carlos r. fadragas arxiv:1405.2465v2 [gr-qc] 12 oct 2014

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iv:1

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Some remarks about non-minimally coupled scalar fieldmodels

Carlos R. FadragasDepartment of Physics, Universidad Central de Las Villas, Santa Clara CP 54830, Cuba

E-mail: [email protected]

Genly LeonInstituto de Fısica, Pontificia Universidad Catolica de Valparaıso, Casilla 4950, Valparaıso,Chile

E-mail: [email protected]

Abstract. Several results related to flat Friedmann-Lemaıtre-Robertson-Walker models inthe conformal (Einstein) frame of scalartensor gravity theories are extended. Scalar fieldswith arbitrary (positive) potentials and arbitrary coupling functions are considered. Mildassumptions under such functions (differentiable class, number of singular points, asymptotes,etc) are introduced in a straightforward manner in order to characterize the asymptotic structureon a phase space. We pay special attention to the possible scaling solutions. Numericalevidence confirming our results is presented.

PACS numbers: 98.80.-k, 98.80.Jk,98.80.Cq., 95.36.+x, 95.30.Sf, 04.20.Ha

http://arxiv.org/abs/1405.2465v2

Some remarks about non-minimally coupled scalar field models 2

1. Introduction

Recent astrophysical observations suggest that the universe is currently experiencing anaccelerated expansion [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,14, 15, 16, 17, 18, 19, 20, 21].To explain this feature of the universe one choice is to introduce the concept of Dark Energy(DE) (see [22, 23, 24] and references therein), which could be a cosmological constant, aquintessence field [25, 26, 27, 28, 29, 30, 31], a phantom field[32, 33, 34, 35, 36, 37],the quintom field [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], among other examples.Another choice is to consider Higher Order Gravity (HOG) theories, say thef(R)- models(see [50, 51, 52, 53, 54, 55, 56, 57, 58] and references therein). Other modified gravitationalscenarios are the extended nonlinear massive gravity scenario [59, 60, 61, 62], the TeleparalellDark Energy (TDE) model [63, 64, 65] and some generalizations of the TDE [66, 67]. All ofthese scenarios have very interesting cosmological features.

Another interesting effective scalar-field model, which isrelated to inhomogeneouscosmologies, is the so-called “morphon” field that arises asan effective scalar field modelfor averaged cosmologies [68, 69]. In this case the scalar field is not interpreted as a sourceof the Einstein equations, but as a mean field description of averaged inhomogeneities.In this case the “backreaction” effects (due to averaged expansion and shear fluctuations,the averaged 3-Ricci curvature, averaged pressure gradients and frame fluctuation terms) isformally equivalent to the dynamics of a homogeneous, minimally coupled scalar field. Thisrelation was completely addressed in [68].

Although our results are more widely applicable if the scopeis widened to effective scalarfields like the morphon field, in this paper the DE contribution is modelled as a conventionalself-interacting quintessence scalar field,φ, with potentialV (φ) [25, 26, 27, 28, 29, 30, 31].

In the inflationary universe scenarios the potentialV (φ) must satisfy some requirementswhich are necessary to lead to the early-time acceleration of the expansion [70, 71, 72, 73, 74].Exponential (de Sitter) expansion arises, for example, as aresult of considering a constantpotentialV (φ) = V0, whereas the power-law inflationary solutions arise when consideringan exponential potentialV (φ) = V0 exp(−λφ) [75, 76]. In the reference [77] have beeninvestigated a minimally coupled scalar field evolving in the quadratic potentialV (φ) =12m

2φ2 in the flat Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric. There wasprovided a global picture of the solution space by means of a regular global dynamical systemdefined in extended compact space. This system is suitable for obtaining global piecewiseapproximations for the late-time attractor solution due tothe large range of convergence forcenter manifold expansions and the associated approximants as compared with the slow-rollapproximation and associated slow-roll approximants [77].

Several gravity theories consider multiple scalar fields with exponential potential,e.g., assisted inflation scenarios [78, 79, 80, 81, 82, 83], quintom dark energy paradigm[39, 40, 42, 43, 45] and others. The potential have been considered as positive and negativeexponential [84], single and double exponential [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95,96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106], etc. Multiple scalar fields can be found at[107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118].

Very interesting cases are non-minimally coupled scalar fields that appear in the stringtheory context [119] or in the Scalar-tensor theories (STT)context [120, 121, 122, 123,124, 125, 126, 127, 128, 129]. Coupled quintessence models were investigated by meansof phase-space studies for example in [94, 95, 96, 98, 100, 102, 130, 131, 132]. Specific non-minimally coupled subclass of Horndeski scalar-tensor theories arising from the decouplinglimit of massive gravity by covariantization were studied in [133, 134]. In the reference[135], it was investigated a flat FLRW scalar field with potential of typesV (φ) = φn


andV (φ) = φn1 + φn2 , conformally coupled to the Ricci scalar,R, through the function−ξB(φ)R, whereξ is the coupling constant andB(φ) = φN . The authors worked in theJordan frame in the absence of matter. It was presented therea global picture of the phasespace by means of compact variables. Some exact solutions, for some choices of the slopesof the potential and the coupling function, were discussed there. In the reference [136], it wasinvestigated a scalar field non-minimally coupled to the Ricci scalar evolving in Higgs-like(quadratic) potentials plus a negative cosmological constant. Double exponential potentialand exponential coupling function were discussed in [91] aswell as in [92, 93, 95, 98], underthe ansatzφ = λH .

In the reference [137], it was performed a detailed dynamical analysis of Kantowski-Sachs, Locally Rotationally Symmetric (LRS) Bianchi I and LRS Bianchi III models bymeans of the method off -devisers presented in [138] and applied in [139] to scalar fieldcosmologies in the framework of a generalized Chaplygin gas. This method, that allows us toperform the whole analysis for a wide range of potentials, isa modification of the method firstintroduced in [140]. The original method was used for investigating flat FLRW scalar fieldcosmologies [31, 140, 141, 142, 143, 144] and it was generalized to several cosmologicalcontexts in [145, 146, 147, 148]. As a drawback of the method of f -devisers it cannotbe applied to some specific inflationary potentials like the logarithmicV (φ) ∝ φp lnq(φ)[149] and the generalized exponential oneV (φ) ∝ φn exp (−λφm) [150] since the resultingf -functions are not single valued, then one should apply asymptotic techniques in order toextract the dominant branch at largeφ-values as in [149, 150].

The idea to obtain general results for scalar field cosmologies only providing generalfeatures of the potentials and coupling functions, is not new. Preceding works for a largevariety of non-negative potentials are [151, 152, 153, 154,155, 156, 157, 158]. In thereference [158], it have been extended many of the results obtained in [156] by consideringarbitrary potentials. In [155] it has been shown that for a large class of flat FLRW cosmologieswith scalar fields with arbitrary potential, the past attractor corresponds to exactly integrablecosmologies with a massless scalar field. A list of integrable models with a minimallycoupled scalar field, including double exponential potentials, in the absence of matter, ispresented in [159]. Integrable non-minimally coupled scalar field models in the Jordanframe were investigated in [160]. In [161] were investigated flat FLRW cosmologies basedon STTs. The new asymptotic expansions for the cosmologicalsolutions near the initialspace-time singularity obtained there contains as particular cases those studied in [155].The proof of the local initial singularity theorem presented in [161] have been improvedin [162]. Additionally in [162] were presented several results corresponding the late-timedynamics for the case of a scalar field non-minimally coupledto dark matter. In thispaper we extent these results by adding a radiation fluid. This leads to a more realisticcosmological model in the framework of the so-calledcomplete cosmological dynamics[163],that is, a viable cosmological model should describe a radiation dominated era (RDE) beforeentering a matter dominated era (MDE), which should be succeeded by the current late-time accelerated expansion [49, 163, 164]. The transition from one era to the next can beunderstood, in the language of dynamical systems, in terms of the so-called heteroclinicsequences [165, 166, 167, 144].

Models arising in the conformal frame ofF (R) theories were considered. e.g., in[168, 169, 170, 171, 162]. In [168] were investigated flat andnegatively curved FLRWmodels with a perfect fluid matter source and a nonminimally coupled scalar fieldφ withpotentialV (φ) (related to theF (R) function). There was proved that for potentials thateventually becomes non-negative asφ → ±∞ and with a finite number of critical points, thenon-negative local minima and the horizontal asymptotes approached from above by it, are


asymptotically stable. For a nondegenerated minimum with zero critical value andγ > 1,there is a transfer of energy from the fluid to the scalar field,which eventually dominatesthe expansion in a generic way. In [170] it was developed a mathematical procedure forinvestigate the dynamics when|φ| → +∞ of non-minimally coupled scalar fields modelsinteracting with Dark Matter in the presence of radiation. There were studied the modifiedgravity modelsf(R) = R + αR2 (quadratic gravity) andF (R) = Rn in the STT frame.For quadratic gravity, the equilibrium point corresponding to de Sitter solution is locallyasymptotically unstable (saddle point).

The aim of the paper is to extent several results in [155, 156,161, 168, 169, 171, 91]for the general case of arbitrary potentials and arbitrary couplings. It is described the earlyand late-time dynamics of the models and we pay special attention to the possible scalingsolutions. We follow the method first introduced in [155] andextended in [170] for theanalysis of the limitφ → +∞. We examine the example of a double exponential potentialand our new results complement those in [91]. Additionally,we revisit the example of apowerlaw coupling function and an Albrecht-Skordis potential, first introduced in [161], andthen extended in section 4.4 of [170].

2. Basic framework

The action for a general class of STT, written in the so-called Einstein frame (EF), is given by[172]:

∫d4x√

|g|1

2R− 1

2gµν∇µφ∇νφ− V (φ) + χ(φ)−2L(µ,∇µ, χ(φ)−1gαβ)

. (1)

We use a system of units in which8πG = c = ~ = 1. In this equationR is the curvaturescalar,φ is the a scalar field‡, ∇α is the covariant derivative,V (φ) is the quintessence self-interaction potential,χ(φ)−2 is the coupling function,L is the matter Lagrangian,µ is acollective name for the matter degrees of freedom.

