Asymptotic behavior of a scalar field with an arbitrary potential trapped on a Randall-Sundrum’s braneworld: the effect of a negative dark radiation term on a Bianchi I brane

Abstract

In this work we present a phase space analysis of a quintessence field and a perfect fluid trapped in a Randall-Sundrum’s Braneworld of type 2. We consider a homogeneous but anisotropic Bianchi I brane geometry. Moreover, we consider the effect of the projection of the five-dimensional Weyl tensor onto the three-brane in the form of a negative Dark Radiation term. For the treatment of the potential we use the “Method of f-devisers” that allows investigating arbitrary potentials in a phase space. We present general conditions on the potential in order to obtain the stability of standard 4D and non-standard 5D de Sitter solutions, and we provide the stability conditions for both scalar field-matter scaling solutions, scalar field-dark radiation solutions and scalar field-dominated solutions. We find that the shear-dominated solutions are unstable (particularly, contracting shear-dominated solutions are of saddle type). As a main difference with our previous work, the traditionally ever-expanding models could potentially re-collapse due to the negativity of the dark radiation. Additionally, our system admits a large class of static solutions that are of saddle type. These kinds of solutions are important at intermediate stages in the evolution of the universe, since they allow the transition from contracting to expanding models and viceversa. New features of our scenario are the existence of a bounce and a turnaround, which lead to cyclic behavior, that are not allowed in Bianchi I branes with positive dark radiation term. Finally, as specific examples we consider the potentials V∝sinh^−α(βϕ) and V∝[cosh(ξϕ)−1] which have simple f-devisers.

Appendices

Appendix A: Stability of static solutions

As we commented before, the system (29)–(33) admits eighteen classes of (curves of) fixed points corresponding to static solutions, i.e., having Q=0, that is with H=0. Their coordinates in the phase space and their existence conditions are displayed in Table 2. In the Table 5 are presented the corresponding eigenvalues, where λ ₁(ϵ,Σ _c ),λ ₂(ϵ,Σ _c ),λ ₃(ϵ,Σ _c ), λ ₄(ϵ,Σ _c ) are the roots of the equation

$$\begin{aligned} &{ \lambda^4+\lambda^3 \sqrt{4-6 \varSigma_c^2} \epsilon f'(0)+\lambda ^2 \bigl[f(0)-4 \bigr]} \\ &{\quad{} -4 \lambda\sqrt{4-6 \varSigma_c^2} \epsilon f'(0)-6 f(0) \varSigma_c^2=0,} \end{aligned}$$

and μ ₁(ϵ,Σ _c ),μ ₂(ϵ,Σ _c ),μ ₃(ϵ,Σ _c ), μ ₄(ϵ,Σ _c ) are the roots of the equation

$$\begin{aligned} &{ \mu^4+\sqrt{2} \mu^3 \varSigma_c \epsilon f'(0)+\mu^2 \bigl[2 \bigl(f(0)+4\bigr) { \varSigma_c}^2-8 \bigr]}\\ &{\quad{}+8 \sqrt{2} \mu \varSigma_c \bigl(\varSigma_c^2-1 \bigr) \epsilon f'(0)+4 f(0) \varSigma_c^2 \bigl( \varSigma_c^2-2 \bigr)=0.} \end{aligned}$$

Table 5 Eigenvalues for the critical points in Table 2. Are used the notations $g_{1}(\gamma, \varOmega_{\lambda},\varSigma)=\frac{\sqrt{-3 \gamma (3 \varSigma^{2}-2 ) (-\varSigma^{2}+\varOmega_{\lambda}+1 )^{2}+18 \varSigma^{6}-44 \varSigma^{4}+2 \varSigma^{2} (\varOmega_{\lambda}(9 \varOmega_{\lambda}+2)+17)-8 (\varOmega_{\lambda}^{2}+1 )}}{-\varSigma^{2}+\varOmega_{\lambda}+1}$, $g_{2}(\gamma, \varOmega_{\lambda},\varSigma)=\frac{\sqrt{f(0)} \sqrt {\varSigma ^{2}+\varOmega_{\lambda}-1} \sqrt{3 \gamma (-\varSigma^{2}+\varOmega _{\lambda}+1 )+6 \varSigma^{2}-4}}{\sqrt{\gamma} \sqrt{-\varSigma^{2}+\varOmega _{\lambda}+1}}$, $g_{3}(\gamma, x, \varOmega_{\lambda},\varSigma)=-\frac{9}{2} (\gamma^{2} (\varOmega_{\lambda}-1) \varOmega_{\lambda}+(\gamma-2)^{2} \varSigma ^{4}-(\gamma-2) \varSigma^{2} (2 \gamma \varOmega_{\lambda}+\gamma-2) )-\frac{9 (\gamma -2)^{2} x^{4} (\varSigma^{4}-2 \varSigma^{2} (\varOmega_{\lambda}+1)+\varOmega _{\lambda}^{2}+1 )}{2 (\varSigma^{2}+\varOmega_{\lambda}-1 )^{2}}-\frac{9 (\gamma -2) x^{2} (-\varOmega_{\lambda}(4 (\gamma-1) \varSigma^{2}+\gamma+2 )+(\gamma-2) (2 \varSigma^{4}-3 \varSigma^{2}+1 )+2 \gamma \varOmega_{\lambda}^{2} )}{2 (\varSigma^{2}+\varOmega_{\lambda}-1 )}$, $g_{4}(\gamma, x, \varOmega_{\lambda})=\frac{3 \sqrt{-\gamma^{2} (\varOmega_{\lambda}-1)^{3} \varOmega_{\lambda}-(\gamma-2)^{2} x^{4} (\varOmega_{\lambda}^{2}+1 )-(\gamma-2) x^{2} (\varOmega_{\lambda}-1) (2 \gamma \varOmega_{\lambda}^{2}-(\gamma+2) \varOmega_{\lambda}+\gamma -2 )}}{\sqrt{2} (\varOmega_{\lambda}-1)}$

Now let us comment on the stability of the first order perturbations of (28)–(33) near the critical points showed in Table 2.

