Crossing of the phantom divide in modified gravity

TL;DR

The paper demonstrates that crossing the phantom divide can be realized within $F(R)$ gravity without introducing phantom fields, yielding a finite-time Big Rip in the Jordan frame that maps to an infinite-time singularity in the Einstein-frame scalar theory. It clarifies the precise relationship between scalar-field descriptions and $F(R)$ gravity via (inverse) conformal transformations, showing non-phantom phases correspond to real $F(R)$ and phantom phases to complex $F(R)$. It then constructs a viable Hu–Sawicki–type $F(R)$ model that allows a transition from de Sitter to phantom, and discusses how non-local terms or auxiliary scalars may be needed to realize such transitions while preserving viability. Finally, it analyzes the stability of these phantom-crossing solutions against conformal-anomaly quantum corrections and argues the crossing remains robust at the crossing epoch, with quantum effects becoming relevant only at very large curvature near a potential Big Rip.

Abstract

We reconstruct an explicit model of modified gravity in which a crossing of the phantom divide can be realized. It is shown that the (finite-time) Big Rip singularity appears in the model of modified gravity (i.e., in the so-called Jordan frame), whereas that in the corresponding scalar field theory obtained through the conformal transformation (i.e., in the so-called Einstein frame) the singularity becomes the infinite-time one. Furthermore, we investigate the relations between the scalar field theories with realizing a crossing of the phantom divide and the corresponding modified gravitational theories by using the inverse conformal transformation. It is demonstrated that the scalar field theories describing the non-phantom phase (phantom one with the Big Rip) can be represented as the theories of real (complex) $F(R)$ gravity through the inverse (complex) conformal transformation. We also study a viable model of modified gravity in which the transition from the de Sitter universe to the phantom phase can occur. In addition, we explore the stability for the obtained solutions of the crossing of the phantom divide under a quantum correction coming from conformal anomaly.

Figures (3)

• Figure 1: Behavior of $t_s^2F(\tilde{R})$ as a function of $\tilde{R}$ for $\gamma =1/2$, $\tilde{p}_+ =-1/t_s^{\beta_+}$, $\tilde{p}_- =0$, $\beta_+ = \left(1+2\sqrt{19}\right)/2$ and $t_s =2t_0$.
• Figure 2: Behavior of $F(\tilde{R})/\left(2 \kappa^2 \right)$ as a function of $\tilde{R}$. Legend is the same as Fig. 3.
• Figure 3: $P(\tilde{R})$ and $Q(\tilde{R})/\nu^2$ as functions of $\tilde{R}$ for $t_s = 2t_0$ and $\bar{\rho} = 0.233 \rho_\mathrm{c}$.