The matter energy-momentum tensor is defined by

Tαβ = − 2√|g|

δ

δgαβ

√|g|χ−2L(µ,∇µ, χ−1gαβ)

. (2)

Let’s define

Qβ ≡ ∇αTαβ = −1

2T

1

χ(φ)

dχ(φ)

dφ∇βφ, T = Tα

α .

Since there is an exchange of energy between the scalar and the background fluids, the energyis not separately conserved for each component. Instead, the continuity equation for each fluidreads [185]:

ρm + 3H(ρm + pm) = Q, (3)

ρDE + 3H(ρDE + pDE) = −Q, (4)

where the dot accounts for derivative with respect to the cosmic time andQ is the interactionterm. Now, defining the total energy density and the total pressure asρT = ρm + ρDE andpT = pm + pDE , respectively, then the total energy density is indeed conserved in the sense

‡ For a discussion about the regularity of the conformal transformation, or the equivalence issue of the two frames,see for example [173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184] and references therein.


ρT + 3H(ρT + pT ) = 0. To specify the general form of the interaction term we can look at ascalar-tensor theory of gravity (1) and the interaction term Q in equations (3) and (4), can bewritten in the following form:

Q = −1

2(4 − 3γ)ρmH

[ad lnχ

da

], (5)

where we have assumed that the couplingχ can be written as a function of the scale factorthroughχ(φ(a)). Comparing this with other interaction terms in the bibliography, one canobtain the functional form of the coupling functionχ at each case. In the reference [186], forinstance,Q = 3Hc2(ρDE+ρm) = 3c2Hρm(r+1)/r, wherec2 denotes the transfer strengthand r ≡ Ωm/ΩDE . If one compares this expression with (5) one obtains the followingcoupling function:

χ(a) = χ0 exp

[− 6

4− 3γ

∫da

a

(r + 1

r

)c2], (6)

whereχ0 is an arbitrary integration constant. Ifc2 = c20 = const. andr = r0 = const., then

χ = χ0 a6c20(r0+1)

(4−3γ)r0 . It is well-known that a suitable coupling can produce scaling solutions,although the way to fix the coupling is not univocally determined. In reference [130, 187], forinstance, the coupling is introduced by hand. In [188, 189, 190] the form of the interactionterm is fixed by the requirement that the ratio of the energy densities of DM and quintessencehas an stable fixed point during the evolution that solves thecoincidence; in [188] a suitableinteraction between the quintessence field and DM leads to a transition from the dominationmatter era to an accelerated expansion epoch for the model proposed in [188, 189, 190]. In[191] the coupling function is chosen as a Fourier expansionaround some minimum of the(dilaton) scalar field.

It is well known that the HOG theories [50, 51, 52, 53, 54, 55, 56, 57, 58] derived fromthe action

S =

∫d4x

√−g

1

2F (R) + L(µ,∇µ, gαβ)

, (7)

and the STT with action

S =

∫d4x√

−g

1

2R− (∇φ)2 − V (φ) + e−2

√2/3φL

(µ,∇µ, e−

√2/3φgαβ

), (8)

are conformally equivalent under the transformation,

gµν = F ′(R)gµν , (9a)

φ =

√3

2lnF ′(R), (9b)

V (R(φ)) =1

2 (F ′(R))2 (RF ′(R)− F (R)) , (9c)

where it is assumed that (9b) can be solved forR to obtain a functionR (φ), in order to getthe potential (9c) as an explicit function ofφ. It is easy to note that the model arising from the

action (8) can be obtained from (1) with the choiceχ(φ) = e√

2/3φ. Thus, the results in [161]and in [168] can be obtained as particular cases by investigating a general class of modelscontaining both STTs andF (R) gravity.


Using the above approach, we obtain that the quadratic gravity modelF (R) = R+αR2

is conformally equivalent to a non-minimally coupled scalar field with the potentialV (φ) =

18α

(1− e−

√2/3φ

)2, α > 0. This potential has only one local minimum atφ = 0, and the

asymptoteV∞ = 18α which is approached by below by the potential asφ → +∞. The zero

minimum atφ = 0 of this potential is stable, but it cannot provide the mechanism for thelate-time acceleration sinceV andH asymptotically approach to zero [156, 157, 192]. On theother hand, concerning the upper asymptote ofV asφ → +∞, it follows that the potentialhas exponential order zero (see definition 2) asφ → +∞. Thus, using Proposition 6 of [170],

follows that the de Sitter configurationφ → +∞, V (φ) = V∞, H(φ) =√

V∞

3 is unstableto perturbations along theφ-axis in a neighborhood of “infinity”, and thus it cannot representthe late-time solution.

In the reference [192] were imposed conditions on the function F (R) withcorresponding potential (9c) using the restrictions onV (φ) obtained in the papers [151, 152,153, 154]. There it was exploited from the mathematical viewpoint the connection betweenF (R)-gravity and nonminimally coupled scalar fields and were proved mathematicallyrigorous results. In [193], it was investigated a generic class off(R) models for theKantowski-Sachs metric using dynamical systems tools.

2.1. The Field Equations

In this section we investigate the action (1) for the flat FLRWmetric:

ds2 = −dt2 + a(t)2(dr2 + r2

(dθ2 + sin2 θdϕ2

)). (10)

wherea(t) is the scale factor. The Hubble expansion scalar isH = a/a, where the dotmeans derivative with respect time. We assume that the energy-momentum tensor (2) isTαβ = diag(−ρm, pm, pm, pm) , whereρm and pm are respectively the isotropic energy

density and the isotropic pressure of barotopic matter withequation of statepm = (γ−1)ρm.We include a radiation source with energy densityρr since we want to investigate the possiblescaling solutions in the radiation regime. We neglect ordinary (uncoupled) barotropic matter.

The cosmological equations with the above “ingredients” are

H = − 1

2

(γρm +

4

3ρr + φ2

), (11a)

ρm = − 3γHρm − 1

2(4 − 3γ)ρmφ

d lnχ(φ)

dφ, (11b)

ρr = − 4Hρr, (11c)

φ = − 3Hφ− dV (φ)

dφ+

1

2(4− 3γ)ρm

d lnχ(φ)

dφ, (11d)

3H2 =1

2φ2 + V (φ) + ρm + ρr. (11e)

We assume the general hypothesisV (φ) ∈ C3, V (φ) > 0, χ(φ) ∈ C3 andχ(φ) > 0 andthus the dynamical system (11) is of classC2. In order to derive our results we shall considerfurther assumptions which shall be clearly stated when necessary. We assumeρm, ρr ≥ 0,and0 < γ < 2, γ 6= 4

3 . The later assumption excludes the possibility that the backgroundmatter behaves as radiation which in rigor is automaticallydecoupled from the scalar field(since the energy-momentum tensor for radiation is traceless).


3. Dynamical systems analysis

In the following, we study the late time behavior of solutions of (11), which are expanding

at some initial time, i. e.,H(0) > 0. The state vector of the system is(φ, φ, ρm, ρr, H

).

Definingy := φ, we rewrite the autonomous system as

H = − 1

2

(γρm +

4

3ρr + y2

), (12a)

ρm = − 3γHρm − 1

2(4− 3γ)ρmy

d lnχ(φ)

dφ, (12b)

ρr = − 4Hρr, (12c)

y = − 3Hy − dV (φ)

dφ+

1

2(4 − 3γ)ρm

d lnχ(φ)

dφ, (12d)

φ = y, (12e)

subject to the constraint

3H2 =1

2y2 + V (φ) + ρm + ρr. (13)

Remark 1 Using standard arguments of ordinary differential equations theory, follows fromequations(12b)and(12c)that the signs ofρm andρr, respectively, are invariant. This meansthat if ρm > 0 andρr > 0 for some initial timet0, thenρm(t) > 0, andρr(t) > 0 throughoutthe solution. From(12a)and (13) and only if additional conditions are assumed, for exampleV (φ) ≥ 0 andV (φ∗) = 0 for someφ∗, follows that the sign ofH is invariant. From(12c)and (12a) follows thatρr andH decreases. Also, definingǫ = 1

2y2 + V (φ), follows from

(12b), (12d)and (12c) that

ǫ+ ρm + ρr = −3H(y2 + γρm +4

3ρr). (14)

Thus, the total energy density contained in the dark sector is decreasing.

The system (12) defines a dynamical system in the phase space

Ω = (H, ρm, ρr, y, φ) ∈ R4|3H2 =

1

2y2 + V (φ) + ρm + ρr. (15)

Let’s assume in first place that the potential function has a local minimumV (0) =0. This implies that the point(0, 0, 0, 0, 0) is a singular point of (12) which impliesthat an initially expanding universe (H > 0) should expand forever. Indeed, the set(H, ρm, ρr, y, φ) ∈ Ω|H = 0 is invariant under the flow of (12). Besides, the sign ofHis invariant. Otherwise, if the sign ofH changes, a trajectory withH(0) > 0 can passingthrough(0, 0, 0, 0, 0), violating the existence an uniqueness theorem for ODEs.

The proposition 2 of [156] can be generalized to this contextas follows.

Proposition 1 Suppose thatV ≥ 0 andV (φ) = 0 ⇔ φ = 0. LetA such thatV bounded inA impliesV ′(φ) is bounded inA. If there exists a constantK, K 6= 0 such that

χ′(φ)/χ(φ) ≤ 2K/(2− γ)(4 − 3γ).

Then,limt→∞

(ρm, ρr, y) = (0, 0, 0).