The line of fixed points E ₁ is non-hyperbolic. However, the points located at the curve behave as saddle points since its matrix of perturbations admits at least two real eigenvalues of different signs.

The one-parametric line of fixed points E ₂ has two real eigenvalues of different signs provided cos(4u)≠1. In this case, although non-hyperbolic, it behaves as a set of saddle points. For $u\in\{\frac{\pi}{2},\frac{3\pi}{2} \}$ all the eigenvalues are zero. In this case we need to resort to a numerical elaboration.

g ₁(γ,Ω _λ ,Σ) is always real-valued for the allowed values of the phase space variables and the allowed range for the free parameters. Then, although non-hyperbolic, the 2D set of fixed points E ₃ is of saddle type.

The eigenvalues associated to the one-parametric curve of fixed points E ₄ are always reals for the allowed values of the phase space variables and the allowed range for the free parameters. Thus, although non-hyperbolic, it behaves as a saddle point. For the allowed values of the phase space variables and the allowed range for the free parameters the expression g ₃(γ,x _c ,Ω _λc ,Σ _c )≥0. If it is strictly positive, then the fixed points located in the 2D invariant set E ₅ behaves as saddle points.

The eigenvalues of the perturbation matrix associated to $E_{6}^{\epsilon}$ are {0,0,λ ₁(ϵ,Σ _c ),λ ₂(ϵ,Σ _c ),λ ₃(ϵ,Σ _c ),λ ₄(ϵ,Σ _c )} where λ ₁(ϵ,Σ _c ),λ ₂(ϵ,Σ _c ),λ ₃(ϵ,Σ _c ) and λ ₄(ϵ,Σ _c ) are the roots of the polynomial equation with real coefficients:

$$\begin{aligned} &{\lambda^4+\lambda^3 \sqrt {4-6 \varSigma_c^2} \epsilon f'(0)+\lambda^2 \bigl[f(0)-4 \bigr]} \\ &{\quad{} -4 \lambda\sqrt {4-6 \varSigma_c^2} \epsilon f'(0)-6 f(0) \varSigma_c^2=0.} \end{aligned}$$

(57)

• For f′(0)≠0,f(0)<0, Eq. (57) has only two changes of signs in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there is zero or two positive roots of (57). Substituting λ by −λ in (57) and applying the same rule we have zero or two negative roots of (57) for f′(0)≠0,f(0)<0. If all of them are complex conjugated, we need to resort to numerical investigation. If none of them are complex conjugated, then the curve of fixed points $E_{6}^{\epsilon}$ consists of saddle points.

• For ϵf′(0)>0,f(0)>0, Eq. (57) has only one change of sign in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there is one positive root of (57). Substituting λ by −λ in (57) and applying the same rule we have three negative roots of (57) for ϵf′(0)>0,f(0)>0. In summary, for ϵf′(0)>0,f(0)>0, at least two eigenvalues of the linear perturbation matrix of $E_{6}^{+}$ are real of different signs. In this case the curve of fixed points $E_{6}^{\epsilon}$ consists of saddle points.

• For ϵf′(0)<0,f(0)>0, Eq. (57) has three changes of sign in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there are three positive roots of (57). Substituting λ by −λ in (57) and applying the same rule we have only one negative root of (57) for ϵf′(0)<0,f(0)>0. In this case the curve of fixed points $E_{6}^{\epsilon}$ consists of saddle points.

• For f′(0)=f(0)=0 the non null eigenvalues are −2,2; for f′(0)=0,f(0)≠0, the non null eigenvalues are:

  $$\begin{aligned} &{\pm\frac{\sqrt{-\sqrt{24 f(0) \varSigma_c^2+(f(0)-4)^2}-f(0)+4}}{\sqrt{2}},}\\ &{\pm\frac{\sqrt{\sqrt{24 f(0) \varSigma_c^2+(f(0)-4)^2}-f(0)+4}}{\sqrt{2}};} \end{aligned}$$

  and for f′(0)≠0,f(0)=0, the non null eigenvalues are $-2, 2, -\epsilon\sqrt{4 - 6 \varSigma_{c}^{2}}f'(0)$. Thus, in both cases $E_{6}^{\pm}$ consists of saddle points.

