Cosmological dynamics of the general non-canonical scalar field models
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Cosmological dynamics of the general non-canonical scalar field models
Jibitesh Dutta a,1,2 , Wompherdeiki Khyllep b,3,4 , Hmar Zonunmawia c,3 Mathematics Division, Department of Basic Sciences and Social Sciences, North Eastern Hill University, Shillong, Meghalaya 793022, India Inter University Centre for Astronomy and Astrophysics, Pune 411 007, India Department of Mathematics, North Eastern Hill University, Shillong, Meghalaya 793022, India Department of Mathematics, St. Anthony’s College, Shillong, Meghalaya 793001, IndiaReceived: date / Accepted: date
Abstract
We extend the investigation of cosmological dynam-ics of the general non-canonical scalar field models by dynam-ical system techniques for a broad class of potentials and cou-pling functions. In other words, we do not restrict the analysisto exponential or power-law potentials and coupling functions.This type of investigation helps in understanding the generalproperties of a class of cosmological models. In order to betterunderstand the phase space of the models, we investigate thevarious special cases and discuss the stability and viability is-sues. Performing a detailed stability analysis, we show that itis possible to describe the cosmic history of the universe at thebackground level namely the early radiation dominated era, in-termediate matter dominated era and the late time dark energydomination. Moreover, we find that we can identify a broadclass of potentials and coupling functions for which it is possi-ble to get an appealing unified description of dark matter anddark energy. The results obtained here, therefore, enlarge theprevious analyses wherein only a specific potential and cou-pling functions describes the unification of dark sectors. Fur-ther, we also observe that a specific scenario can also possiblyexplain the phenomenon of slow-roll inflationary exit.
The source for the observed accelerated expansion of the Uni-verse still remains to be an obscured problem in modern cos-mology. It is widely believed that this phenomenon can be the-oretically explained by an exotic quantity commonly known asdark energy (DE) with large negative pressure [1, 2]. In the lit-erature, there are various candidates of DE and the debate overthe fittest candidate of DE, is yet to be resolved. The detailsand status of various candidates of DE so far are summarisedin an excellent review by Brax [3] recently. Scalar field plays a e-mail: [email protected], [email protected] b e-mail: [email protected] c e-mail: [email protected] a significant role in cosmology as they are sufficiently com-plicated to produce the observed dynamics. They are used tomodel DE and characterise inflation [4]. Motivated by highenergy physics, scalar field models play an important role toexplain the nature of DE due to its simple dynamics [5, 6]. Thesimplest form of a scalar field is the canonical field also knownas quintessence field with potential. However, the canonicalscalar field cannot fully explain several complex cosmologi-cal dynamics of the Universe. For instance, the quintessencemodel cannot explain the crossing of the phantom divide line,bouncing solutions. This leads to a more general descriptionof a scalar field known as non-canonical scalar field [7, 8].Another advantage of the non-canonical setting over canoni-cal is that it can resolve the coincidence problem without anyfine-tuning issues [9].The most general non-canonical form of a scalar field whichinvolves higher order derivatives of a scalar field falls underthe well-known Horndeski Lagrangian [10]. Within this class,a simple form of a non-canonical scalar field model is collec-tively known as k -essence, the first term of the Horndeski La-grangian. Basically, k -essence models are the generalisationof the canonical scalar field and its Lagrangian is a functionof both the scalar field and its kinetic term [11]. An impor-tant feature of the k -essence is that, unlike the canonical scalarfield, its kinetic energy term can also source acceleration. Inthe context of cosmology, the k -essence field was first used todescribe early inflationary epoch in [12] and was later foundthat it can also describe DE by Chiba et al. [13]. In literature,there are several forms of k -essence models depending on thetype of its Lagrangian [8, 14, 15].In the present work, we will consider a general form of k -essence Lagrangian introduced by Melchiorri et al. [16]. Ithas been found that while the potential energy of this typeof non-canonical scalar field models behaves as DE, its ki-netic energy plays the role of dark matter (DM) [17–23]. Thisleads to a unification of DE and DM, allowing structure forma-tion and avoiding the strong integrated Sachs-Wolfe effect in a r X i v : . [ g r- q c ] A p r the CMB anisotropies which usually afflict the unification be-tween dark sectors [24]. Interestingly, the Chaplygin gas andits generalised form can also describe the DE-DM unification[17, 25, 26]. But the Chaplygin gas is associated with gravi-tational clustering problems and therefore cannot describe thereal universe [27, 28]. Another advantage of the non-canonicalmodels is that the adiabatic sound speed is sensitive to the val-ues of a non-canonical parameter which have an effect on thegravitational clustering [22]. Moreover, cosmological obser-vations such as SNIa, CMB and BAO favour the general k -essence models[29].Dynamical system techniques are useful tools to investi-gate the complete asymptotic behaviour of a given cosmolog-ical model which allows us to bypass the difficulty in solv-ing non-linear cosmological equations. These tools also pro-vide a description of the global dynamics of the universe byanalysing the local asymptotic behaviour of critical points ofthe system and relate them with the main cosmological epochsof the universe. For instance, while a late-time DE domina-tion would typically correspond to a stable point, the radiationand matter dominated eras correspond to saddle points. Dy-namical system techniques had been extensively used in liter-ature to analyse the evolution of various cosmological mod-els of DE as well as modified gravity based models [30–42].For a recent and comprehensive review on the applications ofdynamical system tools to various cosmological models seereference [43]. However, one limitation of the dynamical sys-tem approach is the dependence on the choice of variableswhich characterise the solution associated with critical points.It is worth mentioning that the absence of some cosmologi-cal epoch of the Universe does not always imply the inabil-ity of the theory to explain such epoch, but it may be due tothe inability of the associated dynamical system to exhibit thepresence of an epoch.In the non-canonical scalar field settings, the dynamicalsystem