Proof. Consider the trajectory passing through an arbitrary point (H, ρm, ρr, y, φ) ∈ Ωwith H > 0 at t = t0. SinceH is positive and decreasing we have thatlimt→∞ H(t) existsand it is a nonnegative numberη; besides,H(t) ≤ H(t0) for all t ≥ t0. Then, from (15)follows that each termρm, ρr, 1/2y2, andV (φ) is bounded by3H(t0)

2 for all t ≥ t0.Let definedA =

φ : V (φ) ≤ 3H(t0)

2. Then, the trajectory is such thatφ remains in

the interior ofA and additionallyV ′(φ) is bounded forφ ∈ A.From equation (12a) follows

−∫ t

t0

(1

2y2 +

γ

2ρm +

2

3ρr

)dt = H(t)−H(t0).

Taking the limitt → ∞, we obtain

1

2

∫ ∞

t0

(y2 + γρm +

4

3ρr

)dt = H(t0)−η =⇒

∫ ∞

t0

(y2 + γρm +

4

3ρr

)dt < ∞. (16)

Taking the time derivative off(t) = y2 + γρm + 43ρr and making use of the hypothesis

for χ(φ) we obtain

d

dt

(y2 + γρm +

4

3ρr

)≤ y (−2V ′(φ) +Kρm)− 16

3ρrH.

As we have seen,y, ρm, ρr andH are bounded fort ≥ t0, and by the hypothesis forV (φ),V ′(φ) is bounded. From this facts follow that the time derivative of f is bounded. Sincef isa nonnegative function, the convergence of

∫∞t0

f(t)dt implieslimt→∞ f(t) = 0. Hence, wehave that

limt→∞

(ρm, ρr, y) = (0, 0, 0).

The hypotheses in 1 concerning to the scalar field self-interacting potential are not veryrestrictive [156]. The hypothesis forχ(φ) is satisfied by a large class of coupling functionstoo, including the exponential ones.

Under the same hypothesis of proposition 1, we can generalize the proposition 3 in [156].

Proposition 2 Suppose thatV ′(φ) > 0 for φ > 0 andV ′(φ) < 0 for φ < 0. Then, under thesame hypotheses as in proposition 1,limt→∞ φ exists and is equal to+∞, 0 or −∞.

Proof. Using the same argument as in Proposition 1,∃ limt→∞ H(t) = η. If η = 0,then by the restriction (15) we obtainlimt→∞ V (φ(t)) = 0. SinceV is continuous andV (φ) = 0 ⇔ φ = 0 this implies thatlimt→∞ φ(t) = 0.

Suppose thatη > 0. From (15) we obtain thatlimt→∞ V (φ(t)) = 3η2. Therefore, existst′ such thatV (φ) > 3η2/2 for all t > t′. From this fact follows thatφ cannot be zero forsomet > t′ becauseφ = 0 ⇔ V (φ) = 0. Then, the sign ofφ is invariant for allt > t′.

Suppose thatφ is positive for all t > t′. SinceV is an increasing function ofφ in(0,+∞), we have thatlimt→∞ V (φ(t)) = 3η2 ≤ limφ→∞ V (φ). By the continuity andmonotony ofV it is obvious that the equality holds if, and only if,limt→∞ φ(t) = +∞.

If limt→∞ V (φ(t)) < limφ→∞ V (φ), then there existsφ ≥ 0 such that

limt→∞

V (φ(t)) = V (φ).

SinceV is continuous and strictly increasing we have that

limt→∞

φ = φ.


By proposition 1,limt→∞(ρm(t), ρr(t), y(t)) = (0, 0, 0). Besides,H andχ′(φ)/χ(φ)are bounded. Therefore, taking the limit ast → ∞ in (12d) we find that

limt→∞

d

dty = −V ′(φ) < 0.

Hence, there existst′′ > t′ such thatddty < −V ′(φ)/2 for all t ≥ t′′. This implies

y(t)− y(t′′) =

∫ t

t′′

(d

dty

)dt < −V ′(φ)

2(t− t′′),

that is,y(t) takes negative values with arbitrary large modulus ast increases, which is notpossible sincelimt→∞ y(t) = 0.

Hence, ifφ > 0 for all t > t′, we have thatlimt→∞ φ = +∞. Similarly, whenφ < 0for all t > t′, we havelimt→∞ φ = −∞.

From this we conclude that, if initially3H(t0)2 < min limφ→∞ V (φ), limφ→−∞ V (φ) ,

then, limt→∞ H(t) = 0. Indeed, we have thatlimt→∞ φ is equal to+∞, 0 or −∞. Iflimt→∞ φ = +∞, then from the restriction (15), follows

3η2 = limt→∞

V (φ(t)) = limφ→∞

V (φ) > 3H(t0)2.

This is impossible sinceH(t) is a decreasing function andH(t0) ≥ η. In the sameway, limt→∞ φ = −∞ leads to a contradiction. Then,limt→∞ φ = 0 and this implieslimt→∞ V (φ(t)) = 0, and again by (15),limt→∞ H(t) = 0.

Thus, we have proved that if the potential has a local minimumat zero, if the derivativeof the potential is bounded in the same set where the potential itself is, and provided thederivative of the logarithm of the coupling function is bounded by above, then, the energydensities of DM and radiation, and the kinetic energy density of DE tend to zero as the timegoes forward. Hence, the Universe would expand forever in a de Sitter phase. Also we haveproved, in a similar way as in Proposition 3 in [156], that under the additional assumptionof V (φ) being strictly decreasing (increasing) ifφ < 0 (φ > 0), then the scalar field can beeither zero or divergent into the future (the former case holds if the Hubble scalar vanishesasymptotically).

In order to complement the former ideas, let us consider a non-negative potential with nonecessarily a local minimum at(0, 0), and let’s us find conditions for the stability of de Sittersolutions, and let’s characterize the asymptotic properties of the scalar field at late times.

Proposition 3 Suppose that there exists a nonzero constantK, such thatχ′(φ)/χ(φ) ≤2K/(2− γ)(4 − 3γ). LetV be a potential function with the properties:

(i) V ≥ 0 andlimφ→−∞ V (φ) = +∞.

(ii) V ′ is continuous andV ′(φ) < 0.

(iii) If A ⊂ R is such thatV is bounded inA, Then,V ′(φ) is bounded inA.

Then,limt→∞(ρm, ρr, y) = (0, 0, 0), andlimt→∞ φ = +∞.

Proof. From equations (12b) and (12c), follow that the setsρm > 0 andρr > 0 areinvariant under the flow of (12) with restriction (15); besidesρm andρr are different fromzero if ρm(t0) andρr(t0) are they are at the initial time. From this fact we have thatH isnever zero (and thus do not have changes of sign) since by (15), 3H(t)2 ≥ ρm(t) > 0 for all


t > t0, then,H is always nonnegative if initially it is. Besides, from equation (12a), followsthatH is decreasing, then∃ limt→∞ H(t) = η ≥ 0 and

1

2

∫ ∞

t0

(y2 + γρm +

4

3ρr

)dt = H(t0)− η < +∞.

As in proposition 1, the total time derivative ofy2 + γρm + 43ρr is bounded. Hence

limt→∞(ρm, ρr, y) = (0, 0, 0).It can be proved thatlimt→∞ φ = +∞ in the same way as proved in 2.From equation (15) we have thatlimt→∞ V (φ) = 3η2. SinceV is strictly decreasing

with respect toφ; then V (φ) > limφ→∞ V (φ) for all φ, thereforelimt→∞ V (φ(t)) ≥limφ→∞ V (φ). We will consider two cases:

(i) If limt→∞ V (φ(t)) = limφ→∞ V (φ), by the continuity of V is obvious thatlimt→∞ φ = +∞;

(ii) If limt→∞ V (φ(t)) > limφ→∞ V (φ), then, there exists a uniqueφ such that

limt→∞

V (φ(t)) = V (φ).

SinceV is continuous and strictly decreasing follows that

limt→∞

φ = φ.

From equation (12d) follows that

limt→∞

d

dty = −V ′(φ) > 0,

therefore, existst′ such thatddty > −V ′(φ)/2 for all t ≥ t′. From this fact we concludethat

y(t)− y(t′) > −V ′(φ)

2(t− t′),

which is impossible sincelimt→∞ y(t) = 0. Finally limt→∞ φ = +∞.

If additionally, the potential is such thatlimφ→∞ V (φ) = 0, then we conclude thatH → 0 ast → ∞.

The previous results are extensions of the Remark 1 and Propositions 4, 5 and 6 discussedin [162] when the radiation is included in the cosmic budget.

For completeness let’s show our Proposition 3 in [170], which is an extension ofthe Proposition 1 of [168] for flat FLRW models since we have included radiation. Thisproposition gives a characterization of the future attractor of the system (12) under some mildassumptions for the potential.

First, let us formalize notion of degenerate local minimum introduced in [168]:

Definition 1 The functionV (φ) is said to have a degenerate local minimum atφ⋆ if

V ′(φ), V ′′(φ), . . . V (2n−1)

vanish atφ∗, andV (2n)(φ∗) > 0, for some integern.

Then, we have the proposition:


Proposition 4 (Proposition 3 in [170]) Suppose thatV (φ) ∈ C2(R) satisfies the followingconditions§:

(i) The setφ : V (φ) < 0 is bounded;

(ii) The set of singular points ofV (φ) is finite.

Let φ∗ a strict local minimum (possibly degenerate) forV (φ), with V (φ∗) ≥ 0. Then

p∗ :=

(φ∗, y∗ = 0, ρm∗ = 0, ρr = 0, H =

√V (φ∗)

3

)is an asymptotically stable singular

point for the flow of(12).

Proof.The demonstration proceeds in an analogous way as the proof of the Proposition 1 in

[168]. The main difference is that we have considered radiation but flat FLRW geometry(k = 0). In this case the function

W (φ, y, ρm, H) ≡ H2 − 1

3

(1

2y2 + V (φ) + ρm

)=

1

3ρr (17)

evolves likeW = −4HW (18)

which decays more faster to zero than the functionW (φ, y, ρm, H) = −ka−2, k = −1, 0defined in [168] asa → +∞. (The complete proof is offered in [170]).