The eigenvalues of the perturbation matrix associated to $E_{7}^{\epsilon}$ are {0,0,μ ₁(ϵ,Σ _c ),μ ₂(ϵ,Σ _c ),μ ₃(ϵ,Σ _c ),μ ₄(ϵ,Σ _c )} where μ ₁(ϵ,Σ _c ),μ ₂(ϵ,Σ _c ),μ ₃(ϵ,Σ _c ) and μ ₄(ϵ,Σ _c ) are the roots of the polynomial equation with real coefficients:

$$\begin{aligned} &{\mu^4+\sqrt{2} \mu^3 \varSigma_c \epsilon f'(0)+\mu^2 \bigl[2 \bigl(f(0)+4\bigr) {\varSigma_c}^2-8 \bigr] } \\ &{\qquad{}+8 \sqrt{2} \mu\varSigma_c \bigl(\varSigma_c^2-1 \bigr) \epsilon f'(0)+4 f(0) \varSigma_c^2 \bigl( \varSigma_c^2-2 \bigr)} \\ &{\quad{}=0.} \end{aligned}$$

(58)

• For $\epsilon f'(0)<0, f(0)<0, -\frac{1}{\sqrt{2}}<\varSigma_{c} <0$ or $\epsilon f'(0)>0, f(0)<0, 0<\varSigma_{c} <\frac{1}{\sqrt{2}}$, or $\epsilon f'(0)<0, f(0)<0, 0<\varSigma_{c} <\frac{1}{\sqrt{2}}$, or $\epsilon f'(0)>0, f(0)<0, -\frac{1}{\sqrt{2}}<\varSigma_{c} <0$, the Eq. (58) has only two changes of signs in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there is zero or two positive roots of (58). Substituting μ by −μ in (58) and applying the same rule we have zero or two negative roots of (58) for the same values of the free parameters. If all of them are complex conjugated, we need to resort to numerical investigation. If none of them are complex conjugated, then the curve of fixed points $E_{7}^{\epsilon}$ consists of saddle points.

• For $f(0)>0, \epsilon f'(0)>0, 0<\varSigma_{c} <\frac{1}{\sqrt {2}}$ or $f(0)>0, \epsilon f'(0)<0, \frac{1}{\sqrt{2}}<\varSigma_{c} <0$, Eq. (58) has only one change of sign in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there is one positive root of (58). Substituting μ by −μ in (58) and applying the same rule we have three negative roots of (58) for the same values of the parameters. In this case the curve of fixed points $E_{7}^{\epsilon}$ consists of saddle points.

• For $f(0)>0, 0<\varSigma_{c} <\frac{1}{\sqrt{2}}, \epsilon f'(0)<0$ or $f(0)>0, -\frac{1}{\sqrt{2}}<\varSigma_{c} <0, \epsilon f'(0)>0$ Eq. (58) has three changes of sign in the sequence of its coefficients. Hence, using the Descartes’s rule, we conclude that for this range of the parameters, there are three positive roots of (58). Substituting μ by −μ in (58) and applying the same rule we have only one negative root of (58) for the same values of the parameters. In this case the curve of fixed points $E_{7}^{\epsilon}$ consists of saddle points.

• For f′(0)=f(0)=0 the non null eigenvalues are $2 \sqrt{2} \sqrt{1-\varSigma_{c}^{2}},-2 \sqrt{2} \sqrt{1-\varSigma_{c}^{2}}$; for f′(0)=0,f(0)≠0, the non null eigenvalues are $\pm\sqrt{4-(f(0)+4) \varSigma_{c}^{2}-\sqrt{\Delta}}$, and $\pm\sqrt{4-(f(0)+4) \varSigma_{c}^{2}+\sqrt{\Delta}}$, where $\Delta=(f(0) (f(0) +4)+16) \varSigma_{c}^{4}-32 \varSigma_{c}^{2}+16$; and for f′(0)≠0,f(0)=0, the non null eigenvalues are $-\sqrt{2} \varSigma_{c} \epsilon f'(0),\pm2 \sqrt{2-2 \varSigma _{c}^{2}}$. Thus, in both cases $E_{7}^{\pm}$ consists of saddle points.

Observe that g ₄(γ,x,Ω _λ ) is always real-valued for the allowed values of the phase space variables and the allowed range for the free parameters. Then, although non-hyperbolic, E ₈ behaves as a set of saddle points.

When Ω _λc ≠1, $\frac{\sqrt{6 \gamma (\varOmega_{\lambda}+1)^{2}-8 (\varOmega_{\lambda }^{2}+1 )}}{\varOmega_{\lambda}+1}$ is real valued, in this case the fixed points in the line E ₉ behaves as saddle points.

For Σ _c ≠0, $\sqrt{2} \sqrt{\gamma (9 \varSigma^{2}-6 )-4 \varSigma^{2}+4}$ is real-valued, in this case the fixed points in the line E ₁₀ behaves as saddle points.

For Σ _c ≠0, the linear perturbation matrix evaluated at E ₁₁ has at least two real eigenvalues of different signs, thus, the fixed points in the line E ₁₁ behaves as saddle points.

Observe that the line E ₉ and the line E ₁₁ contains the special point with coordinates F: (Q=0,x=0,Ω _m =0,Ω _λ =1,Σ=0) in the first case when Ω _λc →1 whereas in the second case for Σ _c →0. Taking in both cases the proper limits we have that the eigenvalues of the linearization for F are $\{0,0,-\sqrt{6 \gamma-4},\sqrt{6 \gamma-4},0,0 \}$. Hence, for $\gamma> \frac{2}{3}$, F is of saddle type, whereas for $\gamma< \frac{2}{3}$, there are two purely imaginary eigenvalues and the rest of the eigenvalues are zero. In this case we cannot say anything about its stability from the linearization and we need to resort to numerical inspection.