tools have been applied extensively to various sub-classes of k -essence Lagrangian [43]. A dynamical system anal-ysis for a k -essence Lagrangian introduced in [16] has beenanalysed in [44] for a power-law and exponential potential.Further, dynamical system analysis of an interacting DE withinthe framework of the k -essence Lagrangian introduced in [16]with power-law potential has been performed in [45]. In [44,45], the choice of potential is restricted only to power-lawand exponential case due to the choice of dynamical variablesconsidered. However, dynamical analysis of the general non-canonical scalar field beyond specific potentials have neverbeen studied before. On the other hand, there are various stud-ies of the canonical scalar field beyond exponential poten-tial [46–49]. The important reason for studying the dynami-cal properties of DE model beyond a specific potential is thatit helps in understanding general properties of a DE modelwhich in turn helps in predicting properties of a class of DEmodels. Therefore, it is more economic and scientific to studyfor arbitrary potentials. Furthermore, the generalisation to var- ious scalar field potentials is not only important from the math-ematical point of view but also to connect these phenomeno-logical models with the low energy limit of high energy phys-ical theories. Therefore, it is interesting to extend the analysisto encompass the various class of potentials.With this motivation, here we extend the work of [44] byperforming a dynamical analysis for a more general k -essenceLagrangian over a wider class of scalar field potentials. More-over, we shall consider a different choice of dynamical vari-ables from the one defined in [44] so as to encompass thedesired class of potentials. The main goal of this work is todetermine the choice of model parameters which leads to in-teresting background cosmological dynamics such as the well-known unification of DE and DM within the non-canonicalscenario. The present work might also provide a preliminarytest of the general non-canonical models which are of interestfor further investigation.The organisation of the paper is as follows: In Sect. 2, webriefly review the basic equations of the non-canonical scalarfield models. In Sect. 3, we perform a dynamical system anal-ysis for the autonomous system of differential equations ob-tained from the basic cosmological equations. Within this sec-tion, we also discuss the cosmological dynamics by consid-ering a different class of scalar fields by taking various typesof potentials and non-canonical coupling functions in varioussubsections. Finally, the conclusion is given in Sect. 4. Notation : In this work, we shall assume the (+ , − , − , − ) signature convention for the metric. We shall also adopt unitswhere 8 π G = c = The general action of a minimally coupled scalar field modelis given by S = (cid:90) d x √− g (cid:16) R + L ( φ , X ) (cid:17) + S m , (1)where L ( φ , X ) is the Lagrangian density which is an arbi-trary function of a scalar field φ and its kinetic term X ( X = ∂ µ φ ∂ µ φ ), R is the Ricci scalar and g is the determinant ofthe metric g µν . The last term S m represents the action of thebackground matter field.Varying the action (1) with respect to the metric yields theEinstein field equations as G µν = T φµν + T m µν , (2)where G µν is the Einstein tensor, T φµν is the energy-momentumfor φ given by T φµν = ∂ L ∂ X ∂ µ φ ∂ ν φ − g µν L . (3) The energy-momentum tensor T m µν for the matter componentwhich is modeled as a perfect fluid is given by T m µν = ( ρ m + p m ) u µ u ν + p m g µν , (4)where ρ m and p m are the energy density and pressure of thematter component and u µ is the four velocity of the fluid.In the present work, we shall consider a homogeneous,isotropic and spatially flat Friedmann Robertson Walker (FRW)universe which is characterised by the line element ds = dt − a ( t )[ dr + r d θ + r sin θ d φ ] , (5)and the following non-canonical Lagrangian density of thescalar field [16] L ( φ , X ) = f ( φ ) F ( X ) − V ( φ ) , (6)where a ( t ) is the scale factor, t is the cosmic time, V ( φ ) is aself-interacting potential for the scalar field φ , f ( φ ) is an arbi-trary function of φ with f ( φ ) ≥ F ( X ) is an arbitrarycoupling function of X . For a spatially homogeneous FRWmetric (5), X = ˙ φ . The positive semi-definiteness of func-tion f is required to explain various cosmological observa-tions as discussed in [16]. It can be seen that Eq. (6) reduces toquintessence field when f ( φ ) = F ( X ) = X , it reduces tophantom field when f ( φ ) = F ( X ) = − X . It means thateach quintessence or phantom scalar field is equivalent to aparticular k -essence model. This type of scalar field model (6)constitutes an alternative model of DE yielding late time ac-celerated solutions and it is well motivated from high-energyphysics [50, 51].There are several functional forms of F ( X ) proposed sofar in the literature see [8, 14, 15]. However, in order to obtaina concrete description of the cosmological dynamics, we shallconsider a case where F ( X ) = X α i.e. the non-canonical scalarfield models whose Lagrangian density is given by [8, 15] L ( φ , X ) = f ( φ ) X α − V ( φ ) , (7)where α is a dimensionless parameter. This class of F ( X ) is asimple generalisation of the canonical scalar field (for α = F ( X ) also allows a non-canonical field to cluster and depending on values of α , itbehaves either like warm or cold dark matter at small scales[22]. It has been found that the slow-roll conditions can be eas-ily realized for the non-canonical scalar field models ( α (cid:54) = α =
1) [52].Further, non-canonical models reduce the tensor-to-scalar ra-tio than their canonical counterparts, leading to a better agree-ment with CMB observations [52]. These interesting featuresof non-canonical scalar fields motivate us to further investigatethe cosmological dynamics of such fields in a more generalcontext. In terms of Lagrangian density L , the expressions of theenergy density ρ φ and pressure p φ associated with the scalarfield are given by ρ φ = X (cid:16) ∂ L ( φ , X ) ∂ X (cid:17) − L ( φ , X ) , (8) p φ = L ( φ , X ) . (9)Employing the line element (5), the Einstein field equationscan be written as3 H = ( α − ) f ( φ ) X α + V ( φ ) + ρ m , (10)˙ H = − (cid:104) α f ( φ ) X α + ρ m ( + w ) (cid:105) , (11)where the over-dot denotes differentiation with respect to cos-mic time t , H = ˙ aa is the Hubble parameter and w is the equa-tion of state (EoS) of matter ( − ≤ w ≤ ) defined as p m = w ρ m .Substituting for L from (7) into (8) and (9), the energydensity and pressure of the scalar field are respectively givenby ρ φ = ρ X + V , (12) p φ = p X − V , (13)where ρ X = ( α − ) f ( φ ) X α and p X = f ( φ ) X α . Finally, weassume that the scalar field and barotropic fluid energy mo-mentum tensors are conserved separately i.e. there is no ex-change of energy between the scalar field and barotropic fluid.In that case by employing line element (5), the energy conser-vation equations are given by˙ ρ φ + H ( ρ φ + p φ ) = , (14)˙ ρ m + H ρ m ( + w ) = . (15)Using Eq. (12) in Eq. (14), the evolution equation of the scalarfield can be expressed as¨ φ = − (cid:104) α d fd φ ˙ φ f ( φ ) + α ( α − ) dVd φ (cid:16) φ (cid:17) α − f ( φ )+ H ˙ φ α − (cid:105) . (16)As expected, the above equation reduces to the case of a canon-ical field, ¨ φ + H ˙ φ + dVd φ =
