arXiv:1202.3781v2 [gr-qc] 6 Jul 2012 ri ePrint: ArXiv coupling mal Keywords: value. cosmological-constant the to uigteeouintedr nryeuto-fsaepar equation-of-state − energy dark the value evolution speciﬁc the in during the solution, of late-time independently constant, qu additional cosmological an standard possesses to energy similarly dark dark-energy attractor, quintessence-like, Rel late-time the General dark-energy in coupl of nonminimal result a equivalent can for universe also teleparallel allowing the ﬁeld, scalar on canonical based is which Abstract. Xu, Chen Energy Dark Teleparallel of analysis Phase-Space JCAP to submission for Prepared a d c b E-mail: Vill Las de Central Cuba Universidad Mathematics, 76 of TX Waco, Department University, Baylor Department, Physics CASPER, Greece Athens of University Technical National Division, Physics o China University 400065, Chongqing Chongqing Physics, Telecommunications, and Mathematics of College ,oeigago ecito o t bevddnmclbeha dynamical observed its for description good a oﬀering 1, [email protected] epromadtie yaia nlsso h teleparalle the of analysis dynamical detailed a perform We oie rvt,dr nry hno-iiecosn,f( crossing, phantom-divide energy, dark gravity, Modiﬁed a maulN Saridakis N. Emmanuel 1202.3781 , Emmanuel − [email protected] b,c n el Leon Genly and dmntdslto,o otestiﬀ the to or solution, -dominated mtrcnb ihraoeo below or above either be can ameter ftemdlprmtr.Finally, parameters. model the of s 58 orfuCmu,Athens, Campus, Zografou 15780 , n ihgaiy eﬁdta the that ﬁnd We gravity. with ing nesne oee,teleparallel However, intessence. hc akeeg eae iea like behaves energy dark which iradissaiiainclose stabilization its and vior tvt,i hc n dsa adds one which in ativity, , ot and Posts f s at lr P54830, CP Clara Santa as, [email protected] 9-30 USA 798-7310, akeeg scenario, energy dark l )gaiy nonmini- gravity, T) d Contents

1 Introduction 1

2 Teleparallel Dark Energy 2

3 Phase-space analysis 4 3.1 Finite Phase-space analysis 6 3.2 Phase-space analysis at inﬁnity 6

4 Cosmological Implications 8

5 Conclusions 13

A Stability of the ﬁnite critical points 14

B Stability of the critical points at inﬁnity 16

1 Introduction

According to observations, the observable universe experiences an accelerating expansion at late times [1–3]. There are two major ways to explain such a behavior, apart from the simple consideration of a cosmological constant. The ﬁrst is to modify the gravitational sector itself (see [4] for a review and references therein), obtaining a modiﬁed cosmological dynamics. The other approach is to modify the content of the universe introducing the dark energy with negative pressure, which can be based on a canonical scalar ﬁeld (quintessence) [5–17], a phantom ﬁeld [18–23], or on the combination of both these ﬁelds in a uniﬁed scenario called quintom [24–32] (see the reviews [33] and [34]). “Teleparallel” dark energy is a recently proposed scenario that tries to incorporate the dark energy sector [35, 36]. It lies on the framework of the “teleparallel” equivalent of General Relativity (TEGR), that is based on torsion instead of curvature formulation [37, 38], in which one adds a canonical scalar ﬁeld, and the dark energy sector is attributed to this ﬁeld. When the ﬁeld is minimally-coupled to gravity the scenario is completely equivalent to standard quintessence [5–7], both at the background and perturbation levels. However, in the case where one allows for a nonminimal coupling between the ﬁeld and the torsion scalar which is the only suitable gravitational scalar in TEGR, the resulting scenario presents a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing [35]. The interesting cosmological behavior of teleparallel dark energy makes it necessary to perform a phase-space and stability analysis, examining in a systematic way the possible cosmological behaviors, focusing on the late-time stable solutions. In the present work we perform such an approach, which allows us to bypass the high non-linearities of the cosmo- logical equations, which prevent any complete analytical treatment, obtaining a (qualitative) description of the global dynamics of these models, that is independent of the initial condi- tions and the speciﬁc evolution of the universe. Furthermore, in these asymptotic solutions we calculate various observable quantities, such as the deceleration parameter, the eﬀective (total) equation-of-state parameter, and the various density parameters.

– 1 – The plan of the work is the following: In section 2 we brieﬂy review the scenario of teleparallel dark energy and in section 3 we perform a detailed phase-space analysis. In section 4 we discuss the cosmological implications and the physical behavior of the scenario. Finally, section 5 is devoted to the summary of the results.

2 Teleparallel Dark Energy

In this section we review teleparallel dark energy. Such a scenario is based on the “teleparal- lel” equivalent of General Relativity (TEGR)[37, 38], in which instead of using the torsionless Levi-Civita connection one uses the curvatureless Weitzenb¨ock one. The dynamical objects are the four linearly independent, parallel vector ﬁelds called vierbeins, and the advantage of this framework is that the torsion tensor is formed solely from products of ﬁrst derivatives µ of the tetrad. In particular, the vierbein ﬁeld eA(x ) forms an orthonormal basis for the µ tangent space at each point x , that is eA eB = ηAB, where ηAB = diag(1, 1, 1, 1), and · µ− − − furthermore the vector eA can be analyzed with the use of its components eA in a coordinate µ 1 basis, that is eA = eA∂µ . The metric tensor is obtained from the dual vierbein as