Observe that $\sqrt{\gamma (6-9 \varSigma^{2} )+18 \varSigma^{2}-8}$ is real-valued for the allowed values of the phase space variables and the allowed range for the free parameters. Thus, all the points located at the line E ₁₂ behaves as saddle points.

For $0<\gamma<\frac{4}{3}$, $E_{13}^{\pm}$ behaves as a saddle point.

Appendix B: Stability of expanding (contracting) solutions

Now let us comment on the stability of the first order perturbations of (28)–(33) near the critical points showed in Table 3.

The line of singular points $Q_{1}^{\pm}$, although it is non-hyperbolic (actually, normally hyperbolic), behaves like a saddle point in the phase space of the RS model, since they have both nonempty stable and unstable manifolds (see the Table 6). The class $Q_{1}^{+}$ is the analogous to the line denoted by P ₁ in Escobar et al. (2012b).

The singular points $Q_{2}^{\pm}(s^{*})$ and $Q_{3}^{\pm}(s^{*})$ are non-hyperbolic (these points are related to $P_{3}^{\pm}$ investigated in Escobar et al. 2012b), however they behave as saddle points since they have both nonempty stable and unstable manifolds (see the Table 6).

The singular point $Q_{4}^{+}(s^{*})$ is the analogous to P ₄ in Escobar et al. (2012b). It is a stable node in the cases $0<\gamma\leq\frac{2}{9},s^{*}<-\sqrt{3\gamma},f' (s^{*} )<0$ or $\frac{2}{9}<\gamma<\frac{4}{3},-\frac{2 \sqrt{6} \gamma}{\sqrt{9 \gamma -2}}\leq s^{*}<-\sqrt{3\gamma}, f' (s^{*} )<0$, or $0<\gamma\leq\frac{2}{9},s^{*}>\sqrt{3\gamma }, f' (s^{*} )>0$, or $\frac{2}{9}<\gamma<\frac{4}{3},\sqrt{3\gamma }<s^{*}\leq\frac{2 \sqrt{6} \gamma}{\sqrt{9 \gamma-2}}, f' (s^{*} )>0$. It is a spiral stable point for $\frac{2}{9}<\gamma <\frac{4}{3},s^{*}<-\frac{2 \sqrt{6} \gamma}{\sqrt{9 \gamma -2}},f' (s^{*} )<0$ or $\frac{2}{9}<\gamma<\frac{4}{3},s^{*}>\frac{2 \sqrt{6} \gamma}{\sqrt{9 \gamma-2}},f' (s^{*} )>0$. In summary, it is stable for $0<\gamma <\frac{4}{3},s^{*}<-\sqrt{3\gamma},f' (s^{*} )<0$ or $0<\gamma <\frac{4}{3},s^{*}>\sqrt{3\gamma},f' (s^{*} )>0$. Otherwise, it is a saddle point.

The singular point $Q_{4}^{-}(s^{*})$ is a local source under the same conditions for which $Q_{4}^{+}(s^{*})$ is stable.

The singular point $Q_{5}^{+}(s^{*})$ is the analogous to P ₅ in Escobar et al. (2012b). It is not hyperbolic for $s^{*}\in\{0,\pm\sqrt{6},\pm\sqrt{3\gamma},2 \}$ or f′(s ^∗)=0. In the hyperbolic case, $Q_{5}^{+}(s^{*})$ is a stable node for $0<\gamma \leq\frac{4}{3},-\sqrt{3 \gamma}<s^{*}<0,f' (s^{*} )<0$ or $\frac{4}{3}<\gamma\leq2,-2<s^{*}<0,f' (s^{*} )<0$ or $0<\gamma\leq \frac{4}{3},0<s^{*}<\sqrt{3 \gamma},f' (s^{*} )>0$ or $\frac{4}{3}<\gamma\leq2,0<s^{*}<2,f' (s^{*} )>0$; otherwise, it is a saddle point.

The singular point $Q_{5}^{-}(s^{*})$ is a local source under the same conditions for which $Q_{5}^{+}(s^{*})$ is stable.

For evaluating the Jacobian matrix, and obtaining their eigenvalues at the singular points $Q_{6}^{\pm}$, $Q_{7}^{\pm}$ and $Q_{11}^{\pm}$ we need to take the appropriate limits. The details on the stability analysis for $Q_{6}^{+}$ and $Q_{11}^{+}$ are offered in the Appendix C. The analysis of $Q_{7}^{+}$ is essentially the same as for $Q_{6}^{+}$.

The circles of critical points $Q_{8} ^{\pm}$ are non-hyperbolic. Both solutions represents transient states in the evolution of the universe.

The singular point $Q_{9}^{\pm}$ and the line of critical points $Q_{10}^{\pm}$ are non-hyperbolic ($Q_{9}^{+}$ is the analogous of P ₁₀ for Ω _λ =0 while $Q_{10}^{+}$ is the analogous of P ₁₀ for Ω _λ ≠0 in Escobar et al. 2012b). In the Appendix B.1 we analyze the stability of $Q_{9}^{+}$ by introducing a local set of coordinates adapted to the singular points. The non-hyperbolic fixed point $Q_{9}^{-}$, behaves as a past attractor.

The singular point $Q_{11}^{+}$ is the analogous of P ₁₁ in Escobar et al. (2012b), this point represents a 1D set of singular points such that (Ω _λ =1), parametrized by the values of s _c . It is normally hyperbolic.

