^2{sin}^2\left(\omega t\right)}\mathrm{for} t\in \mathbb{R}.$$

(463)

Of course,

$$r\left(t\right)\sim \left|B\right|e^{-\lambda t}\mathrm{for} t\to -\infty ,r\left(t\right)\sim \left|A\right|e^{\lambda t}\mathrm{for} t\to +\infty .$$

(464)

Moreover, if $\omega \gg \lambda$, we can expect an oscillatory behavior of the radius in an interval around the time $t=0$ (with $r\left(t\right)>0$ for each $t\in \mathbb{R}$, due to (460)); oscillations should be particularly evident if $\left|A\right|,\left|B\right|\ll C$.

Similar statements hold for the scale factor $a=a\left(t\right)$ ($t\in \mathbb{R}$), related to r via Equation (335); in a few words, in this cosmological model, the scale factor diverges exponentially for $t\to \mp \infty$ and oscillates (but never vanishes) during a time interval centered at $t=0$. In particular, ${min}_{t\in \mathbb{R}}a\left(t\right)$ 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 $t\mapsto \mathbf{r}\left(t\right)$ 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) $W_1={\lambda }^2$ and $W_2=-{\omega }^2$ (with $\lambda ,\omega \in {\mathbb{R}}^{+})$. Let us recall the evolution equations (452) and the representation (453) and (454) for their (maximal) solutions $t\mapsto \mathbf{r}\left(t\right)=\left(x\right(t),y(t\left)\right)$, 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

$$x\left(t\right)=A{e}^{\lambda t},y\left(t\right)=Csin\left(\omega t\right)\mathrm{for} \mathrm{all} \in \mathbb{R};$$

(465)

for all real t, we have $x\left(t\right)\ne 0$ and, consequently, $\mathbf{r}\left(t\right)\ne \mathbf{0}$. According to Equation (461), the mass density parameter is

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}{\omega }^2{C}^2>0.$$

(466)

The radius function is given by

$$r\left(t\right)=\sqrt{A^2{e}^{2\lambda t}+C^2{sin}^2\left(\omega t\right)}\mathrm{for} \mathrm{all} \in \mathbb{R}.$$

(467)

We have

$$r\left(t\right)=\sqrt{C^2{sin}^2\left(\omega t\right)+o\left(1\right)}\mathrm{for} t\to -\infty ,r\left(t\right)\sim A{e}^{\lambda t}\mathrm{for} t\to +\infty$$

(468)

($o\left(1\right)$ means, as usual, a function vanishing in the limit considered above). Thus, the radius $r\left(t\right)$ oscillates (but never vanishes) for $t\ll 0$ and grows exponentially for $t\gg 0$; similar conclusions hold for the scale factor $a=a\left(t\right)$ ($t\in \mathbb{R}$), 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 $t\mapsto \mathbf{r}\left(t\right)$ and radius $t\mapsto r\left(t\right)$ 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

$$x\left(t\right)=B{e}^{-\lambda t},y\left(t\right)=Csin\left(\omega t\right)\mathrm{for} \mathrm{all} \in \mathbb{R};$$

(469)

for all real t, we have $x\left(t\right)\ne 0$, hence $\mathbf{r}\left(t\right)\ne \mathbf{0}$. Equation (466) holds again. The radius function is

$$r\left(t\right)=\sqrt{B^2{e}^{-2\lambda t}+C^2{sin}^2\left(\omega t\right)}\mathrm{for} \mathrm{all} \in \mathbb{R},$$

(470)

and

$$r\left(t\right)\sim B{e}^{-\lambda t}\mathrm{for} t\to -\infty ,r\left(t\right)=\sqrt{C^2{sin}^2\left(\omega t\right)+o\left(1\right)}\mathrm{for} t\to +\infty .$$

(471)

Thus, the radius grows exponentially for $t\ll 0$ and oscillates (without vanishing) for $t\gg 0$; of course, the scale factor $a=a\left(t\right)$ ($t\in \mathbb{R}$) 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 $t\mapsto r\left(t\right)$, 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 $(\omega /2\pi \lambda )ln(-B/A)=0\in \mathbb{Z}$. Equation (453) gives

$$x\left(t\right)=Fsinh\left(\lambda t\right),y\left(t\right)=Csin\left(\omega t\right)\mathrm{for} \mathrm{all} \in \mathbb{R}.$$