0, when α = f ( φ ) =
1. It isworth noting that for constant potential, the kinetic part ρ X andthe potential part V ( φ ) behave as two non-interacting fluids asthey separately satisfy the conservation equation (14).For theoretical consistency of the Horndeski’s most gen-eral scalar-tensor theories, the conditions Q s > C s ≥ Q t > C t ≥ Q t > C t ≥ Q s and C s are given by Q s = α α ( α − ) H − X α f (17) C s = α − + ( + w ) ρ m α α ( α − ) X α f (18) Therefore in the absence of barotropic matter component, theEq. (18) reduces to the case obtained in [23] given by C s = α − . (19)It is important to note here that we have Q s > C s ≥ α > . However, for α = or at a point where X = f = C s diverges. This result is expected as dis-cussed in Ref. [54] for a typical form of k -essence Lagrangian.The divergence of speed of sound is a well known result in k -essence model and therefore violates causality [55]. Furtherthe existence of superluminal solutions i.e. C s > C s < α > . In the next section, we shallconvert these cosmological equations into an autonomous sys-tem of equations and perform a dynamical system analysis forvarious types of f ( φ ) and V ( φ ) . In order to write the above set of cosmological equations as anautonomous system of ordinary differential equations, we in-troduce the following set of normalised phase space variables x = ˙ φ √ H , y = √ V √ H , z = H ( α − ) f , s = − V dVd φ , v = − f d fd φ . (20)Using these variables (20), the above cosmological equationscan be converted to the following autonomous system as x (cid:48) = √ (cid:104) x v α + y szx ( α − ) α ( α − ) ( α − ) − √ x ( α − ) (cid:105) + x (cid:104) α α x α z + ( + w ) { − ( α − ) x α z ( α − ) − y } (cid:105) , (21) y (cid:48) = − xys + y (cid:104) α α x α z + ( + w ) { − ( α − ) x α z ( α − ) − y } (cid:105) , (22) z (cid:48) = −√ xzv − z ( α − ) (cid:104) α α x α z + ( + w ) { − ( α − ) x α z ( α − ) − y } (cid:105) , (23) s (cid:48) = −√ xg ( s ) , (24) v (cid:48) = −√ xh ( v ) , (25)where g ( s ) = s ( Γ − ) and h ( v ) = v ( Γ − ) with Γ = V d Vd φ (cid:46)(cid:16) dVd φ (cid:17) , Γ = f d fd φ (cid:46)(cid:16) d fd φ (cid:17) and the prime denotes dif-ferentiation with respect to N = ln a . Note that if s = s ( φ ) is invertible, then we can also express φ as function of s . Sincethe parameter Γ is a function of φ only, hence Γ can also beexpressed as Γ = Γ ( s ) . Similarly, we can write Γ as func-tion of v . This does not include any arbitrary V or f but itincludes a specific class of potential V and coupling function f . Thus, for such choices of V and f , the system (21)-(25)constitutes an autonomous system of equations. Here, we ob-served that in the context of cosmological evolution, there arefour basic variables H , φ , ˙ φ and ρ m , which are related by aFriedmann constraint (10). Therefore, there are only three in-dependent variables altogether. However, the introduction oftwo extra variables allows us to track the effect of the cou-pling and potential functions on the overall dynamics. In theliterature, these types of variables are often introduced to studythe class of unknown functions [47, 57]. It is worth noting thatthe variables (20) are different from those defined in [44] andthis choice of variables helps in obtaining cosmologically vi-able critical points which however cannot be captured in [44].More importantly, our choice of variables leads to a viable cos-mological sequence: radiation era → matter era → DE era andalso can possibly explain a unified description of DM and DE.Moreover, the above system (21)-(25) reduces to the corre-sponding system of a canonical case for α = f = y = . Similarly, z = z appears as the denom-inator in Eq. (21). For constant potential V , x = V and f , s = v = s ∗ and v ∗ denote the zeroesof g ( s ) = h ( v ) = dg ( s ) , dh ( v ) denote the derivative of g and h with respect to s and v re-spectively. It is worth noting from the definition (20), y > H > y < H < y → − y ,we shall only analyse the case of expanding universe which isalso cosmologically viable. Further, as we consider expandinguniverse, therefore we have z ≥ The simplest approach to determine the existence of an invariant sub-manifold of R n (here n =
5) is to look at the form of the right-hand side(rhs) of a dynamical equation for a given variable (for e.g. y ). If the dy-namical equation is of the form y (cid:48) = ( y − ξ ) g , where g : R n → R is acontinuous function, then y = ξ is an invariant sub-manifold with respectto a flow corresponding to an autonomous vector field given by (21)-(25)[58]. In terms of variables (20), the relative energy density dueto the scalar field Ω φ , the relative energy density due to thebarotropic matter Ω m and the scalar field EoS w φ can be ex-pressed as Ω φ = ρ φ H = ( α − ) ( α − ) x α z + y , (26) Ω m = ρ m H = − ( α − ) ( α − ) x α z − y , (27) w φ = p φ ρ φ = ( α − ) zx α − y ( α − ) ( α − ) zx α + y . (28)Note that the relative energy density parameters are related bythe Friedmann constraint (10) as Ω m + Ω φ = . (29)Here we see that for α > x ≈ y ≈
1, then w φ ≈ − y ≈ ( α − ) ( α − ) x α z ≈ w φ ≈ α − .This implies that purely kinetic non-canonical models cannotdescribe the cosmic acceleration for α > as reviewed in[59]. Further, we see that the scalar field scales as radiationfor α =
2, stiff matter for α = α . This does not mean that the matter component comes fromthe DE, but both of them are sourced by a single scalar field.Additionally, we can define the effective energy density andpressure as ρ eff = ρ φ + ρ m , (30) p eff = p φ + p m , (31)where we can obtain the effective EoS as w eff = − + α α x α z + ( w + ) [ − α − x α z ( α − ) − y ] , (32)which is usually related to the deceleration parameter q through q = − − ˙ HH = + w eff . (33)Hence for accelerated universe, one must obtain q < w eff < − . From Eq. (29), the physical requirement ρ m ≥ Ω φ ≤ . (34)Hence the five dimensional phase space of the system (21)-(25) in terms of variables (20) is given by Ψ = (cid:110) ( x , y , z ) ∈ R : ( α − ) ( α − ) x α z + y ≤ , y , z ≥ (cid:111) × (cid:8) ( s , v ) ∈ R (cid:9) . (35)To determine the evolution of relevant energy densities nu-merically, we estimate the evolution of the Universe in such a way that it is consistent with the present observational data( Ω m (cid:39) . w eff (cid:39) − .