The singular points $Q_{12}^{\pm}(s^{*})$ are non-hyperbolic for s ^∗=0,±2, γ=4/3 or f′(s ^∗)=0. $Q_{12}^{+}(s^{*})$ is a stable node for $\frac{4}{3}<\gamma\leq2,2<s^{*}\leq\frac{8}{\sqrt{15}}, f' (s^{*} )>0$, a stable spiral for $\frac{4}{3}<\gamma\leq2, s^{*}>\frac{8}{\sqrt{15}}, f' (s^{*} )>0$. In summary, $Q_{12}^{+}(s^{*})$ is stable for $\frac{4}{3}<\gamma\leq2, s^{*}>2, f' (s^{*} )>0$. Otherwise it is a saddle point. $Q_{12}^{-}(s^{*})$ is unstable for the same conditions for which $Q_{12}^{+}(s^{*})$ is stable.

The eigenvalues of the linear perturbation matrix evaluated at each of these critical points are displayed in the Table 6.

Table 6 Eigenvalues for the critical points in Table 3. We use the notations $K^{\pm}(\gamma, s^{*})=-\frac{3}{4} (2-\gamma\pm\frac{\sqrt{2-\gamma} \sqrt{{s^{*}}^{2} (2-9 \gamma )+24 \gamma^{2}}}{s^{*}} )$, $L^{\pm}(\varOmega_{\lambda})=-\frac{1}{2} (3\pm \sqrt{12 f(0) (\varOmega_{\lambda}-1)+9} )$ and $M^{\pm}(s^{*})=\frac{1}{4} (-s^{*}\pm\sqrt{64-15 (s^{*} )^{2}} )$, where s ^∗ denotes an s-value such that f(s ^∗)=0

2.1 B.1 De Sitter solutions

To investigate the de Sitter solution $Q_{9}^{+}$, we introduce the local coordinates:

$$\begin{aligned} &{ \{x,\varOmega_\lambda,Q-1,\varOmega_m,\varSigma,s \}} \\ &{\quad= \epsilon\{\hat{x },\hat{\varOmega}_\lambda,\hat{Q},\hat{ \varOmega}_m,\hat{\varOmega}_\sigma,\hat{s} \}+{\mathcal{O}}( \epsilon)^2,} \end{aligned}$$

(59)

where ϵ is a small constant ϵ≪1. Then, we obtain the approximated system

$$ \begin{aligned} &{\hat{Q}'=-4 \hat{Q},\qquad \hat{x}'=\sqrt{ \frac{3}{2}} \hat{s}-3 \hat{x},\qquad \hat{\varOmega}_\lambda'=0,}\\ &{\hat{\varOmega}_m'=-3 \gamma\hat{ \varOmega}_m, \qquad\hat{\varOmega}_\sigma'=-3 \hat{\varOmega}_\sigma, \qquad\hat{s}'=-\sqrt{6} f(0) \hat{x}.} \end{aligned} $$

(60)

The system (60) admits the exact solution passing by ($\hat {Q}_{0},\hat{x}_{0},\hat{\varOmega}_{\lambda0},\hat{\varOmega}_{m0},\hat {\varOmega }_{\sigma0},\hat{s}_{0}$) at τ=0 given by

$$\begin{aligned} &{\hat{Q}(\tau)=\hat{Q}_0 e^{-4 \tau},} \\ &{\hat{x}(\tau)= \frac{1}{2} \hat{x}_0 e^{-\frac{1}{2} \tau (\vartheta+3)} \bigl(e^{\tau \vartheta }+1 \bigr)} \\ &{\phantom{\hat{x}(\tau)=}{} -\frac{e^{-\frac{1}{2} \tau (\vartheta+3)} (e^{\tau \vartheta}-1 ) (\sqrt{6} \hat{s}_0 (u_c-1)+3 \hat{x}_0 )}{2 \vartheta},} \\ &{ \hat{\varOmega}_\lambda(\tau)=\hat{\varOmega}_{\lambda 0},} \\ &{\hat{ \varOmega}_m(\tau)=\hat{\varOmega}_{m0} e^{-3\gamma\tau},\qquad \hat{\varOmega}_\sigma(\tau)=\hat{\varOmega}_{\sigma0}e^{-3\tau},} \\ &{\hat{s}(\tau)= \hat{s}_0 e^{-\frac{3}{2} \tau} \cosh(\beta \tau)} \\ &{\phantom{\hat{s}(\tau)=}{}+ \frac{3 e^{-\frac{3}{2} \tau} (2 \hat{s}_0-\sqrt{6} \hat{x}_0 ) \sinh (\beta \tau )}{4 \beta}} \\ &{\phantom{\hat{s}(\tau)=}{}+ \sqrt{\frac{2}{3}} \beta e^{-\frac {3}{2} \tau} \hat{x}_0 \sinh(\beta\tau),} \end{aligned}$$

(61)

where $\beta=\frac{1}{2} \sqrt{9-12 f(0)}$.

For the choice $\beta^{2}<\frac{9}{4}$, i.e., f(0)>0, all the perturbations $\hat{Q},\hat{x} ,\hat{\varOmega}_{m },\hat{\varOmega}_{\sigma},\hat {s}$ shrink to zero as τ→+∞, and $\hat{\varOmega}_{\lambda}$ converges to a constant value. For $\beta=\pm\frac{3}{2}$, i.e., for f(0)=0, $\hat{x}\rightarrow \frac{\hat{s}_{0}}{\sqrt{6}}$ and $\hat{s}\rightarrow\hat{s}_{0}$, and the other perturbations shrink to zero as τ→+∞. Combining the above arguments we obtain that for f(0)≥0, $Q_{9}^{+}$ is stable, but not asymptotically stable. For $\beta^{2}>\frac{9}{4}$, i.e., f(0)<0, all the perturbation values, but $\hat{x}$ and $\hat{s}$, which diverges, shrink to zero as τ→+∞. Thus, $Q_{9}^{+}$ is a saddle.

