Dynamical system analysis of quintom dark energy model

The present work deals with dynamical system analysis of a quintom model of dark energy. By suitable transformation of variables the Einstein field equations are converted to an autonomous system. The critical points are determined and stability of hyperbolic critical points are determined by Hartman–Grobman theorem. To analyze non-hyperbolic critical points different tools (notably center manifold theory) are used. Possible bifurcation scenarios have also been explained.


Introduction
An important problem in present cosmology is to comprehend the role of dark energy (DE) which discovered at the turn of last century when two independent observational studies [1,2] from Type Ia Supernovae (SNIa) [3,4] revealed that the universe is going through cosmic acceleration at a fast pace. The other two important evidences to support of the role of DE [5][6][7] are based on the experimental study of cosmic microwave background radiation along with largescale structure surveys (CMB & LSS). The salient quantity of DE is its equation of state (EoS) which explicitly defined as ω DE = p DE ρ DE where p DE and ρ DE are the pressure and energy densities respectively. If we restrict ourselves in four dimensional Einsteins gravity, nearly all DE models can be classified by the behaviors of equations of state (EoS). For example, the case of a non-zero and positive cosmological constant boundary corresponds to ω = −1. In this case, ρ is independent of the scale factor a(t). A quintessence field is dynamical field for which the barotropic parameter of the dark energy equation of state is above the CDM boundary [8,9], that is ω Q > −1. Similarly for a phantom field [10,11], ω p < −1. Interestingly some data analyses suggest the cosmological constant boundary (or phantom divide) is crossed [12][13][14][15][16][17], due to the dynamical behavior of the dark energy a e-mail: sudipcmiiitmath@gmail.com EoS [18,19]. Moreover, the quintessence and phantom models alone cannot explain the evolution of the dark energy equation of state and the possible crossing of the phantom divide line.
According to the null energy condition (NEC), the EoS of normal matter should not be smaller than the cosmological constant boundary. On the other hand, there exists a "no-go theorem" [20][21][22][23] that prevents the EoS of a single scalar field to cross over the cosmological constant boundary. One possible solution to this problem is to introduce a superposition between two dynamical scalar fields-i.e., a canonical field φ and a phantom field σ . Such phenomenological models are known as quintom models which give rise to quintom cosmology [24][25][26][27][28][29]. Curiously, some of the recent observational data show a significant accordance with a dynamical EoS for the dark energy component corresponding to quintom models. In these models, the dark energy equation of state parameter presenting an evolution from a phantom behavior ω p < −1 around present epoch, towards a quintessence behavior ω Q > −1 in the near past [27][28][29][30]. In this regard we need to mention that a dynamically valid dark energy quintom model requires to have at least two degrees of freedom [30,33].
The present work is related to quintom dark energy cosmological model. Due to non-linear coupled system of field equations analytic cosmological solutions are not possible. So dynamical system analysis [31,32] has been discussed here. The plan of the present work is as follows: The basic equations for the quintom cosmological model has been presented in Sect. 2. Autonomous system has been formed and stability analysis of the line of critical points has been discussed in Sect. 3. Also bifurcation scenarios have been examined in this section. The paper ends with a brief discussion on cosmological implications of dynamical system analysis in Sect. 4.

Basic equations
This section is devoted to the basic equations related to quintom model. Here gravity is minimally coupled to a normal scalar field φ and a phantom (i.e. negative kinetic energy) scalar field σ with a coupled potential v(φ, σ ). The action for this model is described by where k 2 = 8π G is the gravitational coupling and L m represents the Lagrangian density of matter fields. In the background of the homogeneous and isotropic flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time, the line element is given by The explicit form of the Lagrangian [33] is Hence 'a(t)' is the usual scale factor, φ = φ(t), σ = σ (t) are the canonical and non-canonical scalar fields and an over dot represents differentiation with respect to the cosmic time t. Now varying the action with respect to the scale factor 'a(t)' (assuming L m = 0) gives the two Friedmann equations and 2ä a +ȧ while variation of the action w.r.t. the scalar fields give their evolution equations as and where H =ȧ a is the usual Hubble parameter. The last two equations (i.e. Eqs. 6 and 7) are also known as energy conservation equations for the scalar fields.
Now combining the two fluids as a single matter part, the effective equation of state can be written as It should be noted that the present effective cosmological model will be characterized as quintessence model if ω e f f −1 i.e.φ σ while it will be phantom in nature if ω e f f < −1 i.e.φ <σ . As we have not assumed any direct coupling between the two scalar fields so in the present work two possible choices of the potential function are considered namely, where v 0 and λ dimensionless constants characterizing the slop of the potential (as defined in Sect. 3).
It is desirable to have w e f f ( = 0) close to the cosmological constant boundary (i.e w e f f = −1) for a feasible quintom model and as a result, the dynamical evolution of both scalar fields results a quintom scenario with a smooth transition across w e f f = −1. Now differentiating Eq. (8) and using the scalar field evolution Eqs. (6 and 7) one gets This equation used as a consistency check when any closed form solution is not possible. Lastly, in cosmology the expansion of the universe is characterized by the deceleration parameter q = −(1 +Ḣ H 2 ) with q < 0 indicating accelerated expansion while q > 0 indicating decelerated expansion.