A B gµν (x)= ηAB eµ (x) eν (x). (2.1)

wλ Furthermore, the torsion tensor of the Weitzenb¨ock connection Γνµ [42] reads

wλ wλ λ λ A A T = Γ Γ = e (∂µe ∂ν e ). (2.2) µν νµ − µν A ν − µ In such a framework all the information concerning the gravitational ﬁeld is included in λ the torsion tensor T µν . The corresponding “teleparallel Lagrangian” can be constructed from this torsion tensor under the assumptions of invariance under general coordinate transfor- mations, global Lorentz transformations, and the parity operation, along with requiring the Lagrangian density to be second order in the torsion tensor [38]. In particular, it is proved to be just the torsion scalar T , namely [37, 38, 43, 44]:

1 ρµν 1 ρµν ρ νµ = T = T Tρµν + T Tνµρ T T . (2.3) L 4 2 − ρµ ν Therefore, the simplest action in a universe governed by teleparallel gravity is

4 T I = d xe + m , (2.4) 2κ2 L Z where e = det(eA) = √ g (one could also include a cosmological constant). If we vary it µ − with respect to the vierbein ﬁelds we obtain the equations of motion, which are exactly the same as those of General Relativity for every geometry choice, and that is why the theory is called “teleparallel equivalent to General Relativity”. At this stage one has two ways of generalizing the action (2.4), inspired by the corre- sponding procedures of standard General Relativity. The ﬁrst is to replace T by an arbitrary function f(T ) [39–41, 45–85], similarly to f(R) extensions of GR, and obtain new interesting

1In this manuscript we follow the notation of [39–41], that is Greek indices µ, ν,... and capital Latin indices A,B,... run over all coordinate and tangent space-time 0, 1, 2, 3, while lower case Latin indices (from the middle of the alphabet) i, j, ... and lower case Latin indices (from the beginning of the alphabet) a,b,... run over spatial and tangent space coordinates 1, 2, 3, respectively. Finally, we use the metric signature (+, −, −, −).

– 2 – terms in the ﬁeld equations. The second, on which we focus in this work, is to add a canon- ical scalar ﬁeld in (2.4), similarly to GR quintessence, allowing for a nonminimal coupling with gravity. Then the dark energy sector will be attributed to this ﬁeld and therefore the corresponding scenario is called “teleparallel dark energy” [35]. In particular, the action will simply read:

4 T 1 µ 2 S = d xe 2 + ∂µφ∂ φ + ξT φ V (φ)+ m . (2.5) "2κ 2 − L # Z Here we mention that since in TEGR, that is in the torsion formulation of GR, the only scalar is the torsion one, the nonminimal coupling will be between this and the scalar ﬁeld (similarly to standard nonminimal quintessence where the scalar ﬁeld couples to the Ricci scalar). Variation of action (2.5) with respect to the vierbein ﬁelds yields the equation of motion

2 2 1 ρ µν λ ρ νµ 1 ν + 2ξφ e− ∂µ(ee Sρ ) e T µλSρ e T κ2 A − A − 4 A ν 1 µ µ ν e ∂µφ∂ φ V (φ) + e ∂ φ∂µφ − A 2 − A em ρ µν ρ ν +4ξeASρ φ (∂µφ)= eA T ρ . (2.6) em ν where T ρ stands for the usual energy-momentum tensor. Thus, imposing a ﬂat Friedmann- Robertson-Walker (FRW) background metric

2 2 2 i j ds = dt a (t) δijdx dx , (2.7) − where t is the cosmic time, xi are the comoving spatial coordinates and a(t) is the scale factor, that is for a vierbein choice of the form

A eµ = diag(1, a, a, a), (2.8) we acquire the corresponding Friedmann equations: κ2 H2 = ρ + ρ , (2.9) 3 φ m κ2 H˙ = ρφ + pφ + ρm + pm , (2.10) − 2 where H =a/a ˙ is the Hubble parameter, with a dot denoting t-diﬀerentiation. In these expressions, ρm and pm are the matter energy density and pressure, respectively, following the standard evolution equationρ ˙m + 3H(1 + wm)ρm = 0, with wm = pm/ρm the matter equation-of-state parameter. Furthermore, we have introduced the eﬀective energy density and pressure of scalar ﬁeld

1 2 2 2 ρφ = φ˙ + V (φ) 3ξH φ , (2.11) 2 − 1 2 2 2 pφ = φ˙ V (φ) + 4ξHφφ˙ + ξ 3H + 2H˙ φ . (2.12) 2 − Moreover, variation of the action with respect to the scalar ﬁeld provides its evolution equa- tion, namely: 2 φ¨ + 3Hφ˙ + 6ξH φ + V ′(φ) = 0. (2.13)

– 3 – Note that in the above relations we have made use of the useful relation

T = 6H2, (2.14) − which straightforwardly arises from the calculation of (2.3) for the ﬂat FRW geometry. In the present cosmological paradigm, similarly to the standard quintessence, dark en- ergy is attributed to the scalar ﬁeld, and thus its equation-of-state parameter reads:

pφ wDE wφ = . (2.15) ≡ ρφ

Finally, one can see that the scalar ﬁeld evolution (2.13) leads to the standard relation

ρ˙φ + 3H(1 + wφ)ρφ = 0. (2.16)

Teleparallel dark energy exhibits very interesting cosmological behavior. In the minimally- coupled case the cosmological equations coincide with those of the standard quintessence, both at the background and perturbation levels. However, in the nonminimal case one can obtain a dark energy sector being quintessence-like, phantom-like, or experiencing the phantom-divide crossing during evolution, a behavior that is much richer comparing to Gen- eral Relativity (GR) with a scalar ﬁeld [35, 36]. Therefore, it is interesting to perform a phase-space analysis, that is to investigate late-time solutions that are independent from the initial conditions and the speciﬁc universe evolution. This is performed in the next section.