7) [60]. This corresponds to points inthe phase space where x = y = ± . z is any value.In order to extract the dynamics of the above system, first,we need to find the critical points of the system by equat-ing the right-hand side of the system (21)-(25) to zero. Thisis followed by perturbing the system near the critical pointsfrom which the stability and type of critical points can be de-termined from the eigenvalues of the corresponding perturbedmatrix. In the present work, since we have arbitrary functions f and V , in what follows, we shall analyse various cases sepa-rately for different choices of f and V .3.1 f ( φ ) =Constant, V ( φ ) = ConstantWe start our study by considering the simplest choice of f and V where both are constant. In this case the variables s = v =
0, hence, the system (21)-(25) reduces to a three di-mensional system in x , y and z . It is worth noting that for thischoice of V and f , the functions Γ , Γ cannot be expressed asexplicit functions of s and v respectively but the three dimen-sional system is still autonomous as the equations involving Γ , Γ are not involved. We note here that the system has three in-variant sub-manifolds x = , y = z =
0. Under this case,we have only one critical point A = ( , , ) and one set ofcritical points A = ( , , z ) . – The critical point A corresponds to a matter dominateduniverse [ Ω m = Ω φ = w eff = w , w φ = α − ] witheigenvalues of the Jacobian matrix (cid:110) λ = ( + w ) , λ = − ( α − )( + w ) , λ = ( α w + α − w − ) ( α − ) (cid:111) of the perturbed matrixat A . Therefore, this point behaves as unstable node for + w + < α <
1, otherwise it behaves as a saddle. Thus,even though point A lies on the intersection of all invari-ant sub-manifolds, it cannot be a global attractor. – The set A corresponds to an accelerated scalar field dom-inated solution [ Ω m = Ω φ = w eff = − , w φ = − ] .It is a non-hyperbolic set with eigenvalues (cid:110) λ = , λ = − α − , λ = − ( + w ) (cid:111) . However, as the dimension ofthe set is same as the number of vanishing eigenvalues, it istherefore a normally hyperbolic set [61]. Therefore, pointson this set behave as stable points for α > but this setcannot be global attractor as points on it corresponds to y (cid:54) = A ),it evolves towards an accelerated DE dominated solution (set A ). Hence, at the background level, this model can success-fully explain the structure formation as well as the late timeevolution of the universe for α > . This is more visible fromFig. 1, where we plot the time evolution of different energydensity parameters along with the effective EoS against the W Φ W m w eff - - - - ln H + z L Fig. 1: The time evolution of the relative matter energy density Ω m , the relative scalar field energy density Ω φ and the effec-tive EoS w eff for the case when f = constant, V = constant.Here, w = α = z = a a − a = α =
2. Here, while z = a = a ) rep-resents the present time of the universe, z ≈ − a → ∞ ) represents an asymptotic future of the uni-verse.3.2 f ( φ ) = ConstantIn this subsection, we investigate the case where only the func-tion f is constant. In this case variable v vanishes and thusthe system (21)-(25) reduces to a four-dimensional system in x , y , z , s . The system contains one invariant sub-manifold y = z =
0. Depending onthe choice of potential, the system may possess another in-variant sub-manifold s =
0. For α =
1, we have checked thatthe behaviour of the system coincides with that of the canoni-cal scalar field [30], therefore, we will not further analyse thiscase. However for α (cid:54) =
1, there are only two set of criticalpoints: B = ( , , z , s ) and B = ( , , z , s ) . Both sets are inde-pendent of the choice of the potential for their existence. Fromthe second term of Eqs. (21) and (18), it can be seen that bothsets demand α < – The set B corresponds to an unaccelerated DM dominatedsolution [ Ω m = Ω φ = w eff = w , w φ = α − ] . It is anormally hyperbolic set of points with eigenvalues (cid:110) λ = ( + w ) , λ = ( − α )( + w ) , λ = , λ = ( α w + α − w − ) ( α − ) (cid:111) .Therefore, points on this set are behaving as unstable nodefor + w + < α <
1, otherwise they are behaving as asaddle. – The set B corresponds to an accelerated scalar field dom-inated solution [ Ω m = Ω φ = w eff = − , w φ = − ] .It is also non-hyperbolic in nature with eigenvalues { λ = , λ = , λ = , λ = − ( + w ) } but not normally hyper-bolic. Usually, the centre manifold theory is employed todetermine the stability of points on this set [62]. However, N x (a) N y (b) N z (c) N s (d) Fig. 2: Projection of the time evolution of phase space trajec-tories along the (a) x -axis, (b) y -axis, (c) z -axis and (d) s -axiswhich determine the stability of set B . Here we take w = α = . V = V φ n with n = f = constant.in this work, we will simply refer to a numerical computa-tional method of perturbation plot to determine the stabil-ity near points of this set. This numerical method has beenfound to be quite successful in literature for determiningthe stability behaviour of critical points of the system [37–41]. For this, we perturb the system near points of this setand plot the projections of phase trajectories along the x , y , z and s axes as given in Fig. 2. We note here that for theseplots, we choose the values of initial conditions that arevery close to the points of the set. It is evident from Figs.2(a) and 2(b), that the phase space trajectories approach x = y = N → ∞ . Further, it can beseen in Fig. 2(c), 2(d), that trajectories starting from anyvalues of z and s remains almost constant. From the be- haviour of the perturbed system near the set B , we canconclude that the set B behaves as a stable set but pointson this set cannot be global attractors as they do not lie onthe intersection of invariant sub-manifolds.Hence, for a restricted choice of α ( < α < B ),it eventually evolves towards an accelerated DE dominated so-lution (set B ).3.3 V ( φ ) = ConstantIn this subsection, we investigate the case where only the po-tential is constant. For this choice of potential, the variable s vanishes, so the system (21)-(25) reduces to a four dimen-sional system in x , y , z and v . The system thus contains threeinvariant sub-manifolds x = , y = z =
0, however, de-pending on the choice of h ( v ) , the system may contains an-other invariant sub-manifold v =
