Phantom Field and the Fate of Universe

TL;DR

This work examines phantom-field dark energy in a flat FRW universe with potentials unbounded above, where the negative kinetic term yields $w < -1$ when the phantom energy dominates. It formulates the model via the phantom Lagrangian, deriving $ ho_{\phi} = -\dot{\phi}^2/2 + V(\phi)$ and $p_{\phi} = -\dot{\phi}^2/2 - V(\phi)$ and analyzing the dynamical equations. The main result is a taxonomy of future cosmological outcomes determined by the steepness of $V(\phi)$: power-law, exponential, and steeper-than-exponential potentials produce distinct singularities, including slow-climb with $w \to -1$, big rip, or $w \to -\infty$ at finite scale factor. These findings illuminate when classical cosmology remains valid and when energy-density blow-ups signal impending breakdowns, shaping our understanding of late-time cosmic evolution with phantom energy.

Abstract

In this paper we analyze the cosmological dynamics of phantom field in a variety of potentials unbounded from above. We demonstrate that the nature of future evolution generically depends upon the steepness of the phantom potential and discuss the fate of Universe accordingly.

Figures (3)

• Figure 1: Evolution of energy density is plotted against the scale factor. Solid line corresponds to the phantom field energy density in case of the model described by the potential: $V(\phi)=m^2 \phi^2$ with $m \simeq 10^{-60} M_p$. The dashed and dotted lines correspond to energy density of radiation and matter. Initially, the phantom field mimics the cosmological constant like behaviour and its energy density is extremely subdominant to the background (the system is numerically evolved starting from the radiation domination era with $\rho_r =1 MeV^4$) and remains to be so for most of the period of evolution. The phantom field $\phi$ continues in the state with $w=-1$ till the moment $\rho_{\phi}$ approaches $\rho_b$. The background ceases now to play the leading role (becomes subdominant) and the phantom field takes over and starts climbing up the potential fast. At late times, the field energy density catches up with the background, overtakes it and starts growing ($w <-1)$ and drives the current (the value of the scale factor $a \simeq 4 \times 10^{9}$ corresponds to the present epoch) accelerated expansion of the Universe with $\Omega_{\phi} \simeq 0.7$ and $\Omega_m \simeq 0.3$. Initial value of the field was tuned to get the present values of $\Omega$. The field then enters into the slow climbing regime allowing the slowing down of growth of $\rho_{\phi}$ and making $w \to -1$ asymptotically which is an attractor in this case.
• Figure 2: Display of the phase portrait (plot of $Y_2 \equiv \dot{\phi}/M_p^2$ versus $Y_1 \equiv \phi/M_p$) for the phantom field in case of the quadratic potential $V(\phi)=m^2 \phi^2$. The figure shows that the trajectories starting anywhere in the phase space move towards a configuration( slow climb regime) with $\dot{\phi} \to {\it const}$ asymptotically(the convenient choice of mass parameter in the potential used here corresponds to the value of the const equal to $2/\sqrt{3}$). This picture is drawn in absence of the background energy density and allows to probe a wider class of initial conditions; the asymptotic regime is independent of the background.
• Figure 3: Equation of state parameter $w$ for the phantom field is plotted against the scale factor. The solid line corresponds to the exponential potential $V(\phi) \sim e^{\lambda \phi/M_p}$, the dotted line to $V(\phi) \sim e^{\lambda \phi^2/M_p^2}$ and the dashed line to $V(\phi) \sim \phi^2$. The exponential potential exhibits a critical behaviour in which case $w$ fast evolves to a constant value less than minus one (numerical value depends upon $\lambda$ which was taken to be equal to one for convenience). For potential steeper than the exponential $w$ fast evolves towards larger and larger negative values (dotted line) leading to a singularity at some finite value of the scale factor. In case of a less steeper potential than the exponential, the equation of state parameter $w$ first decreases fast and then subsequently turns back towards minus one and approaches it asymptotically (dashed line).