Phantom field dynamics in loop quantum cosmology

We consider a dynamical system of phantom scalar field under exponential potential in the background of loop quantum cosmology. In our analysis, there is neither stable node nor repeller unstable node but only two saddle points, hence no big rip singularity. Physical solutions always possess potential energy greater than the magnitude of the negative kinetic energy. We found that the universe bounces after accelerating even in the domination of the phantom field. After bouncing, the universe finally enters the oscillatory regime.

Figure 1

Three-dimensional phase space of $X$, $Y$, and $Z$. The saddle points (a) ($-0.40825$, 1.0801, 0) and (b) ($-0.40825$, $-1.0801$, 0) appear in the figure. $\lambda$ is set to 1. In region $Z<0$, the solutions (red and blue lines) are nonphysical. In this region, $Z\to -\infty$ when $(X,Y)\to (0,0)$. The green lines (class I) are in region $|X|>|Y|$ and $Z>1$ but they are also nonphysical since they correspond to imaginary $H$ values. The only set of physical solutions (class II) is presented with black lines. They are in region $|Y|>|X|$ and range from $0<Z<1$. This is the region above (a) and (b) of which $H$ takes real value. There are separatrices $|X|=|Y|$, $Z=0$, and $Z=1$ in the system (see Sec. 5b).

Figure 2

Phase space of the kinetic part $X$ and potential part $Y$ (top view). The saddle points (a) ($-0.40825$, 1.0801) and (b) ($-0.40825$, $-1.0801$) are shown here. The blue lines and red lines are in the region $Z<0$ which is nonphysical. Green lines are of class I solutions which yield imaginary $H$. Only class II solutions shown as black lines are physical with a real $H$ value.

Figure 3

Phase space of the kinetic part $X$ and potential part $Y$ in standard general relativistic case. The location of points (a) and (b) in Fig. 2 is on the trajectory solutions here. This plot shows the dynamics of a phantom field in standard cosmological background without any other fluids. In the presence of a barotropic fluid with any equation of state, points (a) and (b) correspond to the big rip [23, 25].

Figure 4

Evolution of $H$ with time of a class II solution. Set values are $\lambda =1$, ${\rho }_{\mathrm{lc}}=1.5$, $V_0=1$, and $M_{\mathrm{P}}=2$. The universe undergoes acceleration from the beginning until reaching a turning point at $\rho ={\rho }_{\mathrm{lc}}/2=0.75$, where $H=H_{\max }=0.17678$. Beyond this point, the universe expands with deceleration until halting ($H=0$) at $\rho \approx {\rho }_{\mathrm{lc}}=1.5$. After halting, it undergoes contraction until $H$ bounces. The oscillating in $H$ goes on forever.

Figure 5

Time evolution of potential energy density (P.E.), kinetic energy density (K.E.), and $\rho =\mathrm{K}.\mathrm{E}.+\mathrm{P}.\mathrm{E}.$ of the field for a class II solution. K.E. is always negative and, at late time, it goes to $-\infty$ while P.E. is always positive. $\rho$ is maximum when $\rho \approx {\rho }_{\mathrm{lc}}=1.5$. Other features are discussed as in Fig. 4.

Figure 6

Oscillation in kinetic energy density (K.E.) that contributes to oscillation in $\rho$. This is a zoomed-in portion of Fig. 5.