3 Phase-space analysis

In order to perform the phase-space and stability analysis of the scenario at hand, we have to transform the aforementioned dynamical system into its autonomous form X′ = f(X) [86–90], where X is the column vector constituted by suitable auxiliary variables, f(X) the corresponding column vector of the autonomous equations, and prime denotes derivative with respect to M = ln a. Then we extract its critical points Xc satisfying X′ = 0, and in order to determine the stability properties of these critical points, we expand around Xc, setting X = Xc +U, with U the perturbations of the variables considered as a column vector. Thus, for each critical point we expand the equations for the perturbations up to the ﬁrst order as: U′ = Q U, where the matrix Q contains the coeﬃcients of the perturbation equations. · Finally, for each critical point, the eigenvalues of Q determine its type and stability. In the scenario at hand, we introduce the auxiliary variables:

κφ˙ x = √6H κ V (φ) y = p√3H z = ξ κφ. (3.1) | | Using these variables the Friedmann equationp (2.9) becomes

κ2ρ x2 + y2 z2sgn(ξ)+ m = 1. (3.2) − 3H2

– 4 – This constraint allows for expressing ρm as a function of the auxiliary variables (3.1). There- fore, using (3.2) and (2.11) we can write the density parameters as: 2 κ ρm 2 2 2 Ωm = 1 x y + z sgn(ξ) ≡ 3H2 − − 2 κ ρφ 2 2 2 ΩDE = x + y z sgn(ξ), (3.3) ≡ 3H2 − while for the dark-energy equation-of-state parameter (2.15) we obtain:

2 2 2 2 2 2 2 x y + 4 3 zx ξ sgn(ξ) z wmsgn(ξ) 1 x y + z sgn(ξ) wDE = − | | − − − . (3.4) q [1p + z2 sgn(ξ)] [x2 + y2 z2sgn(ξ)] − In this scenario wDE can be quintessence-like, phantom-like, or experience the phantom divide crossing during the evolution. Without loss of generality, in the following we restrict the analysis in the dust matter case, that is we assume that wm = 0. It is convenient to introduce two additional quantities with great physical signiﬁcance, namely the “total” equation-of-state parameter:

x2 y2 +4 2 zx ξ sgn(ξ) pφ − 3 | | wtot = wDEΩDE = 2 , (3.5) ≡ ρφ + ρm 1+qz sgn(pξ) and the deceleration parameter q: H˙ 1 3 q 1 = + wtot ≡ − − H2 2 2 1 + 3(x2 y2)+ z + 4√6x ξ z sgn(ξ) = − | | . (3.6) 2[1+h z2 sgn(ξ)]p i Finally, concerning the scalar potential V (φ) the usual assumption in the literature is to assume an exponential potential of the form V = V exp( κλφ), (3.7) 0 − since exponential potentials are known to be signiﬁcant in various cosmological models [86–92] (equivalently, but more generally, we could consider potentials satisfying λ = 1 ∂V (φ) − κV (φ) ∂φ ≈ const, which is valid for arbitrary but nearly ﬂat potentials [93–95]). Moreover, note that the exponential potential was used as an example in the initial work on teleparallel dark energy [35]. In summary, using the auxiliary variables (3.1) and considering the exponential potential (3.7), the equations of motion (2.9), (2.10) and (2.13) in the case of dust matter, can be transformed into the following autonomous system:

3 2 3x 2√6z ξ x sgn(ξ) 3x 1 2 3x x′ = + | | + y √6λ 2z2sgn(ξ) + 2 z2sgn(ξ) + 2 − 2 2 − sgn(ξ)z2 + 1 p √6z ξ sgn(ξ) − | | p 3 3yx2 3 y2 2 yx λz 4 ξ zsgn(ξ)+ λ y′ = + y 1 − | | 2sgn(ξ)z2 + 2 2 − z2sgn(ξ) + 1 − q nh 1+ z2sgn(iξ) o p z′ = √6 ξ x, (3.8) | | p

– 5 – where we have used that for every quantity F we acquire F˙ = HF ′. Since ρm is nonnegative, from (3.2) we obtain that x2 + y2 z2sgn(ξ) 1. Thus, we deduce that for ξ < 0 the system − ≤ (3.8) deﬁnes a ﬂow on the compact phase space Ψ = x2 + y2 z2sgn(ξ) 1,y 0 , for − ≤ ≥ ξ = 0 the phase space is compact and it is reduced to the circle Ψ = x2 + y2 1,y 0 , ≤ ≥ while for ξ > 0 the phase space Ψ is unbounded. Before proceeding we make two comments on the degrees of freedom and the choice of auxiliary variables. Firstly, as we have already mentioned, in the minimal coupling case, that is when ξ = 0, the model at hand coincides with standard quintessence, which phase space analysis is well known using two degrees of freedom, namely the variables x and y deﬁned above [86]. On the other hand, when ξ = 0 we have three degrees of freedom and all x, y, z 6 are necessary. Therefore, in order to perform the analysis in a uniﬁed way, we use the three variables deﬁned above, having in mind that for ξ = 0 the variable z becomes zero and thus irrelevant. Secondly, apart from the standard choices of the variables x and y, one must be careful in suitably choosing the variable z in order not to lose dynamical information. For example, although the choice for the variable z = κρm is another reasonable choice, however √3H it still loses the critical points that lie at “inﬁnity”. Therefore, in order to completely cover the phase-space behavior in the following we will additionally use the Poincar´ecentral projection method to investigate the dynamics at “inﬁnity”. The negligence of this point was the reason of the incomplete phase space analysis of teleparallel dark energy performed in [96].