For analyzing the curve of singular points $Q_{10}^{+}$ we consider an arbitrary value Ω _λ =u _c , 0<u _c <1, and introduce the local coordinates

$$\begin{aligned} &{ \{x,\varOmega_\lambda-u_c,Q-1,\varOmega_m, \varSigma,s \}} \\ &{\quad= \epsilon\{ \hat{x},\hat{\varOmega}_\lambda,\hat{Q},\hat{ \varOmega}_m,\hat{\varOmega}_\sigma, \hat{s} \}+{\mathcal{O}}(\epsilon)^2,} \end{aligned}$$

(62)

where ϵ is a small constant ϵ≪1.

Then, we obtain the approximated system

$$\begin{aligned} &{\hat{Q}'=-4 \hat{Q},\qquad \hat{x}'= -3 \hat{x}-\sqrt{\frac{3}{2}} \hat{s} (u_c-1),} \\ &{ \hat{ \varOmega}_\lambda'=-{3 \gamma u_c \hat{ \varOmega}_m},\qquad \hat{\varOmega}_m'=-3 \gamma\hat{\varOmega}_m,}\\ &{ \hat{\varOmega}_\sigma'=-3 \hat{\varOmega}_\sigma,\qquad \hat{s}'=-\sqrt{6} f(0) \hat{x}.} \end{aligned}$$

(63)

The system (63) admits the exact solution passing by ($\hat{Q}_{0},\hat{x}_{0},\hat{\varOmega}_{\lambda0},\hat{\varOmega }_{m0},\hat{\varOmega }_{\sigma0},\hat{s}_{0}$) at τ=0 given by

$$\begin{aligned} &{\hat{Q}(\tau)=\hat{Q}_0 e^{-4 \tau},} \\ &{\hat{x}(\tau)= \frac{e^{-\frac{3}{2} \tau} (\sqrt{6} \hat{s}_0-3 \hat{x}_0 ) \sinh(\beta \tau)}{2 \beta}+e^{-\frac{3}{2} \tau} \hat {x}_0 \cosh(\beta\tau), } \\ &{\hat{\varOmega}_\lambda(\tau) =u_c \hat{ \varOmega}_{m0} \bigl(e^{-3 \gamma \tau}-1 \bigr)+\hat{ \varOmega}_ {\lambda0},} \\ &{\hat{\varOmega}_m(\tau)=\hat{\varOmega}_{m0} e^{-3\gamma\tau}, \hat{\varOmega}_\sigma(\tau)=\hat{ \varOmega}_{\sigma0}e^{-3\tau},}\\ &{ \hat{s}(\tau)= \frac{3 e^{-\frac{1}{2} \tau (\vartheta+3)} (e^{\tau \vartheta}-1 ) (2 \hat{s}_0 (u_c-1)+\sqrt{6} \hat{x}_0 )}{4 (u_c-1) \vartheta}} \\ &{\phantom{\hat{s}(\tau)=}{}+\frac{1}{2} \hat{s}_0 e^{-\frac{1}{2} \tau (\vartheta+3)} \bigl(e^{\tau \vartheta }+1 \bigr)} \\ &{\phantom{\hat{s}(\tau)=}{}-\frac{\hat{x}_0 \vartheta e^{-\frac {1}{2} \tau (\vartheta +3)} (e^{\tau \vartheta}-1 )}{2 \sqrt{6} (u_c-1)},} \end{aligned}$$

(64)

where $\vartheta=\sqrt{12 f(0) (u_{c}-1)+9}$. It is easy to see that for f(0)≥0 the solution is stable, but not asymptotically stable. For f(0)<0 is of saddle type.

It is worthy to mention that the above results could be proved by noticing that $Q_{10}^{+}$—as 1D set—is normaly hyperbolic since the eigen-direction associated with the null eigenvalue, (0,0,1,0,0,0)^T, that is, the Ω _λ -axis, is tangent to the set. Thus, the stability issue can be resolved by analyzing the signs of the non-null eigenvalues. However, let us comment that if we include the variable $y=\frac{V}{3 H^{2}}$, in the analysis, the solution is unstable to perturbations along the y-direction.

Appendix C: Asymptotic analysis on the singular surface Ω _λ +Σ ²=1

In this section we analyze the asymptotic structures of the system (28)–(33) near the critical points at the singular surface Ω _λ +Σ ²=1.