0. Under this case, we havetwo sets of critical points C = ( , , , v ) , C = ( , , z , v ) andthree critical points C = (cid:16) − α √ ( α w + α − w − ) v ∗ ( α − ) , , , v ∗ (cid:17) , C = (cid:16) √ α v ∗ ( α − ) , , , v ∗ (cid:17) and C = (cid:32) − α √ ( α − ) v ∗ ( α − ) , , ( − α ) (cid:104) − α √ ( α − ) v ∗ ( α − ) (cid:105) − α ( α − ) , v ∗ (cid:33) . Note that from the second term ofthe equation (21), the existence of point C specifically de-mands s = C , C , C do not exists when v ∗ = – The set of critical points C corresponds to a matter domi-nated universe [ Ω m = Ω φ = w eff = w , w φ = α − ] witheigenvalues (cid:110) λ = ( + w ) , λ = − ( α − )( + w ) , λ = ( α w + α − w − ) ( α − ) , λ = (cid:111) . Therefore, points on this set be-having as unstable node for + w + < α <
1, otherwisethey are behaving as saddle points. This again implies thatpoints on this set cannot be global attractor. – The set C corresponds to an accelerated scalar field poten-tial dominated solution [ Ω m = Ω φ = w eff = − , w φ = − ] . It is a normally hyperbolic set with eigenvalues (cid:110) λ = , λ = , λ = − ( + w ) , λ = − α − (cid:111) . Therefore, it isalways stable but it cannot be global attractor as y (cid:54) = – The point C corresponds to a matter dominated solutionwith [ Ω m = Ω φ = w eff = w , w φ = α − ] . The eigen-values of the corresponding perturbed matrix are (cid:110) λ = ( + w ) , λ = ( α w − w − ) α − , λ = − ( α w + α − w − ) ( α − ) , λ = α ( α w + α − w − ) dh ( v ∗ ) v ∗ ( α − ) (cid:111) . Since λ > λ , λ arenot both positive, therefore this point is a saddle for anychoice of function f . – The point C corresponds to an accelerated solution domi-nated by the potential term of the scalar field [ Ω m = , Ω φ = , w eff = − , w φ = − ] with eigenvalues (cid:110) λ = − ( + w ) , λ = α − , λ = − α α − , λ = − α dh ( v ∗ ) v ∗ ( α − ) (cid:111) . It is there-fore saddle for α > α < v ∗ dh ( v ∗ ) < α < v ∗ dh ( v ∗ ) >
0. This pointcorresponds to a slow roll behaviour where potential partof the scalar field dominates and its derivative vanishes (asits existence demands s = – The critical point C corresponds to a solution dominatedby the kinetic term of a scalar field [ Ω m = , Ω φ = , w eff = α − , w φ = α − ] . The eigenvalues of the correspondingperturbed matrix are (cid:110) λ = α α − , λ = ( − α ) α − , λ = ( − w ( α − )) α − , λ = α ( α − ) α − dh ( v ∗ ) v ∗ (cid:111) . This point is anunstable node if < α < v ∗ dh ( v ∗ ) <
0, otherwiseit is a saddle. It is worth mentioning that the existence ofthis point is cosmologically very interesting as it describesthe early radiation domination for α =
2, the stiff mattersolution for α = α .Hence, from the above analysis it can be seen that for thosefunction f in which v ∗ (cid:54) =
0, the system evolves from the point C which behaves as a radiation (i.e. w eff = for α =
2) orDM ( w eff ≈ α ) or stiff matter ( w eff = α = C or point C with w eff = C with w eff = −
1. Therefore, a viablesequence of cosmological eras (radiation → matter → dark en-ergy) is achieved by choosing the trajectory: C → C → C or C → C → C for α =
2. Further, the early times radia-tion (for α =
2) or the DM behaviour (for large α ) is specif-ically due to the non-canonical scalar field. This explains thewell-known unified DE-DM behaviour in non-canonical set-ting where the kinetic term of the scalar field plays the roleof DM and the potential plays the role of DE [17–23]. Simi-lar unified description is obtained in the context of generalizedGalileon theories but without any instabilities at the perturba-tive level at all times [63]. In order to clearly see the back-ground cosmological sequence in detail, we plot the time evo-lution of different relative energy densities along with the over-all effective EoS by considering a specific choice of the func-tion f viz. f = f e βφ with β = α = , f is specifically taken sothat the function Γ can be written as function of v . Moreover,it has been found that this choice of f fit with data from thecombination of various datasets such as SNIa, BAO and CMB[29]. From Fig. 3, it can be seen that the background evolutionof the Universe resembles that of the Λ CDM model, with anearly time contribution arising from the non-canonical scalarfield term. However, the presence of a non-canonical param- W Φ W m w eff - - ln H + z L (a) W Φ W m w eff - - - - ln H + z L (b) Fig. 3: The time evolution of the relative matter energy den-sity Ω m , the relative scalar field energy density Ω φ and theeffective EoS w eff . Here we take f ( φ ) = f e βφ , β = V = constant, w = α = α = α in the speed of sound produces a possible deviation ofthis model from Λ CDM at the perturbation level [22].3.4 f ( φ ) = General function, V ( φ ) = General potentialIn this subsection, we shall investigate the case where bothfunction f and potential V are kept general. By general, wemean V and f are such that functions Γ , Γ can be writtenas functions of s and v respectively. It is worth mentioningthat constant potential and constant function f do not belongsto this category as both Γ and Γ cannot be defined. For thiscase, the system possess an invariant sub-manifold y = z =
0. For some choice of V and f , s = v = α (cid:54) = α = α (cid:54) = D = ( , , , s , v ) , D = ( , , z , s , v ) and two critical points: N x (a) N y (b) N z (c) N s (d) N v (e) Fig. 4: Projection of the time evolution of phase space trajec-tories along the (a) x -axis, (b) y -axis, (c) z -axis, (d) s -axis and(e) v -axis which determine the stability of set D . Here, wetake w = α = . V ( φ ) = V ( − e − λφ ) with λ = f ( φ ) = f φ − n with n = Α - s * - v * (a) Α - s * - v * (b) Fig. 5: (a)
Regions of stability of point D : The red shadedregion represents the region of stability for dg ( s ∗ ) > dh ( v ∗ ) >
0, the blue shaded region of stability for the casewhen dg ( s ∗ ) < dh ( v ∗ ) <
0. (b)
Regions of instabilityof point D : The green shaded region represents the region ofunstable node for dg ( s ∗ ) > dh ( v ∗ ) >
0, the blue shadedregion represents the region of unstable node for the case when dg ( s ∗ ) < dh ( v ∗ ) <
0. In both panels, the remainingunshaded regions represent regions where points are saddle.Here, we have taken w = D = (cid:16) − α √ ( α w + α − w − ) v ∗ ( α − ) , , , s ∗ , v ∗ (cid:17) and D = (cid:32) − √ α ( α − ) v ∗ ( α − ) , , (cid:16) − α ( α − ) v ∗ ( α − ) (cid:17) − α − α α − , s ∗ , v ∗ (cid:33) . It can beseen from the second term of Eq. (21) that the existence of set D demands 0 < α < and D is 0 < α <
1. Therefore, the set D is not physically viable as it existence implies C s <