3.1 Finite Phase-space analysis Let us now proceed to the phase-space analysis. The real and physically meaningful (that is corresponding to an expanding universe, i.e possessing H > 0) critical points (xc,yc, zc) of the autonomous system (3.8), obtained by setting the left hand sides of the equations to zero, are presented in Table 1. In the same table we provide their existence conditions. The 3 3 × matrix Q of the linearized perturbation equations of the system (3.8) is shown in Appendix A. For each critical point of Table 1 we examine the sign of the real part of the eigenvalues of Q in order to determine the type and stability of the point. The details of the analysis and the various eigenvalues are presented in Appendix A, and in Table 2 we summarize the 2 results. In addition, for each critical point we calculate the values of ΩDE, wDE, wtot and q given by (3.3), (3.4), (3.5) and (3.6) respectively.

3.2 Phase-space analysis at inﬁnity Due to the fact that the dynamical system (3.8) is non-compact for the choice ξ > 0, there could be features in the asymptotic regime which are non-trivial for the global dynamics. Thus, in order to complete the analysis of the phase space we must extend our study using the Poincar´ecentral projection method [97]. Let us consider the Poincar´evariables

xr = ρ cos θ sin ψ, zr = ρ sin θ sin ψ, yr = ρ cos ψ, (3.9) where ρ = r , r = x2 + y2 + z2, θ [0, 2π], and π ψ π (we restrict the angle √1+r2 ∈ − 2 ≤ ≤ 2 ψ to this interval sincep the physical region is given by y > 0) [97–100]. Thus, the points at 2The critical point D that exists only for ξ = 0, is very close to be a stationary point in the general case (ξ =6 0) too, but with an arbitrarily small value of the slope of the potential λ. It becomes an exact critical point only for λ = 0, namely the point J. Furthermore, for arbitrary large values of λ the singular point E (that exists for ξ = 0) mimics the behavior of the point A (that exists for ξ =6 0).

– 6 – Cr. P. xc yc zc Exists for A 0 0 0 all ξ,λ B 1 0 0 ξ = 0, all λ C -1 0 0 ξ = 0, all λ 2 D λ 1 λ 0 ξ = 0, λ2 6 √6 − 6 ≤ E 3 1 q 3 1 0 ξ = 0, λ2 3 2 λ 2 λ ≥ 2 2 q 2ξ q2√ξ(ξ λ ) hξ √ξ(ξ λ )i√ ξ − − − − | | 2 F 0 λ2 λξ 0 < λ ξ r ≤ 2 2 2ξ+2√ξ(ξ λ ) hξ+√ξ(ξ λ )i√ ξ − − | | 2 G 0 λ2 λξ 0 < λ ξ or ξ < 0 r ≤ J 0 1 0 ξ = 0 and λ = 0 6

Table 1. The real and physically meaningful critical points of the autonomous system (3.8). Existence conditions.

Cr. P. Stability ΩDE wDE wtot q 1 A saddle point 0 arbitrary 0 2 B unstable for λ< √6 saddle point otherwise 1 1 1 2 C unstable for λ> √6 − saddle point otherwise 1 1 1 2 2 2 2 D stable node for λ2 < 3 1 1+ λ 1+ λ 1+ λ − 3 − 3 − 2 saddle point for 3 < λ2 < 6 2 24 3 1 E stable node for 3 < λ < 7 λ2 0 0 2 2 24 stable spiral for λ > 7 F stable node for λ2 <ξ 1 1 1 1 − − − G saddle point 1 1 1 1 3 − − − J stable spiral for 8 <ξ 1 1 1 1 3 − − − stable node for 0 <ξ< 8 saddle point for ξ < 0

Table 2. The real and physically meaningful critical points of the autonomous system (3.8). Stability conditions, and the corresponding values of the dark-energy density parameter ΩDE, of the dark- energy equation-of-state parameter wDE, of the total equation-of-state parameter wtot and of the deceleration parameter q.

“inﬁnity” (r + ) are those having ρ 1. Furthermore, the physical phase space is given → ∞ → by 2 2 2 1 (xr,yr, zr) [ 1, 1] [0, 1] [ 1, 1] z x + y . ∈ − × × − | r ≤ r r ≤ 2 Inverting relations (3.9) and substituting into (3.3),(3.4), we obtain the dark energy

– 7 – Cr. P. xrc yrc zrc Stability ΩDE wDE wtot q K √2 0 √2 unstable for ± ∓ 2 ± 2 0 <ξ< 3 arbitrary arbitrary 1 4 2ξ 2 2√6ξ 8 − 3 − saddle point q 3 for 8 <ξ L √2 0 √2 saddle point ± ± 2 ± 2 2ξ for all ξ > 0 arbitrary arbitrary 1 + 4 3 2 + 2√6ξ q

Table 3. The real and physical critical points of the autonomous system (3.8) at inﬁnity, which exists for ξ > 0, all λ, and the corresponding values of the dark-energy density parameter ΩDE, of the dark-energy equation-of-state parameter wDE, of the total equation-of-state parameter wtot and of the deceleration parameter q. density and equation-of-state parameters as a function of the Poincar´evariables, namely:

2 2 2 xr+yr zr Ω = 2 −2 2 (3.10) DE 1 xr yr zr − − − 2 2 2 2 2 (3xr+4√6√ξzrxr 3yr )(1 xr yr zr ) w = 2 2− 2 2− 2− − , (3.11) DE 3(1 xr yr )(xr+yr zr ) − − − and similarly substituting into (3.5), (3.6) we obtain the corresponding expressions for the total equation-of-state and deceleration parameters:

2 2 3x + 4√6√ξzrxr 3y w = r − r , (3.12) tot 3 (1 x2 y2) − r − r

2 2 1 3 3x + 4√6√ξzrxr 3y q = + r − r . (3.13) 2 2 3 (1 x2 y2) ( − r − r ) Applying the procedure prescribed in Appendix B we conclude that there are four physical critical points at inﬁnity, namely the points K , satisfying cot θ = 1, and the ± − points L satisfying cot θ = 1. These critical points, along with their stability conditions ± are presented in Table 3. In the same Table we include the corresponding values of the observables ΩDE, wDE, wtot and q, calculated using (3.10),(3.11),(3.12),(3.13). As we show in Appendix B, the above critical points correspond to the limit

φ˙ φ , , (3.14) | |→∞ H →∞

x satisfying the rate (ln φ)′ √6 = √6 cot θ. ≡ z 4 Cosmological Implications

Having performed the complete phase-space analysis of teleparallel dark energy, we can now discuss the corresponding cosmological behavior. A ﬁrst remark is that in the minimal case

– 8 – (that is ξ = 0) we do verify that the scenario at hand coincides completely with standard quintessence. Therefore, we will make a brief review on the subject and then focus on the nonminimal case. The points B to E exists only for the minimal case, that is only for ξ = 0. Points B and C are not stable, corresponding to a non-accelerating, dark-energy dominated universe, with a stiﬀ dark-energy equation-of-state parameter equal to 1. Both of them exist in standard quintessence [86]. Point D is a saddle one for 3 < λ2 < 6, however for 0 < λ2 < 3 it is a stable node, and thus it can attract the universe at late times. It corresponds to a dark-energy dominated universe, with a dark-energy equation-of-state parameter lying in the quintessence regime, which can be accelerating or not according to the λ-value. This point exists in standard quintessence [86]. It is the most important one in that scenario, since it is both stable and possesses a wDE compatible with observations. Point E is a stable one for 3 < λ2. It has the advantage that the dark-energy density parameter lies in the interval 0 < ΩDE < 1, that is it can alleviate the coincidence problem, since dark matter and dark energy density parameters can be of the same order (in order to treat the coincidence problem one must explain why the present dark energy and matter den- sity parameters are of the same order of magnitude although they follow diﬀerent evolution behaviors). However, it has the disadvantage that wDE is 0 and the expansion is not accel- erating, which are not favored by observations. This point exists in standard quintessence [86]. Let us now analyze the case ξ = 0. In this case we obtain the critical points A, 6 F , G and J. The point A is saddle point, and thus it cannot be late-time solution of the universe. It corresponds to a non-accelerating, dark-matter dominated universe, with arbitrary dark-energy equation-of-state parameter. Note that this trivial point exists in the standard quintessence model too [86], since it is independent of ξ. The present scenario of teleparallel dark energy, possesses two additional, non-trivial critical points that do not exist in standard quintessence. Thus, they account for the new information of this richer scenario, and as expected they depend on the nonminimal coupling ξ. In particular, point F is stable if λ2 < ξ, and thus it can attract the universe at late times. It corresponds to an accelerating universe with complete dark energy domination, with wDE = 1, that is dark energy behaves like a cosmological constant. We stress that − this wDE value is independent of λ and ξ, which is an important and novel result. Thus, while point D (the important point of standard quintessence) needs to have a very ﬂat, that is quite tuned, potential in order to possess a wDE near the observed value 1, point F − exhibits this behavior for every λ-value, provided that λ2 < ξ. This feature is a signiﬁcant advantage of the scenario at hand, amplifying its generality, and oﬀers a mechanism for the stabilization of wDE close to the cosmological-constant value. Similarly, we have the point G, which has the same cosmological properties with F , however it is a saddle point and thus it cannot be a late-time solution of the universe, but the universe can spend a large period of time near this solution. Finally, when ξ = 0 and for the limiting case λ = 0, that is for a constant potential, the 6 present scenario exhibits the critical point J, which is stable for ξ > 0. It corresponds to a dark-energy dominated, de Sitter universe, in which dark energy behaves like a cosmological constant. Let us now now analyze the critical points at inﬁnity. The points K are saddle for 3 ± ξ > 8 . The fact that they possess arbitrary ΩDE and wDE is a signiﬁcant advantage, since

– 9 – such solutions can alleviate the coincidence problem. Note however that the corresponding wtot and deceleration parameter are not arbitrary, oﬀering a good quantitative description of the cosmological behavior (actually this is the reason we introduced these observables). In 1 3 particular, we conclude that for ξ > 6 we obtain an accelerated expansion, while for ξ > 8 the expansion is super-accelerating (q < 1 that is H˙ > 0 and wtot < 1) that is the universe − − presents a phantom behavior. We mention that since these points are saddle they cannot attract the universe at late times, however the universe can spend a large period of time near these solutions, before approaching the saddle point A or the global attractor F (for potentials with slope λ2 ξ). In that case the above physical features present a transient ≤ character, which is quite signiﬁcant from the observational point of view. Points L are saddle for all ξ > 0. They have arbitrary ΩDE and wDE however they cor- ± respond to a non-accelerating expansion, and thus they could be important only as transient states of the universe. In order to present the aforementioned behavior more transparently, we ﬁrst evolve the autonomous system (3.8) numerically for the choice λ = 1, and ξ = 0, that is in the minimal scenario, and in Fig. 1 we depict the corresponding phase-space behavior, projected in the x y plane. As we can wee, in this case the quintessence-like critical point D is the late-time − solution of the universe. Note that this point exists only for the minimal case (ξ = 0), where