4. Dynamical analysis forφ → +∞.

In this section we will investigate the flow asφ → ∞ following the nomenclature andformalism introduced in [155] (see also [58] and [170]). Analogous results hold asφ → −∞.

Definition 2 (Function well-behaved at infinity [155]) Let V : R → R be a C2 non-negative function. Let there exist someφ0 > 0 for which V (φ) > 0 for all φ > φ0 andsome numberN such that the functionWV : [φ0,∞) → R,

WV (φ) =V ′(φ)

V (φ)−N

satisfieslimφ→∞

WV (φ) = 0. (19)

Then we say thatV is Well Behaved at Infinity (WBI) of exponential orderN .

Theorem 1 (Theorem 2, [155])Let V be a WBI function of exponential orderN, then, forall λ > N,

limφ→+∞

e−λφV (φ) = 0.

In order to classify the smoothness of WBI functions at infinity it is introduced thedefinition

§ Empty set is bounded and finite, and it is not excluded in the hypothesis (i) and (ii).


Definition 3 Let be some coordinate transformationϕ = f(φ) mapping a neighborhood ofinfinity to a neighborhood of the origin. Ifg is a function ofφ, g is the function ofϕ whosedomain is the range off plus the origin, which takes the values;

g(ϕ) =

g(f−1(ϕ)) , ϕ > 0

limφ→∞ g(φ) , ϕ = 0

Definition 4 (Class k WBI functions [155]) ACk functionV is class k WBI if it is WBI andif there existsφ0 > 0 and a coordinate transformationϕ = f(φ) which maps the interval[φ0,∞) onto (0, ǫ], whereǫ = f(φ0) and limφ→∞ f = 0, with the following additionalproperties:

i) f isCk+1 and strictly decreasing.ii) the functionsWV (ϕ) andf ′(ϕ) areCk on the closed interval[0, ǫ].

iii)dWV

dϕ(0) =

df ′

dϕ(0) = 0.

We designate the set of all class k WBI functionsEk+.

By assuming thatV, χ ∈ E3+, with exponential ordersN andM respectively, we can

define a dynamical system well suited to investigate the dynamics near the initial singularity.We will investigate the singular points therein. Particularly those representing scalingsolutions and those associated with the initial singularity [170].

Let’s define the new Hubble-normalized dimensionless variables [170]:

σ1 = φ, σ2 =φ√6H

, σ3 =

√ρm√3H

, σ4 =

√V√3H

, σ5 =

√ρr√3H

(20)

and the time coordinatedτ = 3Hdt. (21)

Using these coordinates the equations (12) recast as an autonomous system satisfying aninequality arising from the Friedmann equation (11e). Thissystem is given by [170]:

σ′1 =

√2

3σ2 (22a)

σ′2 = σ3

2 +1

6

(3γσ2

3 + 4σ25 − 6

)σ2 −

σ24√6

d lnV (σ1)

dσ1+

(4− 3γ)σ23

2√6

d lnχ(σ1)

dσ1, (22b)

σ′3 =

1

6σ3

(6σ2

2 + 3γ(σ23 − 1

)+ 4σ2

5

)− (4− 3γ)σ2σ3

2√6

d lnχ(σ1)

dσ1, (22c)

σ′4 =

1

6σ4

(6σ2

2 + 3γσ23 + 4σ2

5

)+

√6

6σ2σ4

d lnV (σ1)

dσ1, (22d)

σ′5 =

1

6σ5

(6σ2

2 + 3γσ23 + 4σ2

5 − 4). (22e)

The system (22) defines a flow in the phase space

Σ :=

σ ∈ R5 :

5∑

j=2

σ2j = 1, σj ≥ 0, j = 3, 4, 5

. (23)


Let Σǫ =(σ1, σ2, σ3, σ4, σ5) ∈ Σ : σ1 > ǫ−1

whereǫ is any positive constant which

is chosen sufficiently small so as to avoid any points whereV orχ = 0, thereby ensuring thatWV (ϕ) andWχ(ϕ) are well-defined.‖

Let be defined the projection map

π1 : Σǫ → Ωǫ

(σ1, σ2, σ3, σ4, σ5) → (σ1, σ2, σ4, σ5) (24)

whereΩǫ :=

σ ∈ R

4 : σ1 > ǫ−1, σ22 + σ2

4 + σ25 ≤ 1, σj ≥ 0, j = 4, 5

. (25)

Let be defined in Ωǫ the coordinate transformation(σ1, σ2, σ4, σ5)ϕ=f(σ1)−→

(ϕ, σ2, σ4, σ5) wheref(σ1) tends to zero asσ1 tends to+∞ and has been chosen so thatthe conditions i)-iii) of definition 4 are satisfied withk = 2.

The flow of (22) defined onΣǫ is topologically equivalent (underf π1) to the flow ofthe 4-dimensional dynamical system [170]:

ϕ′ =

√2

3f ′σ2, (26a)

σ′2 = σ3

2 +

(2σ2

5

3− 1

)σ2 −

(WV +N

)σ24√

6+

+

((Wχ +M

)(4− 3γ)

2√6

+σ2γ

2

)(1− σ2

2 − σ24 − σ2

5

)(26b)

σ′4 =

1

6σ4

(√6(WV +N

)σ2 + 3(2− γ)σ2

2 + 3γ(1− σ24) + (4− 3γ)σ2

5

), (26c)

σ′5 =

1

6σ5

(3(2− γ)σ2

2 − 3γσ24 − (4− 3γ)(1− σ2

5)), (26d)

defined in the phase space¶

Ωǫ = (ϕ, σ2, σ4, σ5) ∈ R4 : 0 ≤ ϕ ≤ f(ǫ−1), σ2

2 + σ24 + σ2

5 ≤ 1, σ4 ≥ 0, σ5 ≥ 0. (27)

It can be easily proved that (27) defines a manifold with boundary of dimension 4. Itsboundary,∂Ωǫ, is the union of the setsp ∈ Ωǫ : ϕ = 0, p ∈ Ωǫ : ϕ = f(ǫ−1), p ∈Ωǫ : σ4 = 0, p ∈ Ωǫ : σ5 = 0 with the unitary 3-sphere. The coordinates, existenceconditions and stability of the singular points of the flow of(26) in the phase space (27) arepresented in the Appendix A. Finally, the physical description of the solutions and connectionwith observables is discussed in Appendix B. The results discussed in both appendices werefirst published in our reference [170].

5. Examples

In this section we discuss the example of a double exponential potential, presented in [91],applying the procedure of [155, 170]. Our new results complement those in [91]. Next werevisit the example of a powerlaw coupling function and an Albrecht-Skordis potential, firstintroduced in [161], and then extended in section 4.4 of [170].

‖ See 3 for the definition of functions with bar.¶ For notational simplicity we will denote the image ofΩǫ underf by the same symbol.


5.1. Double exponential potential and exponential coupling function

First let’s discuss the case presented in [91]. In this example the potential is double

exponential:V (φ) = V1e−αφ + V2e

−βφ, 0 < α < β, andχ = χ0 exp[

λφ4−3γ

], whereλ is

a constant. Then, forγ < 43 , chooseK ≥ (2−γ)λ

2 , and for 43 < γ ≤ 2, chooseK ≤ (2−γ)λ2 .

Thus, the hypothesis forχ in our results it is easily fulfilled. SinceV1 > 0, V2 > 0, we haveV ′(φ) < 0 < V (φ), ∀φ. Furthermore, under the hypothesis0 < α < β, we have the strongrestriction−βV (φ) < V ′(φ) < −αV (φ), ∀φ. Thus, the hypothesis of our Proposition 3 aresatisfied and we result in the Proposition 1 of [91].

Additionally we have that under the hypothesis0 < α < β, V (φ) = V1e−αφ + V2e

−βφ

is well-behaved atφ → +∞ with exponential orderN = −α. On the other hand the couplingfunction is well behaved of exponential orderM = λ

4−3γ . Under the transformationϕ = φ−1

we have thatV, χ ∈ Ek+, and

Wχ(ϕ) = 0, (28)

WV (ϕ) =

V2(α−β)eαϕ

V1eβϕ +V2e

αϕ

, ϕ > 0

0 , ϕ = 0, (29)

f ′(ϕ) = −ϕ2, (30)

In this example, the evolution equations forϕ, σ2, σ4, andσ5 are given by the equations(26) withM = λ

4−3γ , N = −α, andWχ(ϕ), WV (ϕ), andf ′, given by (28), (29) and (30)respectively. The state space is defined by

Ωǫ = (ϕ, σ2, σ4, σ5) ∈ R4 : 0 ≤ ϕ ≤ ǫ, σ2

2 + σ24 + σ2

5 ≤ 1, σ4 ≥ 0, σ5 ≥ 0.

Now, since the functionWV defined by (29) is a transcendental function forϕ = 0, wecan introduce the new variable

v = − V2(α− β)eαϕ

V1eβϕ + V2e

αϕ

≥ 0 (31)

which is better for numerical integrations. It can be provedthat under the hypothesis0 < α < β, v → 0+ asϕ → 0+.

Using the coordinate transformation (31) we obtain the dynamical system

v′ =

√2

3σ2v(α− β + v), (32a)

σ′2 =

ασ24√6

+1

6σ2

(−3γ

(σ22 + σ2

4 + σ25 − 1

)+ 6σ2

2 + 4σ25 − 6

)− λ

(σ22 + σ2

4 + σ25 − 1

)

2√6

,

(32b)

σ′4 = −1

6σ4

(√6ασ2 + 3(γ − 2)σ2

2 + 3γ(σ24 + σ2

5 − 1)− 4σ2

5

)− σ2σ4v√

6, (32c)

σ′5 =

1

6σ5

(−3γ

(σ22 + σ2

4 + σ25 − 1

)+ 6σ2

2 + 4σ25 − 4

), (32d)

defined in the phase space

Ψǫ = (v, σ2, σ4, σ5) ∈ R4 : 0 ≤ v ≤ V2(β − α)eα/ǫ

V1eβ/ǫ + V2eα/ǫ, σ2

2+σ24+σ2

5 ≤ 1, σ4 ≥ 0, σ5 ≥ 0.