0. Fur-ther, the existence of points D and D demands those func-tions f for which v ∗ (cid:54) =
0. In what follows, we discuss the sta-bility conditions of each physically viable critical points. – The set D corresponds to an accelerated scalar field domi-nated solution [ Ω m = Ω φ = w eff = − , w φ = − ] . It isnon-hyperbolic in nature with eigenvalues (cid:110) λ = − ( + w ) , λ = , λ = , λ = , λ = (cid:111) . The stability behaviourof points on this set can be determined numerically from N x (a) N y (b) N z (c) N s (d) Fig. 6: Projection of the time evolution of phase space trajec-tories along the (a) x -axis, (b) y -axis, (c) z -axis and (d) s -axiswhich determine the stability of set E . Here we take w = α = V = V φ − n with n = f ( φ ) = f e βφ with β = f and V . It can be seen that tra-jectories near this set approach to different points on theset as N → ∞ (see Fig. 4). Therefore, this set behaves as alate time attractor but not global as y (cid:54) = – The point D corresponds to a matter dominated solution [ Ω m = Ω φ = w eff = w , w φ = α − ] . The eigenvaluesof the corresponding perturbed matrix are (cid:110) λ =
32 2 s ∗ ( w + ) α +(( w + ) v ∗ − s ∗ ( w + )) α − v ∗ ( w + ) v ∗ ( α − ) , λ = α ( α w + α − w − ) dg ( s ∗ ) v ∗ ( α − ) , λ = − ( α w + α − w − ) ( α − ) , λ = ( α w − w − ) α − , λ = α ( α w + α − w − ) dh ( v ∗ ) v ∗ ( α − ) (cid:111) . Due to thecomplicated expressions of the eigenvalues on model pa-rameters, we analysed its region of stability for a physi- cally interesting case w =
0. It can be seen that this pointis either stable or saddle by determining the region of sta-bility in ( α , s ∗ , v ∗ ) parameter space for the case w = – The critical point D also corresponds to a solution dom-inated by the kinetic energy part of the scalar field [ Ω m = , Ω φ = , w eff = α − , w φ = α − ] . The eigenvalues of thecorresponding perturbed matrix are (cid:110) λ = α ( s ∗ ( α − )+ v ∗ ) v ∗ ( α − ) , λ = ( − α ) α − , λ = ( − w ( α − )) α − , λ = α ( α − ) α − dg ( s ∗ ) v ∗ , λ = α ( α − ) α − dh ( v ∗ ) v ∗ (cid:111) . It can be either a stable point or saddle orunstable node depending on the choice of α , w , the func-tion f and potential V . However, for the case w =
0, thispoint is either saddle or unstable node only as one of theeigenvalue λ is positive. In Fig. 5(b), we plot the regionof instability of this point in ( α , s ∗ , v ∗ ) parameter spacefor the case w =
0. The saddle behaviour of this pointis cosmologically rich as it shows the contribution of anon-canonical term to describe two important intermediateepochs in the history of the universe: the radiation domi-nated epoch for α = w eff = ) and the matter dominationfor very large values of α ( w eff = f and the value of parameters α , this class of non-canonical scalar field models can successfully describe the be-haviour of the Universe at the background level. For f in which v ∗ (cid:54) = α <
1, the models can describe the evolution of theUniverse from a matter domination (point D ) towards a de-Sitter attractor (set D ). For α =
2, the models can explain theradiation domination (point D ) to matter domination (point D ) only, but these cannot be connected to a DE finite latetime attractor. Thus, in summary, the class of models in which f is such that v ∗ (cid:54) = → DM only without a DE epoch or DM → DE butwithout a radiation epoch. However, for f in which v ∗ =
0, themodels cannot successfully describe the complete evolutionof the Universe as there is no matter dominated critical pointwhich is required to explain the large-scale structure formationat the background level. α = E = ( , , z , s , v ) and E = ( , , z , , v ) . The set E corresponds to a matter domi-nated solution [ Ω m = Ω φ = w eff = w , w φ = α − ] witheigenvalues (cid:110) λ = ( w − ) , λ = ( + w ) , λ = , λ = , λ = (cid:111) . Therefore, points on this set always behave as saddle points.The set E corresponds to an accelerated scalar field dom-inated solution [ Ω m = Ω φ = w eff = − , w φ = − ] . It is non-hyperbolic in nature with eigenvalues (cid:110) λ = , λ = , λ = − (cid:104) + (cid:113) − g ( ) z (cid:105) , λ = − (cid:104) − (cid:113) − g ( ) z (cid:105) , λ = − ( + w ) (cid:111) . However, this set is normally hyperbolicset if g ( ) z (cid:54) =
0. Therefore, points on this set behave as stablenode for 0 < g ( ) z ≤ , they behave as stable focus for g ( ) z > and behave as saddle for g ( ) z <
0. For the case g ( ) z =
0, theset E corresponds to a non-hyperbolic set. In order to con-firm the stability behaviour of this set, we numerically per-turbed the solution near the set and determine the behaviourof trajectories by considering a specific choice of f and V . Itis evident from Figs. 6(a), 6(b) and 6(d) that the phase spacetrajectories approach to x = y = s = N → ∞ . Further it can be seen from Fig. 6(c), that trajec-tories starting from any values of z remains almost constant.From the behaviour of the perturbed system near the set E ,we can conclude that the set E behaves as a stable set evenfor g ( ) z =
0. Note that for the choice of potential considered inFig. 6, we have g ( ) =
0. From the above analysis, we see thatin this case for some choice of potential and initial conditions,the universe passes through a matter domination epoch andevolve towards an accelerated DE dominated epoch. Hence,the model can successfully describe the late time transition ofthe universe.Finally, we end this section with some comments on thechoice of coupling and potential functions. In the context ofinflation, there are huge classes of potentials which give thesame dynamics. Thus in order to connect observations withtheories, it is important to identify possible degeneracy be-tween two classes of potentials. For a detail discussion of de-generacy between non-canonical and canonical models in thecontext of inflation see [64, 65]. Usually, the general k -essenceLagrangian of the form L ( φ , X ) = f ( φ ) X α − V ( φ ) can al-ways be recast into a canonical kinetic term with a modifiedpotential for α = α . In gen-eral, by field redefinition, out of f and V , one of them can bereduced to a trivial form for α =
1. This indicates that theremay be some kind of correspondence between the results of f = constant and V = constant cases for α =
1. This is alsosuggested from the cosmological evolution of both the cases at α =
1, where the universe evolves from a stiff matter to a DMdomination and then eventually evolves to a DE domination.