Figure 1. The projection of the phase-space evolution on the x y plane, for the teleparallel dark energy scenario with λ = 1 and ξ = 0. The trajectories are attracted− by the quintessence-like stable point D, which exists only in the minimal case (ξ =0). our model coincides with standard quintessence (see Fig. 2 of [86]). As we mentioned above, the late-time attractors D and E exists only for the case ξ = 0. To illustrate how sensible is the scenario at hand in slight changes of ξ, in Fig. 2 we depict the projection of the phase-space evolution on the x y plane, for λ = 2 and ξ = 10 3. − − − In this case the point E becomes “quasi-stationary”, since typical trajectories remain very

– 10 – Figure 2. The projection of the phase-space evolution on the x y plane, for the teleparallel dark 3 − energy scenario with λ = 2 and ξ = 10− . Observe that an open set of trajectories remains very close to the quasi-stationary solution −E, before approaching an invariant arc above the saddle point A. The critical point G is a saddle one.

close to it for a large but ﬁnite time interval, while D becomes also quasi-stationary, but it is unstable to perturbations in the z-axis. 3 Finally, to illustrate the dynamics for positive ξ, which is expressed through points at “inﬁnity”, in Fig. 3 we present the three-dimensional Poncar´e(global) phase space for λ = 0.5 and ξ = 1. According to Table 3 the critical points K are saddle ones and exhibit phantom be- ± havior, while L are saddle with a stiﬀ dark energy sector. Thus, far from their basin of ± attraction (but inside the invariant set yr = 0) the orbits departs from them to approach the matter-dominated solution A, while for yr > 0 the orbits are attracted by the cosmological- constant-like solution F . Thus, the epoch sequence K L+ K+ L A F − → → → − → → represents the transition from a universe with a phantom-like dark energy, to a matter- dominated universe with non-phantom dark energy, and then to a dark-energy-dominated, cosmological-constant-like solution. This sequence has a great cosmological signiﬁcance, since it can describe the epoch sequence inﬂation, dark-matter domination, dark-energy domina- tion, in agreement with observations. Additionally, the epoch sequence L K+ F − → → represents the transition from a universe with comparable dark matter and dark energy sectors, to a cosmological-constant-like solution. Before closing this section, let us make a comment on another crucial diﬀerence of teleparallel dark energy, comparing with standard quintessence, that is of the ξ = 0 case 6 comparing to the ξ = 0 one. In particular, when ξ = 0, in which teleparallel dark energy

3Although the singular points B,C,D and E do not exist as critical points for the choices ξ =6 0, in Fig. 2 we mark their locations explicitly, for comparison with the case ξ = 0 of Fig. 1.

– 11 – Figure 3. Poncar´e(global) phase space for the teleparallel dark energy scenario with λ = 0.5 and ξ =1. The points at inﬁnity K ,L are saddles. The trajectories on both ﬁnite and inﬁnite regions are ﬁnally attracted by the cosmological-constant-like± ± stable point F (global attractor), having passed through matter-dominated solutions or solutions with comparable dark matter and dark energy sectors.

coincides with standard quintessence, wDE is always larger that 1, not only at the critical − points, but also throughout the cosmological evolution as well. However, for ξ = 0, during 6 the cosmological evolution wDE can be either above or below 1, and only at the stable − critical point it becomes equal to 1. Such a cosmological behavior is much richer, and very − interesting, both from the theoretical as well as from the observational point of view, since it can explain the dynamical behavior of wDE either above or below the phantom divide, and moreover its stabilization to the cosmological-constant value. In order to present the novel features of the scenario at hand, in Fig. 4 we depict the three-dimensional phase-space behavior for the choice λ = 0.7 and ξ = 1. In this case the universe at late times is attracted by the cosmological-constant-like stable point F . How- ever, during the evolution towards this point the dark-energy equation-of-state parameter wDE presents a very interesting behavior, and in particular, depending on the speciﬁc initial conditions, it can be quintessence-like, phantom-like, or experience the phantom-divide cross- ing. Such a behavior is much richer than standard quintessence, and reveals the capabilities of teleparallel dark energy scenario. Finally, one should also investigate in detail whether the scenario at hand is stable at the perturbation level, especially in the region wDE < 1. Such − an analysis can be heavily based on [39] and one can ﬁnd that f(T ) cosmology and telepar- allel dark energy are stable even in the region wDE < 1. This project is in preparation − [101].

– 12 – Figure 4. The phase-space evolution for the teleparallel dark energy scenario with λ =0.7 and ξ =1. The trajectories are attracted by the cosmological-constant-like stable point F , however the evolution towards it possesses a wDE being quintessence-like, phantom-like, or experiencing the phantom-divide crossing, depending on the speciﬁc initial conditions. The orbits with a thick gray curve on the top left are those with wDE > 1 initially, while the thin black curve corresponds to wDE < 1 initially. − −