Stability of critical points and Bifurcation analysis
In order to reveal the autonomous structure of the cosmological dynamical system described by Eqs. (4)-(7), we introduce the following variables, φ = y, σ = z,ẏ = r ,ż = s. Thus the Einstein field Eqs. (4, 5) and evolution Eqs. (6, 7) turn into an autonomous system as followṡ The over dot represents the differentiation with respect to 't' and m, n are choosing to be positive integer greater than 1.
As equation (40) in [33] authors showed that the potential function is of the reflection symmetry and rotation invariant power law form v(φ, σ ) = v 0 (φ 2 − σ 2 ) + μ 0 by Noether symmetry approach, so we study this case separately.

m = 2 and n = 2 i.e. the potential function
In this case the autonomous system (10)(11)(12)(13)(14) takes the form as followṡ The Jacobian matrix for the above system is The Jacobian Matrix evaluated at the critical points (H c , 0, 0, 0, 0) (subscript c stands for critical point) takes the form as follows . The Jordan form of this matrix is The line of non-hyperbolic critical points (H c , 0, 0, 0, 0) are normally hyperbolic [34,35]. The stability of normally hyperbolic set can be completely classified by considering the sign of the eigenvalues in the remaining directions.
Secondly, when 9H 2 c − 8v 0 < 0, the vector field on the (y, r)-plane and (z, s)-plane near the critical points behaves like stable focus when H c > 0 and unstable focus when H c < 0 ( Table 2).
When H c = 0, then α = √ −2v 0 and β = − √ −2v 0 . The critical points are saddle type for v 0 < 0 and the origin is a center on H = 0 hypersurface for v 0 > 0. We next consider 9H 2 c −8v 0 > 0 to define the transformation of basis by the matrix P as the following:
The flow is same as in sub case 2 only the flow along the eigenvectors [ 0 − 1 3H c 1 0 0 ] T and [ 0 0 0 − 1 3H c 1 ] T are repelling from the CP (Table 4).

m > 2 and n > 2
The Jacobian matrix evaluated at the critical points The center manifold (34) is tangent to [ 0 1 0 0 0 ] T near the origin and the flow along the center manifold is determined bẏ Similarly, the flow near the origin along the center manifold (35) is determined bẏ So the flow near the origin (after shifting the critical point to origin) is saddle for m (or n) is even and H c < 0 as Fig. 2.
On the other hand, the flow near the origin is saddle-node for m (or n) is odd and H c < 0 as Fig. 3. We get the stable node near origin for H c > 0 and m being even positive integer and saddle for H c > 0 and m being odd.
On H the system of Eqs. (38) and (39) are uncouple with (40) and (41). So we analyze them independently. In the Eqs. (38) and (39), the power of y namely n − 1 2 and origin (O) is only critical point. First we choose n is even. So n − 1 is odd, say n − 1 = 2k + 1, for some k 1. So the origin is a focus or a center for (−nv 0 ) < 0 i.e v 0 > 0 (for reference, see theorem 2 and 3 in section 2.11 in [39]) where numerical computations ensure that O is a center (Fig. 4) and for all > 0 there exists a δ > 0 such that for all x ∈ N δ (O) and t 0 we have φ t (x) ∈ N (O). So O is stable. On the other hand, O is a (topological) saddle for v 0 < 0. So O is unstable in this case. If n is odd, then n−1 =2k for some k 1. In this case the origin is a cusp [39] as in Fig. 5. Similarly, replacing n by m, we get the same stability criteria at the origin for the Eqs. (40) and (41) projecting on H (Table 5).

n = 2 (or n > 2) and m > 2 (or m = 2)
For this sub case we can use the above two sub cases to analyze the phase-space.