(472)

For any $t\in \mathbb{R}$, we have $x\left(t\right)=0$⟺$t=0$ and $y\left(t\right)=0$⟺$t=n\pi /\omega$ ($n\in \mathbb{Z}$); consequently, $\mathbf{r}\left(t\right)=\mathbf{0}$⟺$t=0$. We can accept as a time domain for the present cosmological model a (maximal) interval $\mathcal{J}$ where $\mathbf{r}\left(t\right)$ is never vanishing; we take

$$\mathcal{J}=(0,+\infty ).$$

(473)

Let us also remark that, for the mass density parameter, Equation (461) gives in this case

$${\Omega }_{m\ast }={\displaystyle \frac{1}{2}}({\lambda }^2{F}^2+{\omega }^2{C}^2)>0.$$

(474)

The radius function is given by

$$r\left(t\right)=\sqrt{F^2{sinh}^2\left(\lambda t\right)+C^2{sin}^2\left(\omega t\right)}$$

(475)

$$=\sqrt{{\displaystyle \frac{1}{2}}{F}^{2}cosh\left(2\lambda t\right)-{\displaystyle \frac{1}{2}}{C}^{2}cos\left(2\omega t\right)+{\displaystyle \frac{1}{2}}(C^2-F^2)}\mathrm{for} \mathrm{all} t\in (0,+\infty ),$$

and its derivative reads

$$\dot{r}\left(t\right)={\displaystyle \frac{1}{2r\left(t\right)}}\left(F^2\lambda sinh\left(2\lambda t\right)+C^2\omega sin\left(2\omega t\right)\right)={\displaystyle \frac{F^2\lambda }{2r\left(t\right)}}\mathcal{R}\left(2\lambda t;{\displaystyle \frac{C^2\omega }{F^2\lambda }},{\displaystyle \frac{\omega }{\lambda }}\right)\mathrm{for} t\in (0,+\infty ),$$

(476)

where

$$\mathcal{R}(s;\chi ,\varrho ):=sinh\left(s\right)+\chi sin\left(\varrho s\right)\mathrm{for} s,\chi ,\varrho \in (0,+\infty ).$$

(477)

Let us remark that $\mathcal{R}(s;\chi ,\varrho )⩾sinh\left(s\right)-\chi$ for all $s,\chi ,\varrho$, which implies $\mathcal{R}(s;\chi ,\varrho )>0$ for all $\chi ,\varrho \in (0,+\infty )$ and all $s\in (arcsinh\chi ,+\infty )$; depending on the values of $\chi$ and $\varrho$, the function $s\mapsto \mathcal{R}(s;\chi ,\varrho )$ does not possess or possesses zeroes in the interval $(0,arcsinh\chi ]$. Consequently, for all $F,C,\lambda ,\omega \in (0,+\infty )$, the following holds:

$$\dot{r}\left(t\right)>0\mathrm{for} \mathrm{all} t\in \left({\displaystyle \frac{1}{2\lambda }}arcsinh{\displaystyle \frac{C^2\omega }{F^2\lambda }},+\infty \right),$$

(478)

while the function $t\mapsto \dot{r}\left(t\right)$ does not possess or possesses zeroes in the interval $(0,(1/2\lambda )$$arcsinh(C^2\omega /F^2\lambda )]$, depending on the values of the ratios $C^2\omega /F^2\lambda$ and $\omega /\lambda$; if such zeroes exist, we can say that the radius $r\left(t\right)$ oscillates for t in a right neighborhood of 0. Finally, let us remark that

$$r\left(t\right)\to 0^{+}\mathrm{for} t\to 0^{+},r\left(t\right)\sim {\displaystyle \frac{F}{2}}{e}^{\lambda t}\mathrm{for} t\to +\infty .$$

(479)

Similar conclusions hold for the scale factor $a=a\left(t\right)$ ($t\in (0,+\infty )$), given by (335); $a\left(t\right)$ vanishes for $t\to 0^{+}$, thus presenting a Big Bang, and grows exponentially for $t\to +\infty$. 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 $t\mapsto \mathbf{r}\left(t\right)$ and the radius $t\mapsto r\left(t\right)$ for a specific choice of the parameters giving rise to zeroes of $\mathcal{R}$, 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.