5 Conclusions

In the present work we investigated the dynamical behavior of the recently proposed scenario of teleparallel dark energy [35, 36], which is based on the teleparallel equivalent of General Relativity (TEGR), that is on its torsion instead of curvature formulation [37, 38]. In this model one adds a canonical scalar ﬁeld, in which the dark energy sector is attributed, allowing also for a nonminimal coupling between the ﬁeld and the torsion scalar. Thus, although the minimal case is completely equivalent with standard quintessence, the nonminimal scenario has a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing [35, 36]. Performing a detailed phase-space analysis of teleparallel dark energy, we saw that the standard quintessence-like, late-time solution [86] exists only for ξ = 0, and we showed that this solution becomes quasi-stationary if ξ is perturbed even slightly. The same results hold for the other standard quintessence solution, namely the stiﬀ dark-energy late-time attractor, in which dark energy and dark matter can be of the same order, thus alleviating the coincidence problem. Apart from the above late time solutions that exist also in the standard quintessence scenario, teleparallel dark energy has an additional and physically very interesting late-time

– 13 – behavior. In particular, it possesses a late-time attractor in which dark energy behaves like a cosmological constant, independently of the speciﬁc values of the model parameters (provided that the nonminimal coupling ξ is larger than the square of the potential exponent). This feature is a signiﬁcant advantage of the scenario at hand, amplifying its generality, since it provides a natural way for the stabilization of the dark energy equation-of-state parameter to the cosmological constant value, without the need for parameter-tuning. Additionally, we showed that teleparallel dark energy admits critical points at “inﬁn- 3 ity”, which are saddle for ξ > 8 , and thus although they cannot be the late-time solutions, the universe can spend a large period of time near them. Thus, one can obtain the transi- tion from a universe with a phantom-like dark energy, to a matter-dominated universe with non-phantom dark energy, and then to a dark-energy-dominated, cosmological-constant-like solution. This sequence has a great cosmological signiﬁcance, since it can describe the epoch sequence inﬂation, dark-matter domination, dark-energy domination, in agreement with ob- servations Finally, perhaps the most interesting feature of teleparallel dark energy is that in the nonminimal case the dark energy equation-of-state parameter can be either above or below 1, and only at the stable critical points it becomes equal to 1. Such a cosmological − − behavior is much richer comparing to standard quintessence, and very interesting, both from the theoretical as well as from the observational point of view, since it can explain the dynamical behavior of wDE either above or below the phantom divide, and moreover its stabilization to the cosmological-constant value. All these features make the teleparallel dark energy a good candidate for the description of dark energy.

Acknowledgments

The authors would like to thank Yungui Gong for crucial discussions and comments. This work was partially supported by the NNSF key project of China under grant No. 10935013, the National Basic Science Program (Project 973) of China under grant No. 2010CB833004, CQ CSTC under grant No. 2009BA4050 and CQ CMEC under grant No. KJTD201016. ENS wishes to thank C.M.P. of Chongqing University of Posts and Telecommunications, for the hospitality during the preparation of the manuscript. GL would like to thank the MES of Cuba for partial ﬁnancial support of this investigation, and his research was also supported by the National Basic Science Program (PNCB) of Cuba and Territorial CITMA Project (No. 1115).

A Stability of the ﬁnite critical points

For the critical points (xc,yc, zc) of the autonomous system (3.8), the coeﬃcients of the perturbation equations form a 3 3 matrix Q, which reads: ×

– 14 – Cr. Point Exists for ν1 ν2 ν3 A all ξ,λ 3 1 3 √9 96ξ 1 3+ √9 96ξ 2 4 − − − 4 − − B ξ = 0, all λ 3 3 3 λ 0 − 2 q 3 C ξ = 0, all λ 3 3+ 2 λ 0 2 2 1 2 D ξ = 0, λ 6 λ 3 2 λ q 6 0 2 ≤ +− − E ξ = 0, λ 3 α (λ) α−(λ) 0 2 ≥ + F λ ξ 3 β (λ, ξ) β−(λ, ξ) 2 ≤ − + G λ ξ or ξ < 0 3 γ (λ, ξ) γ−(λ, ξ) ≤ − 3+√9 24ξ 3 √9 24ξ J ξ = 0 and λ = 0 3 − − − − − 6 − 2 2

Table 4. The eigenvalues of the matrix Q of the perturbation equations of the autonomous sys- tem (3.8). Points B-E exist only for ξ = 0, and for these points the variable z is zero and thus irrelevant. Although for these points the eigenvalue associated to the z-direction is zero, the sta- bility conditions are obtained by analyzing the eigenvalues of the non-trivial 2 2 submatrix of × 2 2 3 √24λ2 7λ4 3 q9 24√ξ λ ξ Q. We introduce the notations α±(λ) = 1 2− , β±(λ, ξ) = − ± − − and 4 − ± λ 2 3 9+24√ξ2 λ2ξ − ±q − γ±(λ, ξ)= 2 .

9x2 3y2 4√6xz ξ sgn(ξ) 3 Q11 = −2 + 2 | | 2 + 2z sgn(ξ) 2+ zpsgn(ξ) − 2 3x Q = y √6λ 12 − 1+ z2sgn(ξ) 2√6 z2sgn(ξ) 2 ξ x2sgn(ξ) 3 y2 x2 zx sgn(ξ) Q13 = − | | + − √6 ξ sgn(ξ) 2 2 2 2 − [2 + z sgn( pξ)] [1+ z sgn( ξ)] − | | p 3 y λz2sgn(ξ) 4z ξ sgn(ξ)+ λ 3xy 2 − | | Q21 = 1+ z2sgn(ξ) − q h 1+ z2sgn(pξ) i

2 2 3z + √6x 4 ξ λz z sgn(ξ) 3 + 3x √6λx 9y | |− Q22 = − − + 2 + 2z2sgn(ξ) n 2h +p 2z2sgn(ξ)io y sgn(ξ) 3z y2 x2 + 2x√6 ξ 1 z2sgn(ξ) − | | − Q23 = n [1 + z2sgn(ξp)]2 o

Q = √6 ξ 31 | | Q32 = 0 p Q33 = 0.