Bifurcation analysis
For v 0 = 0, on the eigenspace of −3H c , the vector fields are attracting towards the CPs for H c > 0 and repelling for H c < 0. At H c = 0, we get phase portrait as in Fig. 1. So the line of CPs r = 0 is unstable. Thus at H c = 0, the system is structurally unstable as small perturbation at H c = 0, we get different characteristics of the vector fields. So, taking H c as  [38,39]. At H c = 0, for n, m are even and v 0 > 0, the origin is a focus or center. On the other hand, for v 0 < 0, the origin is a (topological) saddle. For v 0 = 0, the vector fields near origin discussed above (Fig. 1). So v 0 = 0 is a bifurcation value (Fig. 6).

v(φ, σ ) = e −λ(φ+σ )
We next choose the potential function v(φ, σ ) = e −λ(φ+σ ) = e κ(φ+σ ) (here −λ = κ, say). Here λ is a dimensionless constant characterizing the slop of the potential for φ and σ . Further we assume λ 0 since we can make them positive through φ → −φ and σ → −σ if some of them are negative. Now we chooseφ = r andσ = s. So we get the system as follows: The Jacobian matrix of the system is The critical points of the system are (H c , r c , s c , v c ) where r c = −s c and 3H c r c = κv c = 0 (H c , r c , s c , v c ∈ R). The Jacobian matrix evaluated at the critical points is The change of basis matrix is By this (P matrix) change of basis, the system (42-45) takes the forṁ

Stability analysis
The orientation of the vector fields of new autonomous system is same as the original one if r c < 0 and reverse if . The flow along the eigenvectors corresponding to the eigenvalue 0 can not be determined using Hartman-Grobman theorem. So we use center manifold theory to find it. The center manifold at the origin is

Bifurcation analysis
The local dynamics of a critical point may depends one or more arbitrary parameters and a subtle continuous change of parameter results dramatic change in the dynamics when the system passes through a structural instability or the parameter of the system crosses the bifurcation value [38][39][40][41][42][43]. The system of Eqs.

Discussion
The couple scalar field dynamical dark energy model (known as quintom model) has been studied in cosmological perspective in formulation of dynamical system analysis. The coupled potential of the quintom model is chosen as a linear combination of the power-law of the two scalar fields and an exponential product form of the scalar fields. For the linear combination of the power law form of the potential several cases have been discussed for different choices of the powers. In most of the cases, there is a line of critical points: (H c , 0, 0, 0, 0) with H c is the value of Hubble parameter whenḢ = 0. The center manifold is characterized byḢ = 0 when powers (m, n) are chosen to be 2. When m > 2, n > 2, the center manifold is determined by Eqs. (34) and (35) and the flow along the center manifold are given by Eqs. (36) and (37). It is found that v 0 = 0 is a bifurcation point but it is not interesting as coupled potential is zero. However, it has been shown that for v 0 > 0, the critical point is a focus or center. On the other hand, for the exponential product form of choice of the potential, the non-hyperbolic critical point is characterized by center manifold given by Eqs. (50) and (51) and it is found that the system is structurally unstable for H c = 0 and it corresponds to a bifurcation point.
Finally, from cosmological point of view, the critical points of the present quintom model can be analysed as follows: The line of critical points (H c , 0, 0, 0, 0) represents the phantom barrier in the cosmological context as w e f f = −1 and q = 1 at this critical point. So as expected it behaves as phantom field evolution. In the autonomous system (29)- (33), for the critical point (H c , y c , 0, z c , 0) one gets w e f f = −1 and q = −1. Thus the quintom model describes cosmic evolution with a cosmological term-i.e., the model describes the CDM era of evolution. Similar cosmic evolution can be obtained for the critical point (H c , r c , s c , ν c ) for the autonomous system (42)-(45). Therefore, from the dynamical system analysis of the present quintom model one may conclude that the present quintom model mostly describes the CDM phase of cosmic evolution.