Can dark energy evolve to the phantom?

Dark energy with the equation of state $w(z)$ rapidly evolving from the dustlike ($w\simeq 0$ at $z\sim 1$) to the phantomlike ($-1.2\lesssim w\lesssim -1$ at $z\simeq 0$) has been recently proposed as the best fit for the supernovae Ia data. Assuming that a dark energy component with an arbitrary scalar-field Lagrangian $p(\phi ,{\nabla }_{\mu }\phi )$ dominates in the flat Friedmann universe, we analyze the possibility of a dynamical transition from the states $(\phi ,\dot{\phi })$ with $w\ge -1$ to those with $w<-1$ or vice versa. We have found that generally such transitions are physically implausible because they are either realized by a discrete set of trajectories in the phase space or are unstable with respect to the cosmological perturbations. This conclusion is confirmed by a comparison of the analytic results with numerical solutions obtained for simple models. Without the assumption of the dark energy domination, this result still holds for a certain class of dark energy Lagrangians, in particular, for Lagrangians quadratic in ${\nabla }_{\mu }\phi$. The result is insensitive to topology of the Friedmann universe as well.

Figure 1

Possible phase curves in the neighborhood of the $\phi$ axis. Only on the curves $1$ and $6$, the system crosses the $\phi$ axis. Curves $2$, $3$, $4$, and $7$ have an attractor as a shared point with the $\phi$ axis, whereas curves $5$ and $8$ have a repulsor. These attractors and repulsors can be fixed-point solutions or singularities.

Figure 2

Phase curves in the neighborhood of the singular point ${\Psi }_c^{+}$ are plotted for the case of the real $\lambda$. At the points $\xi$, the solutions $\phi (t)$ do not exist. These points together with ${\Psi }_c^{+}$ form the curve $\Gamma$ on which ${\epsilon }_{,X}(\Gamma )=0$.

Figure 3

Phase curves in the neighborhood of the singular point ${\Psi }_c^{+}$ are plotted for the case of the pure imaginary $\lambda$. Here we assume that the singular point is a focus.

Figure 4

If $\mathbf{A}$ had the eigenvalues ${\lambda }_1={\lambda }_2$, then the singular point ${\Psi }_c^{+}$ would be a nodal point and there would be a continuous set of trajectories passing through it. To illustrate this we plot here the phase curves in the particular case of a degenerate nodal point. The form of the equation of motion (2.13) excludes such types of singular points and therefore prevents the possibility of such transitions.

Figure 5

The typical behavior of the phase curves in the neighborhood of the critical line where $K(\phi )=0$ (here $\dot{\phi }$ axis) is plotted for the case when $K_c^{\prime }>0$, $V_c^{\prime }<0$. Horizontal dashed lines are the analytically obtained separatrices ${\dot{\phi }}_{\pm }$ and $(0,u_{\pm })$ are the points of transition.

Figure 6

The typical behavior of the phase curves in the neighborhood of the critical line where $K(\phi )=0$ (here $\dot{\phi }$ axis) is plotted for the case when $K_c^{\prime }>0$, $V_c^{\prime }=0$, $V_c^{\prime \prime }>0$.

Figure 7

Numerically obtained trajectories of the dark energy described by a Lagrangian linear in $X$ are plotted for the cases ${\Omega }_{\phi }=10{\Omega }_m$, ${\Omega }_{\phi }={\Omega }_m$, and ${\Omega }_{\phi }=0.1{\Omega }_m$.