In the present work, we have performed a dynamical systemanalysis of a general k -essence Lagrangian. The general k -essence models give interesting cosmological dynamics suchas a viable cosmological transition from radiation to matterdomination and eventually to DE domination. More impor-tantly, we get a scenario in hand which gives the appealingunified description of DE and DM for a broad class of cou- pling functions. The main aim of this work is to analyse thebackground cosmological dynamics for a broad class of cou-pling and potential functions. Dynamical system techniquesallowed us to determine the broad class of coupling and po-tential functions which can lead to interesting cosmologicaldynamics.In the analysis presented above, we have first analysed thecase when both the function f and potential V are constant (seeSect. 3.1). In this case, the system contains only a matter domi-nated solution ( Ω m = , w eff = w ) and a stable deSitter solution( Ω φ = , w eff = − Λ CDM where the Universe evolvesfrom matter domination towards a DE domination. Secondly,we have considered the case when function f is still constantbut the potential is not constant (Sect. 3.2). In both the cases,for all physically viable critical sets (or points), we have x = f isassumed to be such that Γ is a function of v . Here, the DMcontribution comes from the extra barotropic fluid alone ex-cept for the case when the function f is such that v ∗ (cid:54) = v ∗ is a root of v ( Γ ( v ) − ) =
0) along with a constant potential.In the latter case, the early cosmic matter behaviour is mainlydue to the kinetic part of the non-canonical scalar field (point C in Sect. 3.3). Depending on the value of the non-canonicalparameter α , the Universe evolves either from a radiation orDM or stiff matter solution described by the point C towardsa DM dominated epoch and eventually settles as a cosmolog-ical constant (see Fig. 3 in Sect. 3.3). In particular for α = w eff = ] → DM [ w eff = → DE [ w eff = − α affects the dark energy speed of sound, therefore de-pending on the clustering property of a scalar field, the earlynon-canonical scalar field domination behaves either as coldor warm dark matter [22]. This might somehow circumventthe problem of Λ CDM in fitting CMB and weak lensing datasimultaneously [66]. Further, the early dark matter behaviourof the kinetic component of a scalar field could also explainthe formation of seeds for super-massive black holes [67]. In-terestingly, the scenario in hand can also explain a possibleinflationary exit problem (point C ) for a constant potential.This behaviour is also achieved in k -essence model discussedin [44]. To have a clear picture on this, one requires to investi- gate the dynamical behaviour of the potential V and determinethe conditions for evolving towards a slow-roll regime. It de-serves mentioning that a similar form of a solution is howeveralso obtained in a well-known α -attractors inflationary mod-els with non-constant potential [68]. Here, we should remarkthat in α -attractors inflationary models, the slow roll condi-tions are usually satisfied when the coupling function f is verylarge, irrespective of the form of the potential V . In particular,a singular f may correspond to a critical point, which how-ever cannot be captured by variables (20). The critical pointfor such behaviour might be pushed towards infinity and com-pactification techniques may give such a point. Further, it isworth noting that due to the absence of accelerated global at-tractors, the late time evolution from decelerated to an accel-erated phase is not guaranteed but strictly depends on fine-tuning of initial conditions and model parameters. In additionto this, the non-compact form of the phase space demands thecompactification technique of the phase space, so as to have aglobal picture on the cosmological dynamics. However, suchan analytical investigation is beyond the scope of the presentwork.It is worth noting that in the presence of the photons, the ra-diation dominated era would comprise of two components i.e.the scalar field kinetic component and photons that cause thespace to expand. However, only one component i.e. photonsthat are emitted and absorbed by matter. Despite tight con-straints from standard big bang nucleosynthesis, this may sig-nal interesting observational characteristics and perhaps couldbe connected with the puzzle of the 21 cm line anomaly. Thediscussion on the implications of this phenomenon will requirefurther investigations in the future.In summary, we conclude that on employing dynamicalsystem techniques, the non-canonical scalar field models givecontrasting dynamics for different choices of f and V . Moreprecisely, our approach identified a broad class of couplingand potential functions which give interesting cosmologicaldynamics at the background level. In particular, our analysisreveals that a class of coupling functions f where the rootof an equation v ( Γ ( v ) − ) = → DM → DEdominated era) as well as the unified description of DM andDE. It would be of interest in the future to further analysethe behaviour using an alternative choice of dynamical vari-ables which might give a more broad class of f and V to de-scribe the dark sectors unification. This would corroborate thework of [23]. Further, for a various choice of f and V , thismodel can mimic the Λ CDM model in late times at the back-ground level which supports the work of Sahni and Sen [22].However, there may be a possible deviation at the perturbationlevel, which might lead to interesting signatures towards thepresent and future observations [67]. By employing dynami-cal system techniques, analysis at the perturbation level mightgive a general conclusion over a wide range of initial condi- tions for perturbations. Finally, in the light of precise cosmo-logical data, it would be of interest to investigate the degreeof degeneracy of this class of models for higher order kineticterms. Acknowledgements
The authors thank Laur Järv for useful discussions.JD was supported by the Core research grant of SERB, Department ofScience and Technology India (File No.CRG/2018/001035) and the As-sociate program of IUCAA. The authors also thank the anonymous re-viewer for constructive suggestions which lead to the improvement of thework.