Table 1. Location of the singular points of the flow of (32) defined in the invariant setp ∈ Ωǫ : ϕ = 0 for M = λ

4−3γandN = −α. We use the definitionsΓ+(α, γ) =

α±√

α2 + 6γ2 − 12γ, andΥ(γ) =√2√

(4− 3γ)(2 − γ).

Label (σ2, σ4, σ5) Existence Stabilitya

P1 (−1, 0, 0) always unstable ifα > −√6, λ > −

√6(2− γ)

saddle otherwiseP2 (1, 0, 0) always unstable ifα <

√6, λ <

√6(2− γ)

saddle otherwise

P3

(λ√

6(2−γ), 0, 0

)λ2 ≤ 6(2− γ)2 stable for

0 < γ ≤ 23 , and

√6(2− γ)γ < α ≤

√8(2−γ)√4−3γ

, andΓ−(α, γ) < λ < Γ+(α, γ)

or

0 < γ ≤ 23 , and

α >

√8(2−γ)√4−3γ

, andΓ−(α, γ) < λ < Υ(γ)

or

23 < γ < 4

3 , and

α >

√8(2−γ)√4−3γ

, andΓ−(α, γ) < λ < Υ(γ)saddle otherwise

R1 (0, 0, 1) always saddle

P4

(α√6,√1− α2

6 , 0

)α2 < 6 stable for0 < α < 2, λ > 2α2−6γ

α

saddle otherwise

P5,6

( √6γ

2α−λ ,±√

−2αλ−6(γ−2)γ+λ2

λ−2α , 0

)α(λ−2α)+6γ

(λ−2α)2 ≤ 0 Numerical inspection

R2

(√23 (4−3γ)

λ , 0,

√−6γ2+20γ+λ2−16

|λ|

)γ < 4

3 , λ2 ≥ Υ(γ)2 stable for

γ < 43 ,

α >√

8(2−γ)4−3γ ,

Υ(γ) < λ < 12α(4− 3γ)

saddle otherwise

R3

(2√

23

α , 2√3α

,√α2−4α

)α ≥ 2 stable ifα > 2, λ > 1

2α(4− 3γ),

saddle otherwise

a The stability is analyzed for the flow restricted to the invariant setv = 0.

In table 1 are summarized the location, existence conditions and stability of the singularpoints of the system (32). The stability is analyzed for the flow restricted to the invariant setv = 0, i.e., we are not taking into account perturbations along thev-axis.

Let us discuss some physical properties of the cosmologicalsolutions associated to thesingular points displayed in table 1.

• P1,2 represent kinetic-dominated cosmological solutions. They behave as stiff-likematter. The associated cosmological solution satisfiesH = 1

3t−c1, a = 3

√3t− c1c2, φ =


c3 ±√

23 ln (3t− c1) , wherecj , j = 1, 2, 3 are integration constants. These solutions

are associated with the local past attractors of the system for an open set of values of theparametersα andλ.

• P3 represents a matter-kinetic scaling solution. The associated asymptoticcosmological solutions satisfy the ratesH = 8−4γ

4(γ−2)c1+t(λ2−6(γ−2)γ) , a =

c2(4(γ − 2)c1 + t

(λ2 − 6(γ − 2)γ

))− 4(γ−2)

λ2−6(γ−2)γ , ρm = 48(γ−2)2−8λ2

(t(6(γ−2)γ−λ2)−4(γ−2)c1)2+

c3, φ = ln[(4(γ − 2)c1 + t

(λ2 − 6(γ − 2)γ

)) 4λλ2

−6(γ−2)γ

]+ c4, where cj , j =

1, 2, 3, 4 are integration constants. These solutions are stable or saddle depending onthe parametersα andγ.

• PointR1 represents a radiation dominated solution, ans asymptotically the cosmologicalsolutions satisfy the ratesH = 1

2t−c1, a = c2

√2t− c1, ρr = c3 − 6

c1−2t . It is always asaddle point.

• P4 represents power-law scalar-field dominated inflationary cosmological solutions.As φ → +∞ the potential behaves asV ∼ V1 exp[−αφ]. Thus it is easy toobtain the asymptotic exact solution:H = 2

α2t−2c1, a = c2

(α2t− 2c1

) 2α2 , φ =

ln[(α2t− 2c1

) 2α

]+ c3.

• P5,6 represent matter-kinetic-potential scaling solutions. In the limit φ →+∞ we have the asymptotic expansions:H = 2α−λ

−2αc1+c1λ+3αγt , a =

c2 (c1(λ− 2α) + 3αγt)2α−λ3αγ , ρm =

6(2α2−αλ−6γ)(c1(λ−2α)+3αγt)2+c3, φ = ln

[(c1(λ− 2α) + 3αγt)

2α

]+

c4.

• R3 represents a radiation-matter-scalar field scaling solution. In the limitφ → +∞we have the asymptotic expansions:H = 1

2t−c1, a = c2

√2t− c1, ρm = 16−12γ

λ2(c1−2t)2 +

c3, ρr = c4 −6(−6γ2+20γ+λ2−16)

λ2(c1−2t) , φ = ln[(λ (2t− c1))

(4−3γ)λ

]+ c5.

• R3 represents radiation-kinetic-potential scaling solutions. The associated cosmologicalsolutions satisfy the asymptotic expansions:H = 1

2t−c1, a = c2

√2t− c1, ρr =

c3 −6(α2−4)α2(c1−2t) , φ = ln

[(2αt− αc1)

2α

]+ c4 asφ → +∞.

In order to illustrate our analytical results we proceed to some numerical experimenta-tion.

In the figures 1 and 2 are presented projections of the flow of (32) restricted to theinvariant setσ5 = 0 ⊂ Ψǫ on the planes(σ2, v) and(σ2, σ4), respectively, for the potential

V (φ) = V1e−αφ + V2e

−βφ, and the coupling functionχ = χ0 exp[

λφ4−3γ

]. We select the

values of the parameters:V1 = 1, V2 = 2, α = 1.5, β = 2, λ = −5 andγ = 1. Thesenumerical simulations confirms the stability ofP6 for this choice of parameters.

In figure 3 we show some orbits in the invariant setσ22 + σ2

4 + σ25 ≤ 1, ϕ = 0 for the

potentialV (φ) = V1e−αφ + V2e

−βφ, and the coupling functionχ = χ0 exp[

λφ4−3γ

]. For the

choiceV1 = 1, V2 = 2, α = 2.4, β = 3, λ = 1.1 andγ = 1 results that powerlaw solutionP6

is the attractor.P1 andP2 are sources whereasP3, P4, R1 andR3 are saddles.R2 does notexists.


Figure 1. Projection in the plane(σ2, v) of the flow of (32) restricted to the invariant setσ5 = 0 ⊂ Ψǫ for the potentialV (φ) = V1e

−αφ + V2e−βφ, and the coupling function

χ = χ0 exp[

λφ4−3γ

]

. We select the values of the parameters:V1 = 1, V2 = 2, α =

1.5, β = 2, λ = −5 andγ = 1. In the figure it is illustrated the stability ofP6 along thev-axis.

5.2. Coupling Functions and Potentials of Exponential OrdersM = 0 andN = −µ 6= 0,Respectively

As an example let us considerχ, V ∈ E2+ of exponential ordersM = 0 andN = −µ,

respectively [170]. This class of potentials contains the cases investigated in [85, 197] (thereare not considered coupling to matter, i.e.,χ(φ) ≡ 1, in the second case, for flat FLRWcosmologies), the case investigated in [143] (for positivepotentials and standard flat FLRWdynamics), the example examined in [161], etc.

In table 2 are summarized the location, existence conditions and stability of the singularpoints for the flow at the invariant setϕ = 0.

Let us discuss some physical properties of the cosmologicalsolutions associated to thesingular points displayed in table 2 [170]:

• P1,2 represent kinetic-dominated cosmological solutions. They behave as stiff-likematter and can be local past attractors of the systems for an open set of values of theparameterµ. The same rates fora,H, andφ presented in section (5.1) applies here too.

• P3 represents matter-dominated cosmological solutions thatsatisfyH = 23tγ−2c1

, a =

(3tγ − 2c1)23γ c2, ρm = 12

(3tγ−2c1)2+ c3.

• R1 represents a radiation-dominated cosmological solutionssatisfyingH = 12t−c1

, a =√2t− c1c2, ρr =

3(2t−c1)2

+ c3.

• P4 represents power-law scalar-field dominated inflationary cosmological solutions.As φ → +∞ the potential behaves asV ∼ V0 exp[−µφ]. Thus it is easy to


Figure 2. Projection in the plane(σ2, σ4) of the flow of (32) restricted to the invariant setσ5 = 0 ⊂ Ψǫ for the potentialV (φ) = V1e

−αφ + V2e−βφ, and the coupling function

χ = χ0 exp[

λφ4−3γ

]

. We select the values of the parameters:V1 = 1, V2 = 2, α =

1.5, β = 2, λ = −5 andγ = 1. In the figure it is illustrated the stability ofP6. P1 is a sourceandP2 andP4 are saddles.P3 does not exists.

obtain the asymptotic exact solution:H = 2tµ2−2c1

, a =(tµ2 − 2c1

) 2µ2 c2, φ ∼

1µ ln

[V0(tµ

2−2c1)2

2(6−µ2)

].