3.1 C.1 Shear-dominated solution

Observe that the one of the singular points at the surface Ω _λ +Σ ²=1 that represents a shear-dominated solution is $Q_{6}^{+}$. To investigate its stability we introduce the local coordinates:

$$\begin{aligned} &{ \{x,\varOmega_\lambda,Q-1,\varOmega_m, \varSigma+1,s-s_c \}} \\ &{\quad= \epsilon\{ \hat{x},\hat{\varOmega}_\lambda, \hat{Q},\hat{\varOmega}_m,\hat{\varOmega}_\sigma, \hat{s} \} +{\mathcal{O}}(\epsilon)^2,} \end{aligned}$$

(65)

where ϵ is a constant satisfying ϵ≪1 and s _c is an arbitrary real value for s. Taylor expanding the resulting system and truncating the second order terms we obtain the approximated system

$$\begin{aligned} &{\hat{Q}'=2\hat{Q}, \hat{x}'=-\sqrt{ \frac{3}{2}}s_c (-2 \hat{\varOmega}_\sigma+{\hat{ \varOmega}_\lambda}+{\hat{\varOmega}_m} ),} \\ &{ \hat{ \varOmega}_\lambda'=6 \hat{\varOmega}_\lambda \biggl(\frac{\gamma \hat{\varOmega}_m}{\hat{\varOmega}_\lambda-2 \hat{\varOmega}_ \sigma}+1 \biggr),} \\ &{\hat{\varOmega}_m'=3(2- \gamma)\hat{\varOmega}_m,} \\ &{ \hat{\varOmega}_\sigma'=\frac{3 \gamma \hat{\varOmega }_\lambda \hat{\varOmega}_m}{\hat{\varOmega}_\lambda-2 \hat{\varOmega}_\sigma}+ \frac{3}{2} (4\hat{\varOmega}_\sigma-\gamma\hat{ \varOmega}_m),} \\ &{\hat{s}'=-\sqrt{6}\hat{x} f(s_c)} . \end{aligned}$$

(66)

Let us define the rates:

$$ \{r_m,r_\sigma\}=\frac{1}{\hat{\varOmega}_\lambda}\{\hat{ \varOmega}_m, \hat{\varOmega}_\sigma\}. $$

(67)

Let us assume γ>0. With the above assumptions we can define the new time variable N=γlna which preserves the time arrow. Then we deduce the differential equations

$$ r_m'= \frac{6 r_m^2}{2 r_\sigma-1}-3 r_m, \qquad r_\sigma'=\frac{3 r_m}{2}, $$

(68)

where the comma denotes derivatives with respect to N.

The system admits the general solution:

$$ \begin{aligned} &r_m=\frac{1}{-2 c_1 \cosh (3 N-c_2 )-2 c_1}, \\ & r_\sigma=\frac{2 c_1-1+\tanh (\frac{1}{2} (3 N-c_2 ) )}{4 c_1}, \end{aligned} $$

(69)

or

$$ \begin{aligned} &r_m=\frac{1}{-2 c_1 \cosh (3 N-c_2 )-2 c_1}, \\ & r_\sigma=\frac{2 c_1-1-\tanh (\frac{1}{2} (3 N-c_2 ) )}{4 c_1}, \end{aligned} $$

(70)

where c ₁≠0 and c ₂ are arbitrary constants. In the general case we have that r _m →0 and r _σ tends to a constant as N→−∞.

Thus, as τ→−∞ the system (66) has the asymptotic structure:

$$ \begin{aligned} &\hat{Q}'=2\hat{Q},\quad\hat{x}'=-\sqrt{ \frac{3}{2}} s_c (\hat{\varOmega}_\lambda-2 \hat{ \varOmega}_\sigma),\\ & \hat{\varOmega}_\lambda'=6 \hat{\varOmega}_\lambda,\qquad\hat{\varOmega}_m'=3(2- \gamma)\hat{\varOmega}_m, \qquad\hat{\varOmega}_\sigma'=6 \hat{\varOmega}_\sigma,\\ & \hat{s}'=-\sqrt{6}\hat{x} f(s_c). \end{aligned} $$

(71)

The system (71) admits the exact solution passing by (${Q}_{0},\hat{x}_{0},\hat{\varOmega}_{\lambda0},\hat{\varOmega}_{m0},\hat {\varOmega}_{ \sigma0},\hat{s}_{0}$) at τ=0 given by

$$\begin{aligned} &{{Q}(\tau)=\hat{Q}_0 e^{2\tau},} \\ &{\hat{x}(\tau)= \hat{x}_0-\frac{s_c (e^{6 \tau}-1 ) (\hat{\varOmega}_{\lambda0}-2 \hat{\varOmega}_{\sigma0})}{2 \sqrt {6}},} \\ &{ \hat{\varOmega}_\lambda( \tau)=\hat{\varOmega}_{\lambda0}e^{6\tau},\qquad\hat{ \varOmega}_m(\tau)=\hat{\varOmega}_{m0} e^{3(2-\gamma)\tau},} \\ \\ &{\hat{\varOmega}_\sigma(\tau)=\hat{\varOmega}_{\sigma0}e^{6\tau},} \\ &{ \hat{s}(\tau)= \frac{1}{12} s_c \bigl(-6 \tau +e^{6 \tau}-1 \bigr) f(s_c)} \\ &{\phantom{\hat{s}(\tau)=}{}\times(\hat{\varOmega}_{\lambda0}-2 \hat{\varOmega}_{\sigma0}) -\sqrt{6} \tau\hat{x}_0 f(s_c)+\hat{s}_0.} \end{aligned}$$

(72)

Thus, the energy density perturbations goes to zero, and s diverge as τ→−∞. For that reason, $Q_{6}^{+}$ cannot be the future attractor of the system, thus $Q_{6}^{+}$ has a large probability to be a past attractor. Using the same approach for $Q_{6}^{-}$ we obtain that as τ→+∞, the energy density perturbations goes to zero, but s diverge. Thus, although $Q_{6}^{-}$ have a large probability to be a late time attractor, actually it is of saddle type. For $Q_{7}^{\pm}$ we have similar results as for $Q_{6}^{\pm}$. Since the procedure is essentially the same as for $Q_{6}^{\pm}$, we omit the details.

