Properties of singularities in (phantom) dark energy universe

TL;DR

The paper investigates the fate of a phantom dark energy universe by modeling dark energy with $p=-\rho-f(\rho)$ and classifying finite-time future singularities into four types (Big Rip, sudden, Type III, and Type IV). Using an autonomous two-fluid system and various explicit $f(\rho)$ forms, it demonstrates a stable late-time attractor with $w<-1$ and, in coupled phantom-dark-matter scenarios, shows how a transition from $w>-1$ to $w<-1$ can arise. A key contribution is the systematic mapping between the EOS function $f(\rho)$ and the singularity structure, including analytic and numerical results across multiple parameter regimes. The authors further explore quantum corrections from conformal anomaly, showing that these effects typically moderate or even remove singularities, suggesting possible late-time de Sitter-like behavior and urging consideration of higher-order (stringy) corrections for a complete fate assessment.

Abstract

The properties of future singularities are investigated in the universe dominated by dark energy including the phantom-type fluid. We classify the finite-time singularities into four classes and explicitly present the models which give rise to these singularities by assuming the form of the equation of state of dark energy. We show the existence of a stable fixed point with an equation of state $w<-1$ and numerically confirm that this is actually a late-time attractor in the phantom-dominated universe. We also construct a phantom dark energy scenario coupled to dark matter that reproduces singular behaviors of the Big Rip type for the energy density and the curvature of the universe. The effect of quantum corrections coming from conformal anomaly can be important when the curvature grows large, which typically moderates the finite-time singularities.

Figures (6)

• Figure 1: The phase plane for the plus sign of the model (\ref{['precon']}) with $n=2$. The allowed range corresponds to $y \le 1-x$, $y>5x$ and $x \ge 0$. The point A is a saddle point given by $x=1/(3n)=1/6$ and $y=1-1/(3n)=5/6$. The solutions approach the region $x=0$, corresponding to the equation of state: $w=-1$.
• Figure 2: The phase plane for the minus sign of the model (\ref{['precon']}) with $n=2$. The allowed range corresponds to $y \le 1-x$, $y>-7x$ and $x \le 0$. The point B is a stable node given by $x=-1/(3n)=-1/6$ and $y=1+1/(3n)=7/6$, which corresponds to the late-time attractor. The equation of state at this point is a constant: $w=-1-2/(3n)=-4/3$.
• Figure 3: The phase plane for the model (\ref{['EOS15']}) with $\beta=1.1$, $\alpha=2\beta- 1$, $A=2$ and $B=1$. The solutions approach "instantaneous" critical points: $x=(1+p'(\rho))/2$ and $y=(1-p'(\rho))/2$, which diverge as $x \to -\infty$ and $y \to \infty$ at the type III singularity.
• Figure 4: The phase plane for the model (\ref{['EOS15']}) with $\beta=0.85$, $\alpha=2\beta- 1$, $A=2$ and $B=1$. The solutions approach the type I singularity at which the equation of state is $w=-1$ with $x=0$ and $y =1$.
• Figure 5: The evolution of $H$ for the model $f(\rho)=B\rho^{\beta}$ with $\beta=2$ and $B>0$. The case (a) corresponds to the one in which quantum corrections are taken into account with coefficients $b=0.5$, $b'=-0.1$, and $b"=0$, whereas the case (b) does not implement such effects. The Hubble rate approaches $H=0$ with a finite time in the case (a), while it diverges in the case (b).