• P5,6 represent matter-kinetic-potential scaling solutions. As before, in the limitφ →+∞ we obtain the asymptotic expansions:H = 2

3tγ−2c1, a = (3tγ − 2c1)

23γ c2, φ ∼

1µ ln

[V0µ

2(3tγ−2c1)2

18(2−γ)γ

].

• R3 represent radiation-kinetic-potential scaling solutions. As before are deducedthe following asymptotic expansions:H = 1

2t−c1, a =

√2t− c1c2, φ ∼

1µ ln

[v0µ

2(2t−c1)2

4

].

5.2.1. Powerlaw coupling and Albrecht-Skordis potential.Now, let’s revisit the example ofa powerlaw coupling function and an Albrecht-Skordis potential, first introduced in [161], andthen extended in section 4.4 of [170].

Let us consider the coupling function

χ(φ) =

(3α

8

) 1α

χ0(φ− φ0)2α , α > 0, const., φ0 ≥ 0. (33)

and the Albrecht-Skordis potential [198]:

V (φ) = e−µφ(A+ (φ−B)2

). (34)


Figure 3. Some orbits in the invariant setσ22 + σ2

4 + σ25 ≤ 1 for the flow of (32) restricted

to the invariant setv = 0 ⊂ Ψǫ for the potentialV (φ) = V1e−αφ + V2e

−βφ,

and the coupling functionχ = χ0 exp[

λφ4−3γ

]

. We select the values of the parameters:

V1 = 1, V2 = 2, α = 2.4, β = 3, λ = 1.1 andγ = 1. In the figure it is illustrated thestability ofP6. P1 andP2 are sources whereasP3, P4, R1 andR3 are saddles.R2 does notexists.

The coupling function (33) and the potential (34) are WBI of exponential ordersM = 0andN = −µ, respectively.

It is easy to prove that Power-law coupling and the Albrecht-Skordis potential are at leastE2+, under the admissible coordinate transformation [170]:+

ϕ = φ− 12 = f(φ). (35)

Using the above coordinate transformation we find

Wχ(ϕ) =2ϕ2

α(1 − ϕ2φ0). (36)

WV (ϕ) = − 2ϕ2(Bϕ2 − 1)

Aϕ4 + (Bϕ2 − 1)2. (37)

+ We fix here an error in formulas B6-B9 in [161]. With the choiceϕ = φ−1 the resulting barred functions givenby B7-B9 there, are not of the desired differentiable class.


Table 2. Location of the singular points of the flow of (26) defined in the invariant setp ∈ Ωǫ : ϕ = 0 for M = 0 andN = −µ. (Taken from [170]).

Label (σ2, σ4, σ5) Existence Stabilitya

P1 (−1, 0, 0) always unstable ifµ > −√6

saddle otherwiseP2 (1, 0, 0) always unstable ifµ <

√6

saddle otherwiseP3 (0, 0, 0) always saddle

R1 (0, 0, 1) always saddle

P4

(µ√6,√1− µ2

6 , 0

)µ2 < 6 stable for

0 < γ < 4

3 , µ2 < 3γ, or43 < γ < 2, µ2 < 2

,

saddle otherwise

P5,6

(√32γµ ,± 1

µ

√32 (2− γ)γ, 0

)µ2 > 3γ stable for

0 < γ < 29 , µ

2 > 3γ, or29 < γ < 4

3 , 3γ < µ2 < 24γ2

9γ−2 , or29 < γ < 4

3 , µ2 > 24γ2

9γ−2

,

saddle otherwise

R3

(2√

23

µ , 2√3|µ| ,

√µ2−4

|µ|

)|µ| > 2 stable if 43 < γ < 2, saddle otherwise

a The stability is analyzed for the flow restricted to the invariant setϕ = 0.

and

f ′(ϕ) = −1

2ϕ3. (38)

In this example, the evolution equations forϕ, σ2, σ4, andσ5 are given by the equations(26) withM = 0, N = −µ, andWχ(ϕ), WV (ϕ) = 0, andf ′, given by (36), (37) and (38)respectively. The state space is defined by

Ωǫ = (ϕ, σ2, σ4, σ5) ∈ R4 : 0 ≤ ϕ ≤

√ǫ, σ2

2 + σ24 + σ2

5 ≤ 1, σ4 ≥ 0, σ5 ≥ 0.

Finally, let us discuss some numerical simulations.In the figure 4 are presented some orbits in the invariant setσ3 = 0, σ5 = 0 ⊂ Σǫ for

the model with coupling function (33) potential (34). We select the values of the parameters:ǫ = 1.00, µ = 2.00, A = 0.50, α = 0.33, B = 0.5, φ0 = 0, andγ = 1. Observe that almostall the orbits are past asymptotic toP1; P2 is a saddle, and the center manifold ofP4 attracts allthe orbits in theσ3 = 0. However, it is not an attractor in the invariant setσ3 > 0, ϕ = 0.In figure 5 are presented some orbits in the invariant setϕ = 0, σ5 = 0 ⊂ Σǫ for the samevalues of the parameters as before.P1,2 are local past attractors, butP1 is the global pastattractor;P3,4 are saddles, andP5 is a local future attractor.

In the figure 6 are displayed some orbits in the invariant setσ22 + σ2

4 + σ25 ≤ 1 for the

choice ofϕ = 0 for the model with coupling function (33) potential (34). Weselect the valuesof the parameters:γ = 1, ǫ = 1.00, µ = 2.10, A = 0.50, α = 0.33, B = 0.5, andφ0 = 0. Inthis caseP5 is the local sink in this invariant set.R3 exists and it is a saddle.


P4 P2P1

ϕ

σ2

Figure 4. Orbits in the invariant setσ3 = 0, σ5 = 0 ⊂ Σǫ for the model withcoupling function (33) potential (34). We select the valuesof the parameters:ǫ = 1.00,µ = 2.00, A = 0.50, α = 0.33, B = 0.5, φ0 = 0, andγ = 1. Observe that i) almost all theorbits are past asymptotic toP1; ii) P2 is a saddle, and iii) the center manifold ofP4 attracts allthe orbits in theσ3 = 0. However, it is not an attractor in the invariant setσ3 > 0, ϕ = 0(see figure 5). (Taken from [162]).

6. Conclusion

In this paper we have extended several results related to flatFLRW models in theconformal (Einstein) frame of scalar-tensor gravity theories. We have considered scalarfields with arbitrary (positive) potentials and arbitrary coupling functions. Then, we havestraightforwardly introduced mild assumptions under suchfunctions (differentiable class,number of singular points, asymptotes, etc.) in order to characterize the asymptotic structureon a phase-space. Also, we have presented several numericalevidences that confirm some ofthese results.

Our main results are the following.

(i) Proposition 1 states that for non-negative potentials with a local zero minimum atφ = 0;such that its derivative is bounded in the same set where the potential is; and provided thederivative of the logarithm of the coupling function has an upper bound, then the energydensities o matter and radiation as well as the kinetic term tend to zero when the timegoes forward. Thus, the Universe would expand forever in a deSitter phase in the future.This result is an extension of the Proposition 2 in [156] to the non-minimal couplingcontext. It is also an extension of Proposition 4 in [162] when the radiation is includedin the cosmic budget.

(ii) Under the same hypotheses as in 1 and provided thatV (φ) is strictly decreasing(increasing) for negative (positive) values of the scalar field, then the scalar field divergesinto the future or it equals to zero (the last case holds only if the Hubble scalar vanishtowards the future). This Proposition 2 is an extension of proposition 3 in [156] and ofProposition 5 in [162] when the radiation is included in the background.


P4 P2P1

P5

P3

σ3

σ2

Figure 5. Orbits in the invariant setϕ = 0, σ5 = 0 ⊂ Σǫ for the model withcoupling function (33) and potential (34). We select the values of the parameters:ǫ = 1.00,µ = 2.00, A = 0.50, α = 0.33, B = 0.5, φ0 = 0, andγ = 1. In the figure i)P1,2 are localpast attractors, butP1 is the global past attractor; ii)P3,4 are saddles, and iii)P5 is a localfuture attractor. (Taken from [162]).

(iii) Assuming that the potential is non-negative (with notnecessarily a local minimum at(0, 0)), such that forφ → +∞ it is unbounded. If its derivative is continuous andbounded on a setA where the potential is bounded. Then the cosmological modelevolves to a late-time de Sitter solution characterized by the divergence of the scalarfield. Additionally, if the potential vanish asymptotically, the Hubble scalar vanishes too(see Proposition 3). Proposition 3 is an extension forρr > 0 of Proposition 6 discussedin [162].

(iv) We have formulated and proved the Proposition 4 (Proposition 3 in [170]) generalizinganalogous result in [168]. Our result states that if the potential V (φ) is such that the(possibly empty set) where it is negative is bounded and the (possibly empty) set ofsingular points ofV (φ) is finite, then, the singular point

p∗ :=

(φ∗, y∗ = 0, ρm∗ = 0, ρr = 0, H =

√V (φ∗)

3

),

whereφ∗ is a strict local minimum forV (φ), corresponding to ade Sittersolution, is anasymptotically stable singular point for the flow.

(v) For the analysis of the system asφ → ∞ we have defined a suitable changeof variables to bring a neighborhood ofφ = ∞ in a bounded set. In thisregime we found: radiation-dominated cosmological solutions; power-law scalar-fielddominated inflationary cosmological solutions; matter-kinetic-potential scaling solutionsand radiation-kinetic-potential scaling solutions. The rigorous mathematical apparatuswas developed in section 4.


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

σ2

σ4

σ 5

P1

P2

P3

P5

P6

P4

R1

R3

Figure 6. Some orbits in the invariant setσ22 + σ2

4 + σ25 ≤ 1 for the choice ofϕ = 0 for

the model with coupling function (33) potential (34). We select the values of the parameters:γ = 1, ǫ = 1.00, µ = 2.10, A = 0.50, α = 0.33, B = 0.5, andφ0 = 0. (Taken from[162]).