Despite the above complicated form, we can straightforwardly see that using the explicit critical points presented in Table 1, the matrix Q acquires a simple form that allows for an easy calculation of its eigenvalues. The corresponding eigenvalues νi (i = 1, 2, 3) for each critical point are presented in table 4.

– 15 – Thus, by determining the sign of the real parts of these eigenvalues, we can classify the corresponding critical point. In particular, if all the eigenvalues of a critical point have negative real parts then the corresponding point is stable, if they all have positive real parts then it is unstable, and if they change sign then it is a saddle point.

B Stability of the critical points at inﬁnity

Cr. Point Exists for ν1 ν2 K ξ > 0, all λ 1 3 96ξ 24√6ξ + 9 1 3+ 96ξ 24√6ξ + 9 ± 2 − − 2 − L ξ > 0, all λ 1 3 p96ξ + 24√6ξ + 9 1 3+ p96ξ + 24√6ξ + 9 ± 2 − 2 p p Table 5. The eigenvalues of the matrix Q of the perturbation equations of the autonomous system (B.2), calculated for the four critical points at inﬁnity. Since we are restricted to the invariant set yr = 0, Q is a 2 2 matrix. × Let us consider the Poincar´evariables

xr = ρ cos θ sin ψ, zr = ρ sin θ sin ψ, yr = ρ cos ψ, (B.1) where ρ = r , r = x2 + y2 + z2, θ [0, 2π], and π ψ π (we restrict the angle √1+r2 ∈ − 2 ≤ ≤ 2 ψ to this interval sincep the physical region is given by y > 0) [97–100]. Thus, the points at “inﬁnite” (r + ) are those having ρ 1. Furthermore, the physical phase-space is → ∞ → given by the intersection of the regions 2(x2 + y2) 1, x2 + y2 z2 0 and the circle r r ≤ r r − r ≥ x2 + y2 + z2 1, that is it is the region r r r ≤

2 2 2 1 (xr,yr, zr) [ 1, 1] [0, 1] [ 1, 1] z x + y . ∈ − × × − | r ≤ r r ≤ 2 Performing the transformation (B.1), in terms of the Poincar´evariables the system (3.8) becomes:

2 2 2 2 2 2√6ξ xr 1 zrxr 3 2 2 √6ξ 2yr 1 xr yr + 1 zr x′ = − + 2x 2y 1 xr + − − r z2 + 2 (x2 + y2 1) 2 r − r − x2 + y2 1 r r r − r r − 3 2 2 λyr + , x2 q y2 z2 + 1 − r − r − r 2 2 2 3 2√p6ξyrzr xr yr zr + 1 xr 1 2 2 yr′ = 2 2 − 2− −2 2 + yr 6xr + 4 6ξzrxr 6yr + 3 (xr + yr 1) [zr + 2 (xr + yr 1)] 2 − − − 3 p 2 λyrxr , − x2q y2 z2 + 1 − r − r − r 2 2 3 2 2 4√p6ξ xr + yr 1 xr 2√6ξyr zr xr 3 z′ = − + + 6ξ 2x + xr r − z2 + 2 (x2 + y2 1) x2 + y2 1 r r r r − r r − 4 2 4 2 p 3 2x x 2y + y zr + r − r − r r . (B.2) 2 (x2 + y2 1) r r −

– 16 – Transforming to spherical coordinates and taking the limit ρ 1, the leading terms in (B.2) → are

ρ′ 0, → 2 3 1 ρ θ′ λcos ψ cot ψ sin θ, − →−r2 p 2 3 1 ρ ψ′ λcos θ cos ψ. (B.3) − → r2 p Since the equation for ρ decouples, it is adequate to investigate the subsystem of the angular variables. Therefore, the singular points at inﬁnity satisfy

xr = cos θ, ± zr = sin θ, ± yr = 0, (B.4) for some particular θ [0, 2π] that are determined as follows. Substituting (B.4) in (B.2) ∈ and solving for θ′ we obtain 1 θ′ = cos(2θ) 3 cot θ + 2√6 ξ . (B.5) −2 p This equation admits the parametric solution

1 1 τ(θ)= c + 4 6ξ tanh− (tan θ) 1 9 24ξ − n p +3 ln[cos(2θ)] 6 ln 3 cos θ + 2 6ξ sin θ . − h p io Thus, the singular solutions with θ [0, 2π] are those satisfying θ = π , 3π , and cot θ = ∈ 4 4 2 2 ξ, which due to (B.4) lead to simple expressions for xr,yr,zr. These results are sum- − 3 marizedq in Table 3. In summary there are four physical critical points at inﬁnity. The points K , satisfying ± θ = 3π and the points L satisfying θ = π . Thus, according to (B.1), these critical points 4 ± 4 correspond to the limit φ , φ/H˙ satisfying the rate → ±∞ → ±∞ x (ln φ)′ √6 = √6 cot θ. ≡ z Concerning the stability analysis of the singular points K and L , we take advantage ± ± that they are located in the invariant set yr = 0 and we examine their stability for the reduced 2D system xr, zr. The eigenvalues of the linearized 2D subsystem evaluated at K and L ± ± are displayed in Table 5, while the results of the stability analysis are presented in Table 3. Finally, we mention that mathematically there are two more critical points, namely the points P satisfying cot θ = 2 2 ξ. However, they have no physical meaning since they ± − 3 give a divergent ΩDE in (3.10), namelyq sign[8ξ 3] , and thus we do not consider them − ·∞ in the cosmological discussion.

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