Dynamical system analysis of quintom dark energy model
Abstract
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 nonhyperbolic critical points different tools (notably center manifold theory) are used. Possible bifurcation scenarios have also been explained.
1 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 \(\omega _{DE}=\frac{p_{DE}}{\rho _{DE}}\) where \(p_{DE}\) and \(\rho _{DE}\) 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 nonzero and positive cosmological constant boundary corresponds to \(\omega _{\Lambda }=1\). In this case, \(\rho _{\Lambda }\) 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 \(\Lambda \)CDM boundary [8, 9], that is \(\omega _Q>1\). Similarly for a phantom field [10, 11], \(\omega _p<1\). 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 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 “nogo 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 \(\phi \) and a phantom field \(\sigma \). 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 \(\omega _p<1\) around present epoch, towards a quintessence behavior \(\omega _Q >1\) 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 nonlinear 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.
2 Basic equations
 i.
\(v(\phi ,\sigma ) = v_0(\phi ^n\sigma ^m) + \mu _0\), (\(\mu _0\) is an arbitrary constant \(>0\))
 ii.
\(v(\phi ,\sigma )=e^{\lambda (\phi + \sigma )}\)
3 Stability of critical points and Bifurcation analysis
3.1 \(v(\phi ,\sigma ) = v_0(\phi ^n\sigma ^m) + \mu _0\)
3.1.1 m = 2 and n = 2 i.e. the potential function \(v(\phi ,\sigma )=v_0(\phi ^2\sigma ^2)+\mu _0\).
(a) \(v_0 \ne 0\).
Stability analysis (\(v_0 \ne 0\), \(9H_c^28v_0=0\))
Critical points  \(H_c \ne 0\) (as \(v_0 \ne 0\))  Stability 

\((H_c,0,0,0,0)\)  \(H_c > 0\)  Stable node (4dimensional) 
\(H_c < 0\)  Saddle node (4dimensional) 
Stability analysis (\(v_0 \ne 0\), \(9H_c^28v_0<0\))
Critical points  \(H_c\)  Stability 

\((H_c,0,0,0,0)\)  \(H_c > 0\)  Stable focus (2dimensional) 
\(H_c < 0\)  Unstable focus (2dimensional)  
\(H_c=0\), \(v_0>0\)  Center (2dimensional) 
When \(H_c=0\), then \(\alpha =\sqrt{2v_0}\) and \(\beta =\sqrt{2v_0}\). The critical points are saddle type for \(v_0<0\) and the origin is a center on H = 0 hypersurface for \(v_0>0\).
Stability analysis (for \(v_0 \ne 0\), \(9H_c^28v_0>0\))
Critical points  \(\alpha \) , \(\beta \) (\(\ne 0\))  Stability 

\((H_c,0,0,0,0)\)  \(\alpha <0\) , \(\beta <0\)  Stable node (4dim.) 
\(\alpha <0\) (or \(>0\)), \(\beta >0\)(or \(<0\))  Saddle node (4dim.)  
\(\alpha >0\), \(\beta >0\)  Unstable node (4dim.) 
(b) \(v_0 = 0\).

\(H_c=0\), \(v_0=0\) implies \(\alpha =0\) and \(\beta =0\). In this sub case the flow is undetermined analytically. But numerically we can plot the vector fields. First we plot the vector fields on the yr plane as in Fig. 1. The vector fields for zaxis vs saxis is exactly same as Fig. 1.
 \(H_c>0\), \(v_0=0\) implies \(3H_c<0\). In this sub case, the system (15)–(19) reduces to$$\begin{aligned} \dot{H}= & {} \frac{1}{2}r^2+\frac{1}{2}s^2 \end{aligned}$$(29)$$\begin{aligned} \dot{y}= & {} r \end{aligned}$$(30)$$\begin{aligned} \dot{r}= & {} 3Hr \end{aligned}$$(31)$$\begin{aligned} \dot{z}= & {} s \end{aligned}$$(32)In this system the critical points are \((H_c,y_c,0,z_c,0)\) where \(H_c, y_c, z_c \in \mathbb {R}\). So, no flow or vector fields along the eigenvectors correspond to the zero eigenvalue as they are line of critical points. This argument also gets support in terms of center manifold theory and the center manifold is r=s=0 which indicates \(\dot{H}=\dot{y}=\dot{z}=0\). The flow along the eigenvectors \([\ 0\ \frac{1}{3H_c}\ 1\ 0\ 0\ ]^T\) and \([\ 0\ 0\ 0\ \frac{1}{3H_c}\ 1\ ]^T\) are attracting to the CP (Table 4).$$\begin{aligned} \dot{s}= & {} 3Hs \end{aligned}$$(33)

\(H_c<0\), \(v_0=0\) implies \(3H_c>0\). In this sub case, The flow is same as in sub case 2 only the flow along the eigenvectors \([\ 0\ \frac{1}{3H_c}\ 1\ 0\ 0\ ]^T\) and \([\ 0\ 0\ 0\ \frac{1}{3H_c}\ 1\ ]^T\) are repelling from the CP (Table 4).
Stability analysis (for \(v_0=0\))
Critical points  \(H_c\)  Stability 

\((H_c,0,0,0,0)\)  \(=0\)  Unstable (saddle) (4dimensional) 
\((H_c,y_c,0,z_c,0)\)  \(>0\)  Stable node (4dimensional) 
\(<0\)  Unstable node (4dimensional) 
3.1.2 \(m>2\) and \(n>2\)
Stability analysis (for \(m>2\) and \(n>2\))
Critical points  \(H_c\)  Stability 

m = even, n = even  \(H_c>0\), \(v_0>0\)  Stable node (4dimensional) 
m = odd, n = odd  \(H_c>0\), \(v_0>0\)  Saddle node (4dimensional) 
m = odd, n = even or m = even, n = odd  \(H_c>0\), \(v_0>0\)  Saddle (4dimensional) 
3.1.3 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 phasespace.
3.1.4 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 a parameter, the bifurcation value is \(H_c=0\) and bifurcation point is the origin [38, 39].
3.2 \(v(\phi ,\sigma )=e^{\lambda (\phi + \sigma )}\)
3.2.1 Stability analysis
3.2.2 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. (42)–(45) is structurally unstable when \(H_c=0\). Thus taking \(r_c\) and \(v_c\) fixed, the values of the parameter \(\kappa \) for which \(H_c=0\) (by the relation \(3H_cr_c=\kappa v_c\)) are the bifurcation values where origin is the bifurcation point. So for each fixed \(r_c\) and \(v_c\) we get different bifurcation values.
4 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 powerlaw 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 when \(\dot{H}=0\). The center manifold is characterized by \(\dot{H}=0\) 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 nonhyperbolic 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 \(w_{eff}=1\) 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 \(w_{eff}=1\) and \(q=1\). Thus the quintom model describes cosmic evolution with a cosmological term i.e., the model describes the \(\Lambda \)CDM era of evolution. Similar cosmic evolution can be obtained for the critical point \((H_c,r_c,s_c,\nu _c)\) 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 \(\Lambda \)CDM phase of cosmic evolution.
Notes
Acknowledgements
The author S. Mishra is grateful to CSIR, Govt. of India for giving Junior Research Fellowship (CSIR Award No: 09/096 (0890)/2017EMRI) for the Ph.D work.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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