(vi) Using the above procedure we have investigated the behavior at the limitφ → +∞ forthe following models:(i) a double exponential potentialV (φ) = V1e

−αφ + V2e−βφ, 0 < α < β, and the

coupling functionχ = χ0 exp[

λφ4−3γ

], whereλ is a constant discussed in [91] and

(ii) a general class of potentials containing the cases investigated in [85, 197] and in[198]. We have re-examined the toy model with power-law coupling and Albrecht-Skordis potentialV (φ) = e−µφ

(A+ (φ−B)2

)[198] investigated in [161] in presence

of radiation.

Acknowledgements

CRF wish to thank the MES of Cuba for partial financial supportof this investigation.GL was supported by COMISION NACIONAL DE CIENCIAS Y TECNOLOGIA throughProyecto FONDECYT DE POSTDOCTORADO 2014 grant 3140244 and by DI-PUCV grant123.730/2013. The authors wish to thank to two anonymous referees for helpful suggestions.

Appendix A. Singular points of the flow of (26) in the phase space(27).

The system (26) admits the following singular points (takenfrom [170]).

(i) The singular pointP1 with coordinatesϕ = 0, σ2 = −1, σ4 = 0, σ5 = 0 always exists.The eigenvalues of the linearisation around the singular point are0, 1

3 , 1− N√6, M(4−3γ)√

6−

γ + 2. Thus,


(a) P1 has a 1-dimensional center manifold tangent to theϕ-axis providedN 6=√6

andM 6= −√6(γ−2)3γ−4 (otherwise the center manifold would be 2- or 3-dimensional).

(b) P1 admits a 3-dimensional unstable manifold and a 1-dimensional center manifold

for N <√6, 0 < γ < 4

3 , M > −√6(γ−2)3γ−4 ; or N <

√6, 4

3 < γ < 2, M <

−√6(γ−2)3γ−4 . In this case the center manifold ofP1 acts as a local source for an open

set of orbits in (27).(c) P1 admits a 2-dimensional unstable manifold, a 1-dimensionalstable manifold and

a 1-dimensional center ifN >√6, 0 < γ < 4

3 , M > −√6(γ−2)3γ−4 ; or N >

√6, 4

3 < γ < 2, M < −√6(γ−2)3γ−4 ; or N <

√6, 0 < γ < 4

3 , M < −√6(γ−2)3γ−4 ; or

N <√6, 4

3 < γ < 2, M > −√6(γ−2)3γ−4 .

(d) P1 admits a 1-dimensional unstable manifold, a 2-dimensionalstable manifold and

a 1-dimensional center manifold forN >√6, 0 < γ < 4

3 , M < −√6(γ−2)3γ−4 ; or

N >√6, 4

3 < γ < 2, M > −√6(γ−2)3γ−4 .

(ii) The singular pointP2 with coordinatesϕ = 0, σ2 = 1, σ4 = 0, σ5 = 0 always exists.The eigenvalues of the linearization around the singular point are0, 1

3 , 1 + N√6,−γ +

M(3γ−4)√6

+ 2. As before, let us determine conditions on the free parameters for theexistence of center, unstable and stable manifolds forP2.

(a) If N 6= −√6 andM 6=

√6(γ−2)3γ−4 there exists a 1-dimensional center manifold

tangent to theϕ-axis, otherwise the center manifold would be 2- or 3-dimensional.(b) P2 has a 3-dimensional unstable manifold a a 1-dimensional center manifold

(tangent theϕ-axis) if N > −√6, 0 < γ < 4

3 , M <√6(γ−2)3γ−4 ; or N > −

√6, 4

3 <

γ < 2, M >√6(γ−2)3γ−4 . In this case the center manifold ofP2 acts as a local source

for an open set of orbits in (27).(c) P2 has a 2-dimensional unstable manifold a 1-dimensional stable and a 1-

dimensional center manifold ifN < −√6, 0 < γ < 4

3 , M <√6(γ−2)3γ−4 ; or N <

−√6, 4

3 < γ < 2, M >√6(γ−2)3γ−4 ; or N > −

√6, 0 < γ < 4

3 , M >√6(γ−2)3γ−4 ; or

N > −√6, 4

3 < γ < 2, M <√6(γ−2)3γ−4 .

(d) P2 has a 1-dimensional unstable manifold a 2-dimensional stable and a 1-

dimensional center manifold ifN < −√6, 0 < γ < 4

3 , M >√6(γ−2)3γ−4 ; or

N < −√6, 4

3 < γ < 2, M <√6(γ−2)3γ−4 .

(iii) The singular pointP3 with coordinatesϕ = 0, σ2 = M(3γ−4)√6(γ−2)

, σ4 = 0, σ5 = 0 exists for

(a) 0 < γ < 43 , −

√6(γ−2)3γ−4 ≤ M ≤

√6(γ−2)3γ−4 ; or

(b) 43 < γ < 2,

√6(γ−2)3γ−4 ≤ M ≤ −

√6(γ−2)3γ−4 .

The eigenvalues of the linearization are0, λ1 = − (3γ−4)((3γ−4)M2−2γ+4)12(γ−2) , λ2 =

−M2(4−3γ)2+6(γ−2)γ+2MN(3γ−4)12(γ−2) , λ3 = 6(γ−2)2−M2(4−3γ)2

12(γ−2) . As before, let usdetermine conditions on the free parameters for the existence of center, unstable andstable manifolds forP3.

(a) For γ,N andM such thatλ1 6= 0, λ2 6= 0, λ3 6= 0 the center manifold is 1-dimensional and tangent to theϕ-axis. Otherwise the center manifold coud be 2-,or 3-dimensional (it is never 4-dimensional).


(b) P3 admits a 1-dimensional center manifold and a 3-dimensionalstable manifold for

0 < γ < 43 , −

√2√γ−2√

3γ−4< M < 0, N > M2(4−3γ)2−6(γ−2)γ

2M(3γ−4) ; or 0 < γ < 43 , 0 <

M <√2√γ−2√

3γ−4, N < M2(4−3γ)2−6(γ−2)γ

2M(3γ−4) .

(c) The unstable manifold ofP3 is 2-dimensional (thus its stable and center manifolds

are both 1-dimensional) in the cases0 < γ < 43 , −

√6(γ−2)3γ−4 < M <

−√2√γ−2√

3γ−4, N < M2(4−3γ)2−6(γ−2)γ

2M(3γ−4) ; or 0 < γ < 43 ,

√2√γ−2√

3γ−4< M <

√6(γ−2)3γ−4 , N > M2(4−3γ)2−6(γ−2)γ

2M(3γ−4) ; or 43 < γ < 2,

√6(γ−2)3γ−4 < M < 0, N >

M2(4−3γ)2−6(γ−2)γ2M(3γ−4) ; or 4

3 < γ < 2,M = 0, N ∈ R; or 43 < γ < 2, 0 < M <

−√6(γ−2)3γ−4 , N < M2(4−3γ)2−6(γ−2)γ

2M(3γ−4) , Otherwise,P3 has a 1-dimensional unstablemanifold. Thus, it is never a local source since its unstablemanifold is of dimensionless than3.

(iv) The singular pointR1 with coordinatesϕ = 0, σ2 = 0, σ4 = 0, σ5 = 1 alwaysexists. The eigenvalues of the linearization are0, 2

3 ,− 13 ,

43 − γ. The center manifold is

1-dimensional and tangent to theϕ-axis. The unstable (stable) manifold is 1-dimensional(2-dimensional) ifγ > 4

3 otherwise it is 2-dimensional (1-dimensional).

(v) The singular pointR2 with coordinatesσ2 =

√23

M , σ4 = 0, σ5 =

√

4−2γ

M2 +3γ−4√3γ−4

exists for 0 < γ < 43 , M

2 ≥ 2(γ−2)3γ−4 . The eigenvalues of the linearization are

0,−M+√

3M2(4γ−5)−8(γ−2)

6M ,

√3M2(4γ−5)−8(γ−2)−M

6M , 13

(NM + 2

). Let us determine

conditions on the free parameters for the existence of center, unstable and stablemanifolds forR2.

(a) R2 has a 3-dimensional stable manifold and a 1-dimensional center manifold if

0 < γ < 54 ,−2

√23

√γ−24γ−5 ≤ M < −

√2√

γ−23γ−4 , N > −2M ; or 0 < γ <

54 ,√2√

γ−23γ−4 < M ≤ 2

√23

√γ−24γ−5 , N < −2M ; or 5

4 ≤ γ < 43 ,M <

−√2√

γ−23γ−4 , N > −2M ; or 5

4 ≤ γ < 43 ,M >

√2√

γ−23γ−4 , N < −2M ;

or 0 < γ < 54 ,M < −2

√23

√γ−24γ−5 , N > −2M ; or 0 < γ < 5

4 ,M >

2√

23

√γ−24γ−5 , N < −2M.

(b) By reversing the sign of the last inequality, i.e., the inequality solved forN , inthe previous six cases we obtain conditions forR2 having a 2-dimensional stablemanifold, a 1-dimensional unstable manifold and a 1-dimensional center manifold.

(vi) The singular point P4 with coordinates ϕ = 0, σ2 = − N√6, σ4 =

√1− N2

6 , σ5 = 0 exists wheneverN2 < 6. The eigenvalues of the linearization are

0, 16

(N2 − 6

), 16

(N2 − 4

), 13N(2M + N) − 1

2 (MN + 2)γ. The conditions for theexistence of stable, unstable and center manifolds is as follows.

(a) The center manifold is 1-dimensional and the stable manifold is 3-dimensional

providedN = 0, M ∈ R, γ 6= 43 ; or 0 < γ < 4

3 ,−2 < N < 0,M >2(N2−3γ)N(3γ−4) ;

or 0 < γ < 43 , 0 < N < 2,M <

2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2,−2 < N < 0,M <

2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2, 0 < N < 2,M >2(N2−3γ)N(3γ−4) .