3.2 C.2 Solution with 5D-corrections

Another singular points at the surface Ω _λ +Σ ²=1 are the isotropic solutions with Ω _λ =1 (critical points $Q_{11}^{\pm}$). For the stability analysis of $Q_{11}^{+}$ , we introduce the local coordinates:

$$\begin{aligned} &{ \{x,\varOmega_\lambda-1,Q-1,\varOmega_m, \varSigma,s-s_c \}} \\ &{\quad= \epsilon\{\hat{x},\hat{\varOmega}_\lambda, \hat{Q},\hat{\varOmega}_m,\hat{\varOmega}_\sigma,\hat{s} \}+{\mathcal{O}}(\epsilon)^2,} \end{aligned}$$

(73)

where ϵ is a constant satisfying ϵ≪1 and Q _c ,s _c are arbitrary real values of Q,s respectively. Then, we obtain the approximated system

$$ \begin{aligned} & \hat{Q}'=-2 \hat{Q} (3 \gamma r+2),\\ & \hat{x}'= \frac{1}{2} \bigl(-6 \hat{x} (\gamma r+1)-\sqrt{6} s_c (\hat{ \varOmega}_\lambda+\hat{\varOmega}_m) \bigr), \\ &\hat{ \varOmega}_\lambda'=-3 \gamma r \hat{\varOmega}_\lambda, \\ & \hat{\varOmega}_m'=-3 \gamma(2 r+1) \hat{ \varOmega}_m, \\ & \hat{\varOmega}_\sigma'=-3 \hat{\varOmega}_\sigma(\gamma r+1),\qquad\hat{s}'=-\sqrt{6} \hat{x} f(s_c), \end{aligned} $$

(74)

where we have defined the rate $r=\frac{\hat{\varOmega}_{m}}{\hat {\varOmega}_{\lambda}}$. The evolution equation for r is given by

$$ r'=-3 \gamma r (r+1). $$

(75)

The Eq. (75) has two singular points r=0 with eigenvalue $\frac{dr'}{dr}|_{r=0}=-3\gamma$ and r=−1 with eigenvalue $\frac {dr'}{dr}|_{r=-1}=3\gamma$. Then, assuming γ>0 we have that r→−1 as τ→−∞. This argument allow us to prove that as τ→−∞, the system (74) has the asymptotic structure

$$ \begin{aligned} &\hat{Q}'=2 (3 \gamma-2)\hat{Q},\qquad\hat{x}'=3 \hat{x} (\gamma-1), \\ & \hat{\varOmega}_\lambda'=3 \gamma\hat{ \varOmega}_\lambda,\qquad\hat{\varOmega}_m'=3 \gamma \hat{\varOmega}_m, \\ & \hat{\varOmega}_\sigma'=3 \hat{\varOmega}_\sigma(\gamma-1),\qquad\hat{s}'=-\sqrt{6} \hat{x} f(s_c). \end{aligned} $$

(76)

The system (76) admits the exact solution passing by (${Q}_{0},\hat{x}_{0},\hat{\varOmega}_{\lambda0},\hat{\varOmega}_{m0},\hat {\varOmega}_{\sigma0},\hat{s}_{0}$) at τ=0 given by

$$ \begin{aligned} &\hat{Q}(\tau)=\hat{Q}_0 e^{2(3\gamma-2)\tau},\qquad\hat{x}(\tau)= \hat{x}_0 e^{3 (\gamma-1)\tau},\\ & \hat{\varOmega}_\lambda(\tau)= \hat{\varOmega}_{\lambda0}e^{3\gamma\tau},\\ & \hat{\varOmega}_m( \tau)=\hat{\varOmega}_{m0} e^{3\gamma\tau},\qquad\hat{\varOmega}_\sigma( \tau)=\hat{\varOmega}_{\sigma0}e^{3(\gamma-1)\tau}, \\ & \hat{s}(\tau)= \hat{s}_0-\frac{\sqrt{\frac{2}{3}} \hat{x}_0 f(s_c) (e^{3 (\gamma-1) \tau}-1 )}{(\gamma-1)}. \end{aligned} $$

(77)

Hence, for γ>1, the perturbations $\hat{Q},\hat{x}, \hat {\varOmega}_{\lambda},\hat{\varOmega}_{m},\hat{\varOmega}_{\sigma}$ shrink to zero as τ→−∞. In that limit s tends to a finite value. In this case, the solution under investigation has a 4D unstable manifold. The numerical simulations suggest that the singular point at the surface Ω _λ +Σ ²=1 with Ω _λ =1 is a local source. On the other hand, for γ<1, the solution behave as a saddle point, since the perturbation values $\hat{x}$ and $\hat{\varOmega }_{\sigma}$ do not tend to zero as τ→−∞. Using the same approach for $Q_{11}^{-}$ we obtain that for γ>1, $Q_{11}^{-}$ is stable as τ→+∞, whereas for γ<1 it is of saddle type.

These results for $Q_{11}^{+}$ and for $Q_{11}^{-}$ are in agreement with the analogous results for m ₊ and for m ₋ in the reference Campos and Sopuerta (2001a).