Tachyon field in loop quantum cosmology: An example of traversable singularity

Loop quantum cosmology (LQC) predicts a nonsingular evolution of the universe through a bounce in the high energy region. But LQC has an ambiguity about the quantization scheme. Recently, the authors in [Phys. Rev. D 77, 124008 (2008)] proposed a new quantization scheme. Similar to others, this new quantization scheme also replaces the big bang singularity with the quantum bounce. More interestingly, it introduces a quantum singularity, which is traversable. We investigate this novel dynamics quantitatively with a tachyon scalar field, which gives us a concrete example. Our result shows that our universe can evolve through the quantum singularity regularly, which is different from the classical big bang singularity. So this singularity is only a weak singularity.

Figure 1

The dotted line, dashed line and solid line correspond to the classical, the $\overline{\mu }$ scheme and the ${\mu }_{MS}$ scheme dynamics, respectively. Left panel: Hubble parameter versus the energy density of matter. When ${\rho }_{\varphi }\ll 1$, the quantum correction becomes negligible in the $\overline{\mu }$ scheme, i.e., the behavior of universe goes back to classical one. But for the ${\mu }_{MS}$ the quantum correction is strong even when ${\rho }_{\varphi }\ll 1$. Right panel: The phase portrait for all admissible $\varphi$ and $\dot{\varphi }$ and the graph shows the region boundary. These boundaries are nothing but the contour lines of ${\rho }_{\varphi }$ in $\dot{\varphi }-\varphi$ space. The dashed line, dotted line and solid line correspond to $\infty$, $3{\rho }_c$ and ${\rho }_c$ respectively. The admissible states for classical system are between the two dashed lines. The admissible states for quantum system are the right part of the corresponding lines.

Figure 2

Comparison of $\ddot{\varphi }$ in the expanding universe. The horizontal axis is $\varphi$, and vertical axis is $\dot{\varphi }$. The lines are contour lines for $\ddot{\varphi }$. The upper row is comparison of three dynamical systems near the bounce region of the $\overline{\mu }$ quantization scheme. The lower row is comparison of the classical system and the ${\mu }_{MS}$ scheme. In the three subfigures of upper row we only show the admissible region for the $\overline{\mu }$ scheme, which is the common region for three dynamical systems. The lines are contour plot of $\ddot{\varphi }$. The figure does not show this singularity properly due to the numerical resolution. But the envelopes of islands in this figure are true. At present, we are not clear what physical phenomena these islands will result in. Near the bounce region, the behavior of $\ddot{\varphi }$ is similar to the one of the $\overline{\mu }$ scheme in its corresponding bounce region (compared middle subfigure in the upper row with the one of last subfigure in the lower row).

Figure 3

Left panel, comparing of the dynamical trajectories for the classical (dotted lines) and the ${\mu }_{MS}$ quantization scheme (solid lines) under the expanding universe scenario in $\dot{\varphi }-\varphi$ space. The left C shape line is the boundary of admissible states for the ${\mu }_{MS}$ scheme, which is also the iso line of ${\rho }_{\varphi }=3{\rho }_c$. The right C shape line corresponds to the singularity (the iso line of ${\rho }_{\varphi }=1.5{\rho }_c$) for the ${\mu }_{MS}$ scheme, which is also the interface for hyperinflationary and deflationary states. Right panel, the concrete example of the dynamical behavior of $\ddot{\varphi }$ and the Hubble parameter $H$ near singularity for the ${\mu }_{MS}$ quantization scheme, which corresponds to any line which passes through the singularity for the ${\mu }_{MS}$ scheme in the left panel. The behavior of other ones are similar on the singularity. The solid line shows behavior of $\ddot{\varphi }$, while the dotted line shows behavior of the Hubble parameter $H=\frac{\dot{a}}{a}$. This subfigure is a blowup near singularity.