References
1. A. G. Riess, et al. [Supernova Search Team] Astron. J. , 1009(1998)2. S. Perlmutter, et al. [Supernova Cosmology Project Collaboration]Astrophys. J. , 565 (1999)3. P. Brax, Rept. Prog. Phys. , 016902 (2018)4. A. R. Liddle, D. H. Lyth, Cosmological inflation and large scalestructure , Cambridge, UK: Univ. Pr. (2000)5. E. J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D ,1753 (2006)6. S. Tsujikawa, Class. Quant. Grav. , 214003 (2013)7. W. Fang, H. Q. Lu, Z. G. Huang, Class. Quant. Grav. , 3799(2007)8. V. F. Mukhanov, A. Vikman, J. Cosmol. Astropart. Phys. , 004(2006)9. J. Lee, T. H. Lee, P. Oh, J. Overduin, Phys. Rev. D , 123003(2014)10. G. W. Horndeski, Int. J. Theor. Phys. , 363 (1974)11. C. Armendariz-Picon, V. F. Mukhanov, P. J. Steinhardt, Phys. Rev.D , 103510 (2001)12. C. Armendariz-Picon, T. Damour, V. F. Mukhanov, Phys. Lett. B , 209 (1999)13. T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D , 023511(2000)14. C. Armendariz-Picon, E. A. Lim, J. Cosmol. Astropart. Phys. ,007 (2005)15. S. Unnikrishnan, Phys. Rev. D , 063007 (2008)16. A. Melchiorri, L. Mersini-Houghton, C. J. Odman, M. Trodden,Phys. Rev. D , 043509 (2003)17. A. Y. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B ,265 (2001)18. T. Padmanabhan, Phys. Rev. D , 021301 (2002)19. R. J. Scherrer, Phys. Rev. Lett. , 011301 (2004)20. A. Diez-Tejedor, A. Feinstein, Phys. Rev. D , 023530 (2006)21. J. De-Santiago, J. L. Cervantes-Cota, Phys. Rev. D , 063502(2011)22. V. Sahni, A. A Sen, Eur. Phys. J. C , 225 (2017)23. S. S. Mishra, V. Sahni, arXiv:1803.09767 [gr-qc] (2018)24. D. Bertacca, N. Bartolo, A. Diaferio, S. Matarrese, J. Cosmol. As-tropart. Phys. , 023 (2008)25. N. Bilic, G. B. Tupper, R. D. Viollier, Phys. Lett. B , 17 (2002)26. M. C. Bento, O. Bertolami, A. A. Sen, Phys. Rev. D , 043507(2002)27. D. Bertacca, M. Bruni, O. F. Piatella, D. Pietrobon, J. Cosmol. As-tropart. Phys. , 018 (2011)28. S. Kumar, A. A. Sen, J. Cosmol. Astropart. Phys. , 036 (2014)29. A. Al Mamon, S. Das, Eur. Phys. J. C , 135 (2016)30. E. J. Copeland, A. R. Liddle, D. Wands, Phys. Rev. D , 4686(1998)31. S. Carloni, S. Capozziello, J. A. Leach, P. K. S. Dunsby, Class.Quant. Grav. , 035008 (2008) 32. G. Leon, E. N. Saridakis, J. Cosmol. Astropart. Phys. , 025(2013)33. G. Leon, J. Saavedra, E. N. Saridakis, Class. Quant. Grav. ,135001 (2013)34. S. Carloni, F. S. N. Lobo, G. Otalora, E. N. Saridakis, Phys. Rev. D , 024034 (2016)35. C. G. Boehmer, N. Tamanini, M. Wright, Phys. Rev. D , 123002(2015)36. C. G. Boehmer, N. Tamanini, M. Wright, Phys. Rev. D , 123003(2015)37. J. Dutta, W. Khyllep, N. Tamanini, Phys. Rev. D , 063004 (2016)38. J. Dutta, W. Khyllep, N. Tamanini, Phys. Rev. D , 023515 (2017)39. H. Zonunmawia, W. Khyllep, N. Roy, J. Dutta, N. Tamanini, Phys.Rev. D , 083527 (2017)40. J. Dutta, W. Khyllep, N. Tamanini, J. Cosmol. Astropart. Phys. , 038 (2018)41. J. Dutta, W. Khyllep, E. N. Saridakis, N. Tamanini, S. Vagnozzi, J.Cosmol. Astropart. Phys. , 041 (2018)42. M. Hohmann, L. Järv, U. Ualikhanova, Phys. Rev. D , 043508,(2017)43. S. Bahamonde, C. G. Boehmer, S. Carloni, E. J. Copeland, W.Fang, N. Tamanini, Phys. Rept. , 777 (2018)44. J. De-Santiago, J. L. Cervantes-Cota, D. Wands, Phys. Rev. D ,023502 (2013)45. S. Das, A. Al Mamon, Astrophys. Space Sci. , 371 (2015)46. S. Y. Zhou, Phys. Lett. B , 7 (2008)47. W. Fang, Y. Li, K. Zhang, H. Q. Lu, Classical Quantum Gravity ,155005 (2009)48. T. Matos, J. R. Luevano, I. Quiros, L. A. Urena-Lopez, J. A.Vazquez, Phys. Rev. D , 123521 (2009)49. L. A. Urena-Lopez, J. Cosmol. Astropart. Phys. , 035 (2012)50. S. Tsujikawa, M. Sami, Phys. Lett. B , 113 (2004)51. F. Piazza, S. Tsujikawa, J. Cosmol. Astropart. Phys. , 004(2004)52. S. Unnikrishnan, V. Sahni and A. Toporensky, J. Cosmol. Astropart.Phys. , 018 (2012)53. A. De Felice and S. Tsujikawa, J. Cosmol. Astropart. Phys. ,007 (2012)54. L. P. Chimento and R. Lazkoz, Phys. Rev. D , 023505 (2005)55. C. Bonvin, C. Caprini and R. Durrer, Phys. Rev. Lett. , 081303(2006)56. D. A. Easson, I. Sawicki, A. Vikman, J. Cosmol. Astropart. Phys. , 014 (2013)57. Y. Leyva, D. Gonzalez, T. Gonzalez, T. Matos, I. Quiros, Phys. Rev.D , 044026 (2009)58. R. Tavakol, Introduction to Dynamical Systems (ch. 4, Part one),Cambridge University Press, Cambridge, (1997)59. Y. F. Cai, E. N. Saridakis, M. R. Setare, J. Q. Xia, Phys. Rep. ,1 (2010)60. P
LANCK collaboration, P. A. R. Ade et al., Astron. Astrophys. ,A13 (2016)61. A. A. Coley,
Dynamical Systems and Cosmology , Kluwer, Dor-drecht, (2003)62. S. Wiggins,
Introduction to applied nonlinear dynamical systemsand chaos , Springer, New York U.S.A. and Heidelberg Berlin Ger-many, (1990)63. G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos, E. N. Saridakis,J. Cosmol. Astropart. Phys. , 003 (2018)64. R. Gwyn, M. Rummel, A. Westphal, J. Cosmol. Astropart. Phys. , 010 (2013)65. L. Järv, K. Kannike, L. Marzola, A. Racioppi, M. Raidal, M. Rün-kla, M. Saal, H. Veermäe, Phys. Rev. Lett. , 151302 (2017)66. M. Kunz, S. Nesseris, I. Sawicki, Phys. Rev. D , 063006 (2015)67. I. Sawicki, V. Marra, W. Valkenburg, Phys. Rev. D , 083520(2013)68. A. Alho, C. Uggla, Phys. Rev. D95