(b) The stable manifold is 2-dimensional, the unstable manifold is 1-dimensional and


the center manifold is 1-dimensional provided0 < γ < 43 ,−

√6 < N < −2,M >

2(N2−3γ)N(3γ−4) ; or 0 < γ < 4

3 ,−2 < N < 0,M <2(N2−3γ)N(3γ−4) ; or 0 < γ < 4

3 , 0 <

N < 2,M >2(N2−3γ)N(3γ−4) ; or 0 < γ < 4

3 , 2 < N <√6,M <

2(N2−3γ)N(3γ−4) ;

or 43 < γ < 2,−

√6 < N < −2,M <

2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2,−2 <

N < 0,M >2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2, 0 < N < 2,M <2(N2−3γ)N(3γ−4) ; or

43 < γ < 2, 2 < N <

√6,M >

2(N2−3γ)N(3γ−4) .

(c) The stable manifold is 1-dimensional, the unstable manifold is 2-dimensional andthe center manifold is 1-dimensional provided0 < γ < 4

3 ,−√6 < N < −2,M <

2(N2−3γ)N(3γ−4) ; or 0 < γ < 4

3 , 2 < N <√6,M >

2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2,−√6 <

N < −2,M >2(N2−3γ)N(3γ−4) ; or 4

3 < γ < 2, 2 < N <√6,M <

2(N2−3γ)N(3γ−4) .

(vii) The singular point R3 with coordinates ϕ = 0, σ2 = − 2√

23

N , σ4 =2√3|N | , σ5 =

√N2−4|N | exists for N2 ≥ 4. The eigenvalues of the lineariza-

tion are0, 16

(−

√64N2−15N4

N2 − 1), 16

(√64N2−15N4

N2 − 1),− (2M+N)(3γ−4)

3N . The con-

ditions for the existence of stable, unstable and center manifolds are as follows.

(a) The stable manifold is 3-dimensional and the center manifold is 1-dimensionalprovided0 < γ < 4

3 , N < − 8√15,M > −N

2 ; or 0 < γ < 43 ,− 8√

15≤

N < −2,M > −N2 ; or 0 < γ < 4

3 , 2 < N ≤ 8√15,M < −N

2 ; or

0 < γ < 43 , N > 8√

15,M < −N

2 ; or 43 < γ < 2, N < − 8√

15,M < −N

2 ; or43 < γ < 2,− 8√

15≤ N < −2,M < −N

2 ; or 43 < γ < 2, 2 < N ≤ 8√

15,M >

−N2 ; or 4

3 < γ < 2, N > 8√15,M > −N

2 .

(b) By reversing the sign of the last inequality, i.e., the inequality solved forM , inthe previous eight cases we obtain conditions forR3 having a 2-dimensional stablemanifold, a 1-dimensional unstable manifold and a 1-dimensional center manifold.

(viii) The singular pointP5 with coordinates

ϕ = 0, σ2 =√6γ

M(3γ−4)−2N , σ4 =

√M2(4−3γ)2+MN(8−6γ)−6(γ−2)γ

2N+M(4−3γ) , σ5 = 0 existsfor 2(2M + N) > 3Mγ, M(3γ − 4)(M(3γ − 4) − 2N) ≥ 6(γ − 2)γ, and3(MN+2)γ−2N(2M+N)

(2N+M(4−3γ))2 ≤ 0. The eigenvalues of the linearization are

0,12M+6N−3(3M+N)γ+

√3√

f(γ,M,N)

6(M(3γ−4)−2N) ,3N(γ−2)+3M(3γ−4)+

√3√

f(γ,M,N)

6(2N+M(4−3γ)) , (2M+N)(3γ−4)6N+3M(4−3γ) ,

wheref(γ,M,N) = 2M3N(3γ − 4)3 + 2MN(4N2 − 6γ2 + 3γ − 6

)(3γ − 4) −

M2(8N2 − 12γ − 3

)(4− 3γ)2+3(γ− 2)

(N2(9γ − 2)− 24γ2

). The stability condi-

tions ofP5 are very complicated to display them here. Thus we must rely on numericalexperimentation. We can obtain, however, some analytic results. For instance, there ex-ists at least a 1-dimensional center manifold. The unstablemanifold is always of dimen-sion lower than 3. Thus the singular point is never a local source. If all the eigenvalues,apart form the zero one, have negative reals parts, then the center manifold ofP5 actsas a local sink. This means that the orbits in the stable manifold approach the centermanifold ofP5 when the time goes forward.

(ix) The singular pointP6 with coordinates

ϕ = 0, σ2 =√6γ

M(3γ−4)−2N , σ4 = −√

M2(4−3γ)2+MN(8−6γ)−6(γ−2)γ

2N+M(4−3γ) , σ5 = 0


exists forM(3γ − 4)(M(3γ − 4) − 2N) ≥ 6(γ − 2)γ, 2(2M + N) < 3Mγ, and3(MN+2)γ−2N(2M+N)

(2N+M(4−3γ))2 ≤ 0. The eigenvalues of the linearization are the same displayedin the previous point. However the stability conditions arerather different (since theexistence conditions are different from those ofP5). As before, the stability conditionsare very complicated to display them here, but similar conclusions concerning the centerand unstable manifold, as forP5, are obtained. For get further information about itsstability we must to resort to numerical experimentation.

Appendix B. Physical description of the solutions and connection with observables.

Let us now present the formalism of obtaining the physical description of a singular point, andalso connect with the basic observables relevant for a physical discussion (taken from [170]).

Firstly, around a singular point we obtain first-order expansions forH, a, φ, andρm andρr in terms oft, considering equations: (11a); the definition of the scale factora in terms ofthe Hubble factorH ; the definition ofσ2; the matter conservation equations (11b) and (11c),respectively, given by

2H(t) = H(t)2(3(γ − 2)σ⋆

22 + 3γ

(σ⋆42 + σ⋆

52 − 1

)− 4σ⋆

52),

a(t) = a(t)H(t),

φ(t) =√6σ⋆

2H(t),

ρm(t) = −3

2H(t)3

(√6M(3γ − 4)σ⋆

2 − 6γ)(

σ⋆22 + σ⋆

42 + σ⋆

52 − 1

),

ρr(t) = −12σ⋆52H(t)3, (B.1)

where the star-superscript denotes the evaluation at a specific singular point. The equation

φ(t) =3

2H(t)2

(M(3γ − 4)

(σ⋆22 + σ⋆

42 + σ⋆

52 − 1

)− 2

(Nσ⋆

42 +

√6σ⋆

2

)), (B.2)

derived from the equation of motion for the scalar field (11d)should be used as a consistencytest for the above procedure. Solving the differential equations (B.1) and substituting theresulting expressions in the equation (B.2) results in

−6M(3γ − 4)(σ⋆22 + σ⋆

42 + σ⋆

52 − 1

)+ 12Nσ⋆

42+

+2√6σ⋆

2

(3γ(σ⋆22 + σ⋆

42 + σ⋆

52 − 1

)− 6σ⋆

22 − 4σ⋆

52 + 6

)= 0. (B.3)

This integrability condition should be (at least asymptotically) fulfilled.Instead of apply this procedure to a generic singular point here, we submit the reader to

section 5 for some worked examples where this procedure has been applied. However we willdiscuss on some cosmological observables.

We can calculate the deceleration parameterq defined as usual as [165]

q = −aa

a2. (B.4)

Additionally, we can calculate the effective (total) equation-of-state parameter of the universeweff, defined conventionally as

weff ≡ptot

ρtot, (B.5)


Table B1. Observable cosmological quantities, and physical behavior of the solutions, at the

singular points of the cosmological system. We use the notationsM1(γ) =

√2γ(3γ−8)+8

4−3γ,

M2(γ) =√

6√

(γ−3)γ+2

4−3γ. (Taken from [170]).

Cr.P. q weff Solution/description

P1 2 1 Decelerating.P2 2 1 Decelerating.

P3−M2(4−3γ)2+2γ(3γ−8)+8

4(γ−2) −M2(4−3γ)2

6(γ−2) + γ − 1 Accelerating for0 < γ < 2

3 , −M1(γ) < M < M1(γ)

P412

(N2 − 2

)13

(N2 − 3

)Accelerating for−√2 < N <

√2

powerlaw-inflationary

P53(M+N)γ−2(2M+N)

2N+M(4−3γ)M(4−3γ)−2N(γ−1)

M(3γ−4)−2N Accelerating for3(M+N)γ−2(2M+N)

2N+M(4−3γ) < 0

matter-kinetic-potential scaling

P63(M+N)γ−2(2M+N)

2N+M(4−3γ)M(4−3γ)−2N(γ−1)

M(3γ−4)−2N Accelerating for3(M+N)γ−2(2M+N)

2N+M(4−3γ) < 0

matter-kinetic-potential scalingR1 1 1

3 Decelerating. Radiation-dominated.R2 1 1

3 Decelerating.radiation-kinetic-potential scaling

R3 1 13 Decelerating.

radiation-kinetic-potential scaling.

whereptot andρtot are respectively the total isotropic pressure and the totalenergy density.Therefore, in terms of the auxiliary variables we have

q = − 3

2(γ − 2)σ2

2 −3γσ2

4

2+

1

2(4− 3γ)σ2

5 +1

2(3γ − 2) (B.6)

weff = (2 − γ)σ22 − γσ2

4 +1

3(4− 3γ)σ2

5 + γ − 1. (B.7)

First of all, for each singular point described in the last section we calculate the effective(total) equation-of-state parameter of the universeweff using (B.7), and the decelerationparameterq using (B.6). The results are presented in Table B1. Furthermore, as usual, for anexpanding universeq < 0 corresponds to accelerating expansion andq > 0 to deceleratingexpansion.

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